LastUpdate 4/18/2024

Japanese (“ú–{Œę”Ĺ)



@"Grapes" and "Function View" are famous as excellent free software that draws very easily just by substituting formulas of functions.
@In high school mathematics classes , when teaching graphs of quadratic functions , students create an xy correspondence table , plot the points on the xy-coordinate plane , and draw a graph by connecting the points with a smooth curve.
@"Grapes" and "Function View"draw graphs simply by substituting function expressions.@Therefore , there is little sense of creating an xy correspondence table , plotting the points , and connecting them with a smooth curve.
@However , Excel is the software that makes it possible on a computer to create an xy correspondence table , plot points , and draw a graoh with a sense of connecting them with a smooth curve. Excel is a spreadsheet sofrware that we use on a daily basis.
@When I went to a public library in Higashi-Omi , I happened to come across a book titled "Easy Mathematics Learning with Excel (from Trigonometory to Calculus) , Co-authered by Yukihisa Takahashi and Ha Watanabe , Ohmusha Publishinng". I found a description that made me realize that how to draw a graph of a funcion using "Excel" and the way of thinking about numerical sequences is exactly the copy function of "Excel".
@In addition , I introduced what can be done using "Excel" in "Simulation" , "Prime and Perfect Numbers" , and "Polor Equations" in Table of Contents 12-14. Please try it. You should be able to apply it to the other things as well.
@By using "Grapes" , "Function View" , and "Excel" according to the senses , skills , and knowledge you want students to acqure , I believe that more effective lessons can be organized.




http://kn-makkun.com/MakkunWp/excels.html
No Table of contents
‚P @Quadatic function
‚Q @Trigonometric function
‚R @Exponential function
‚S @Logarithmic function
‚T @Varius curves
‚U @Differentiation (Part 1)
‚V @Differentiation (Part 2)
‚W @Integral
‚X @Number sequence
‚P‚O @Complex number
‚P‚P @Application of square root
‚P‚Q @Simulation @
‚P‚R @Prime snd perfect numbers@
‚P‚S @Polar equation@
‚P‚T @Download@
‚P‚U @How to use the sample data

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@To table of contents@
@y‚P‚O‚Pz Drwa a graph of the quadratic functions of y=x^2 , y=2x^2 , and y=3x^2.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 x y=x^2 y=2x^2 y=3x^2
row 12 -4 16 32 48
row 13 -3 9 18 27
row 14 -2 4 8 12
row 15 -1 1 2 3
row 16 0 0 0 0
row 17 1 1 2 3
row 18 2 4 8 12
row 19 3 9 18 27
row 20 4 16 32 48

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -4 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B20 , right-click , and select Paste.
@@@@@@@Enter =B12^2 in half-width in cell C12.
@@@@@@@Enter =2*B12^2 in half-width in cell D12.
@@@@@@@Enter =3*B12^2 in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C20 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E20 in the correspondence table of (1) above.
@@@@@@‡A Left-click [Insert]¨[Scatter plot]¨[Smooth line and marker].

@@@(3) Arrange the xy coordinate plane on which the graoh is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add grid lines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 5.0
@@@@@@¨ Close

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click [Insert]¨[Scatter plot]¨[Smooth line].

@@@@@@

@@@ƒConsideration 1„
@@@@@@Compare the spread of graphs of y=x^2 , y=2x^2 , and y=3x^2.@

@@@ƒConsideration 2„
@@@@@@In general , how does the spread of the graph of y=ax^2 change depending on the value of a ?

@y‚P‚O‚Qz Draw a graph of the quadratic functions of y=2x^2 and y=-2x^2.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚P‚O‚Pz how to enter the worksheet.

Column B Column C Column D
row 11 x y=2x^2 y=-2x^2
row 12 -4 32 -32
row 13 -3 18 -18
row 14 -2 8 -8
row 15 -1 2 -2
row 16 0 0 0
row 17 1 2 -2
row 18 2 8 -8
row 19 3 18 -18
row 20 4 32 -32

@@@ (2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correspondence table of (1) above.
@@@@@@‡A Left click mInsertn¨mScatter ployn¨mSmooth line and Markern‚đś¸ŘŻ¸‚ˇ‚éB

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 5.0
@@@@@@¨ Close

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it .
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@
@@@@@@
@@@@@@
@@@ƒConsideration 1„
@@@@@@State the positional relationship between the graphs of y=2x^2 and y=-2x^2.@

@@@ƒConsideration 2„
@@@@@@In general , what is the positional relationship between the graphs of y=ax^2 and y=-ax^2 ?

@y‚P‚O‚Rz Draw a graph of the quadratic functions of y=2x^‚Q , y=2x^2+3 , and y=2x^2-3.
@ƒProduce„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚P‚O‚Pz how to complete the worksheet.

Column B Column C Column D Column E
row 11 x y=2x^2 y=2x^2+3 y=2x^2-3
row 12 -4 32 35 29
row 13 -3 18 21 15
row 14 -2 8 11 5
row 15 -1 2 5 -1
row 16 0 0 3 -3
row 17 1 2 5 -1
row 18 2 8 11 5
row 19 3 18 21 15
row 20 4 32 35 29

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Markern.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 5.0
@@@@@@¨ Close

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph-area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line]

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Compare the spread of the graphs of y=2x^2 , y=2x^2+3 , and y=2x^2-3.

@@@ƒConsideration 2„
@@@@@@@State the positional relationship of the graphs of y=2x^2 , y=2x^2+3 , and y=2x^2-3.

@@@ƒConsideration 3„
@@@@@@@In general , what is the positional relationship between graphs of y=ax^2 and y=ax^2+b ?

@y‚P‚O‚Sz Draw a graph of the quadratic functions of y=2x^2 , y=2(x+2)^2 , and y=2(x-2)^2.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚P‚O‚Pz how to enter the worksheet.

column B column C column D column E
row 11 x y=2x^2 y=2(x+2)^2 y=2(x-2)^2
row 12 -4 32 8 72
row 13 -3 18 2 50
row 14 -2 8 0 32
row 15 -1 2 2 18
row 16 0 0 8 8
row 17 1 2 18 2
row 18 2 8 32 0
row 19 3 18 50 2
row 20 4 32 72 8

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Markern

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 5.0
@@@@@@¨ Close

@@@(4) Deleate points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Compare the spread of the graphs of Y=ax^2 , y=2(x+2)^2 , and y=2(x-2)^2.

@@@ƒConsideration 2„
@@@@@@@State the positional relationship of the graphs of y=2x^2 , y=2(x+2)^2 , and y=2(x-2)^2.

@@@ƒConsideration 3„
@@@@@@@In general , what is the positional relationship between graphs of y=ax^2 and y=a(x-b)^2 ?

@y‚P‚O‚Tz Draw a graph of the quadratic functions of y=2x^2 and y=2(x-2)^2+3.
@ƒProcedure„
@@@(1) Create the folllowing correspondence table on the worksheet.

@@@ƒRefer„
@@@@@@Please refer toy‚P‚O‚Pz how to enter the worksheet.

column B column C column D
row 11 x y=2x^2 y=2(x-2)^2+3
row 12 -4 32 75
row 13 -3 18 53
row 14 -2 8 35
row 15 -1 2 21
row 16 0 0 11
row 17 1 2 5
row 18 2 8 3
row 19 3 18 5
row 20 4 32 11

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correspondence table of (1) above.
@@@@@@‡A Left -click mInsertn¨mScatter plotn¨mSmooth line and Marker].

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Compare the spread of the graphs of y=2x^2 and y=2(x-2)^2+3.

@@@ƒConsideration 2„
@@@@@@@State the positional relationship of the graphs of y=2x^2 and y=2(x-2)^2+3.

@@@ƒConsideration 3„
@@@@@@@In general , what is the positional relationship between the graphs of y=ax^2 and y=a(x-b)^2+c.

@y‚P‚O‚Uz Draw a graph of the quadratic functions of y=-x^2 and y=-x^2+6x-7.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Please refer to @y‚P‚O‚Pz how to enter the worksheet.

column B column C column D
row 11 x y=-x^2 y=-x^2+6x-7
row 12 -4 -16 -47
row 13 -3 -9 -34
row 14 -2 -4 -23
row 15 -1 -1 -14
row 16 0 0 -7
row 17 1 -1 -2
row 18 2 -4 1
row 19 3 -9 2
row 20 4 -16 1

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Markern

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Compare the spread of the graphs of y=-x^2 and y=-x^2+6x-7.

@@@ƒConsideration 2„
@@@@@@@State the positional relationship of the graphs of y=-x^2 and y=-x^2+6x-7.



@To table of contents@
@y‚Q‚O‚Pz Draw a graph of the trigonometric functions of ‚™‚“‚‰‚Ž‚˜ , ‚™‚“‚‰‚Ž‚Q‚˜ , and ‚™‚“‚‰‚Ž‚R‚˜.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 x y=sinx y=sin2x y=sin3x
row 12 0 0 0 0
row 13 2 0.0348ĽĽĽ 0.0697ĽĽĽ 0.1045ĽĽĽ
row 14 4 0.0697ĽĽĽ 0.1391ĽĽĽ 0.2079ĽĽĽ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 190 356 -0.0697ĽĽĽ -0.1391ĽĽĽ -0.2079ĽĽĽ
row 191 358 -0.0348ĽĽĽ -0.0697ĽĽĽ -0.1045ĽĽĽ
row 192 360 -2.4503ĽĽĽ -4.9005ĽĽĽ -7.3508ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B192 , right-click , and select Paste.
@@@@@@@Enter = SIN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter = SIN(RADIANS(2*B12)) in half-width in cell D12
@@@@@@@Enter = SIN(RADIANS(3*B12)) in half-width in cell E12
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of ‚™‚“‚‰‚Ž‚˜ from the graph.

@@@ƒConsideration 2„
@@@@@@@@Find the period of the graph of ‚™‚“‚‰‚Ž2x from the graph.

@@@ƒConsideration 3„
@@@@@@@@Find the period of the graph of ‚™‚“‚‰‚Ž‚R‚˜ from the graph.@

@@@ƒConsideration 4„
@@@@@@@@In general , what can be said about the period of the graph of y=sinax ?

@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D
row 11 x y=sinx y=sin(x/2)
row 12 0 0 0
row 13 2 0.0348ĽĽĽ 0.0174ĽĽĽ
row 14 4 0.0697ĽĽĽ 0.0348ĽĽĽ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 370 716 -0.0697ĽĽĽ -0.0348ĽĽĽ
row 371 718 -0.0348ĽĽĽ -0.0174ĽĽĽ
row 372 720 -4.900ĽĽĽ -2.4503ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B372 , right-click , and select Paste.
@@@@@@@Enter =SIN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =SIN(RADIANS(B12/2)) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:D372 , right-click , and select Paste

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D372 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Find the period of the graph of y=sin(x/2) from the graph ?

@@@ƒConsideration 2„
@@@@@@@In general ,
@@@@@@@what can be said about the period of yhe graph of y=sin(x/a) ?

@
@y‚Q‚O‚Rz Draw a graph of the trigonometric functions of y=sinx , y= 2sinx , and y=3sinx ?
@ƒProcedure„
@@@(1) Create the following correspondence table on the woksheet.

column B column C column D column E
row 11 x y=sinx y=2sinx y=3sinx
row 12 0 0 0 0
row 13 2 0.0348ĽĽĽ 0.0697ĽĽĽ 0.1046ĽĽĽ
row 14 4 0.0697ĽĽĽ 0.1395ĽĽ 0.2092ĽĽĽ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 190 356 -0.0697ĽĽĽ -0.1395ĽĽĽ -0.2092ĽĽĽ
row 191 358 -0.0348ĽĽĽ -0.0697ĽĽĽ -0.1046ĽĽĽ
row 192 360 -2.450.ĽĽĽ -4.9005ĽĽĽ -7.3508ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B192 , right-click , and elect Paste.
@@@@@@@Enter =SIN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =2*SIN(RADIANS(B12)) in half-width in cell D12.
@@@@@@@Enter =3*SIN(RADIANS(B12)) in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and elect Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and elect Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of y=2sinx from the graph.

@@@ƒConsideration 2„
@@@@@@@@Find the period of the graph of y=3sinx from the graph.

@@@ƒConsideration 3„
@@@@@@@@State the difference between graphs y=sinx , y=2sinx , and y=3sinx.

@@@ƒConsideration 4„
@@@@@@@@In general , what can be said about the amplitude of the graph of y=asinx ?

@y‚Q‚O‚Sz Draw a graph of the trigonometric function y=2sin(x-60‹).
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D
row 11 x y=2sinx y=2sin(x-60‹)
row 12 0 0 -1.7320ĽĽĽ
row 13 2 0.0697ĽĽĽ -1.6960ĽĽĽ
row 14 4 0.1395ĽĽĽ -1.6580ĽĽĽ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 220 416 1.6580ĽĽĽ -0.1395ĽĽĽ
row 221 418 1.6960ĽĽĽ -0.0697ĽĽĽ
row 222 420 1.7320ĽĽĽ -4.9005ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet above.t
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B222 , right-click , and select Paste.
@@@@@@@Enter =2*SIN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =2*SIN(RADIANS(B12-60)) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C222 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D222 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that y=2sin(x-60‹) graph is the y=2sinx graph translated +60‹ along the x-axis.

@y‚Q‚O‚Tz Draw a graph of the trigonometric functions y=cosx , y=cos2x , and y=cos3x.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

Column B Column C Column D Column E
row 11 x y=cosx y=cos2x y=cos3x
row 12 0 1 1 1
row 13 2 0.9993ĽĽĽ 0.9975ĽĽĽ 0.9945ĽĽĽ
row 14 4 0.9975ĽĽĽ 0.9902ĽĽĽ 0.9781ĽĽĽ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 190 356 0.9975ĽĽĽ 0.9902ĽĽĽ 0.9781ĽĽĽ
row 191 358 0.9993ĽĽĽ 0.9975ĽĽĽ 0.9945ĽĽĽ
row 192 360 1 1 1

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B192 , right-click , and select Paste.
@@@@@@@Enter =COS(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =COS(RADIANS(2*B12)) in half-width in cell D12.
@@@@@@@Enter =COS(RADIANS(3*B12)) in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E192 in the correspondence table of (1) above
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area by horizontally by dragging the right and lrft edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of y=cosx from the graph..@

@@@ƒConsideration 2„
@@@@@@@@Find the period of the graph of y=cos2x from the graph.

@@@ƒConsideration 3„
@@@@@@@@Find the period of the graph of y=cos3x from the graph.

@@@ƒConsideration 4„
@@@@@@@@In general , what can be said about the period of the graph of y=cosax ?

@y‚Q‚O‚Uz Draw a graph of the trigonometric functions y=cosx and y=cos(x/2).
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D
row 11 x y=cosx y=cos(x/2)
row 12 0 1 1
row 13 2 0.9993ĽĽĽ 0.9998ĽĽĽ
row 14 4 0.9975ĽĽĽ 0.9993ĽĽĽ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 370 716 0.9975ĽĽĽ 0.9993ĽĽĽ
row 371 718 0.9993ĽĽĽ 0.9998ĽĽĽ
row 372 720 1 1

@@@ƒProcedure„
@@@@@sHow to complete the worksheet above.t
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B372 , right-click and select Paste.
@@@@@@@Enter =COS(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =COS(RADIANS(B12/2)) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click and select Copy.
@@@@@@@Drag the cell range C13:C372 , right-click and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D372 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 30.00
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale intercal Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Find the period of the graph of y=cos(x/2) .

@@@ƒConsideration 2„
@@@@@@@In general,
@@@@@@@what can be said about yhr period of the gaph of y=cos(x/a) ?
@y‚Q‚O‚Vz Draw a graph of the trigonometric functions y=cosx , y=2cosx , and y=3cosx.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 x y=cosx y=2cosx y=3cosx
row 12 0 1 2 3
row 13 2 0.9993ĽĽĽ 1.9987ĽĽĽ 2.9981ĽĽĽ
row 14 4 0.9975ĽĽĽ 1.9951ĽĽĽ 2.9926ĽĽĽ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 190 356 0.9975ĽĽĽ 1.9951ĽĽĽ 2.9926ĽĽĽ
row 191 358 0.9993ĽĽĽ 1.9987ĽĽĽ 2.9981ĽĽĽ
row 192 360 1 2 3

@@@ƒReference„
@@@@@sHow to complete the worksheet above.t
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 , right-click , and select Copy.
@@@@@@@Drag the cell range B14:B192 , right-click , and select Paste.
@@@@@@@Enter =COS(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =2*COS(RADIANS(B12)) in half-width in cell D12.
@@@@@@@Enter =3*COS(RADIANS(B12)) in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and select Paste.

@@@(2) Drag a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the grapf area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliay gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of y=2cosx from the graph.

@@@ƒConsideration 2„
@@@@@@@@Find the period of the graph of y=3cosx from the graph

@@@ƒConsideration 3„
@@@@@@@@State the difference between graph ‚™‚ƒ‚‚“‚˜ , graph ‚™‚Q‚ƒ‚‚“‚˜ , and graph ‚™‚R‚ƒ‚‚“‚˜.

@@@ƒConsideration 4„
@@@@@@@@In general , what can be said about of the implitude of the graph of ‚™‚‚ƒ‚‚“‚˜ ?

@y‚Q‚O‚Wz Draw a graph of the trigonometric function y=2cos(x+60‹).
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@
column B column C column D
row 11 x y=2cosx y=2cos(x+60‹)
row 12 -60 1 2
row 13 -58 1.0598ĽĽĽ 1.9987ĽĽĽ
row 14 -56 1.1183ĽĽĽ 1.9951ĽĽĽ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 220 356 1.9951ĽĽĽ 1.1183ĽĽĽ
row 221 358 1.9987ĽĽĽ 1.0598ĽĽĽ
row 222 360 2 1

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -60 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B222 , right-click , and select Paste.
@@@@@@@Enter =2*cos(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enter =2*cos(RADIANS(B12+60)) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12, right-click , and select Copy.
@@@@@@@Drag the cell range C13:C222 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D222 on the correspondence table of (1) above.
@@@@@@‡A Left -click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that the graph of y=2cos(x+60‹) @is the graph of y=2cosx that has been translated by -60 ‹in the x-axis direction.

@y‚Q‚O‚Xz Draw a graph of the trigonometric function y=tanx.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C
row 11 x y=tanx
row 12 -180 1.2251ĽĽĽ
row 13 -178 0.0349ĽĽĽ
row 14 -176 0.0699ĽĽĽ
Ľ Ľ Ľ
row 54 -96 9.5143ĽĽĽ
row 55 -94 14.3006ĽĽĽ
row 56 -92 28.6362ĽĽĽ
row 57 ‹ó”’ ‹ó”’
row 58 -88 -28.6362ĽĽĽ
row 59 -86 -14.3006ĽĽĽ
row 60 -84 -9.5143ĽĽĽ
Ľ Ľ Ľ
row 144 84 9.5143ĽĽĽ
row 145 86 14.3006ĽĽĽ
row 146 88 28.6362ĽĽĽ
row 147 ‹ó”’ ‹ó”’
row 148 92 -28.6362ĽĽĽ
row 149 94 -14..3006ĽĽĽ
row 150 96 -9.5143ĽĽĽ
Ľ Ľ Ľ
row 190 176 -0.0699ĽĽĽ
row 191 178 -0.0349ĽĽĽ
row 192 180 -1.2251ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -180 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B56 , right-click , and select Paste.
@@@@@@@Enter -88 in half-width in cell B58.
@@@@@@@Enter =B58+2 in half-width in cell B59.
@@@@@@@Right-click cell B59 and select Copy.
@@@@@@@Drag the cell range B60:B146 , right-click , and select Paste.
@@@@@@@Enter 92 in half-width in cell B148.
@@@@@@@Enter =B148+2 in half-width in cell B149.
@@@@@@@Right-click cell B149 and select Copy.
@@@@@@@Drag the cell range B150:B192 , right-click , and select Paste.
@@@@@@@Enter =TAN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Right-click cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) avobe.
@@@@@@‡@ Drag and select the cell range B11:C192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of y=tanx from the graph.

@@@ƒConsideration 2„
@@@@@@@@Make sure that the straight lines x=90‹and x=-90‹are asymptote.@

@y‚Q‚P‚Oz Draw a graph of the trigonometric function of y=tan2x.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C
row 11 x y=tan2x
row 12 -90 1.2251ĽĽĽ
row 13 -89 0.0349ĽĽĽ
row 14 -88 0.0699ĽĽĽ
Ľ Ľ Ľ
row 54 -48 9.5143ĽĽĽ
row 55 -47 14.3006ĽĽĽ
row 56 -46 28.6362ĽĽĽ
row 57 ‹ó”’ ‹ó”’
row 58 -44 -28.6362ĽĽĽ
row 59 -43 -14.3006ĽĽĽ
row 60 -42 -9.5143ĽĽĽ
Ľ Ľ Ľ
row 144 42 9.5143ĽĽĽ
row 145 43 14.3006ĽĽĽ
row 146 44 28.6362ĽĽĽ
row 147 ‹ó”’ ‹ó”’
row 148 46 -28.6362ĽĽĽ
row 149 47 -14.3006ĽĽĽ
row 150 48 -9.5143ĽĽĽ
Ľ Ľ Ľ
row 190 88 -0.0699ĽĽĽ
row 191 89 -0.0349ĽĽĽ
row 192 90 -1.2251ĽĽĽ

@ƒReference„
@@@@@sHow to complete the woksheet abovet
@@@@@@@Enter -90‹in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B56 , right-click , and select Paste.
@@@@@@@Enter -44 in half-width in cell B58.
@@@@@@@Enter =B58+1 in half-width in cell B59.
@@@@@@@Right-click cell B59 and select Copy.
@@@@@@@Drag the cell range B60:B146 , right-click , and select Paste.
@@@@@@@Enter 46 in half-width in cell B148.
@@@@@@@Enter =B148+1 in half-width in cell B149.
@@@@@@@Right-click cell B149 and select Copy.
@@@@@@@Drag the cell range B150:B192 , right-click , and select Paste.
@@@@@@@Enter =TAN(RADIANS(2*B12)) in half-width in cell C12.
@@@@@@@Right-click cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C192 in the correspnndencetable of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval 15.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval 5.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of y=tan2x from the graph.@

@@@ƒConsideration 2„
@@@@@@@@Make sure that the straiht lines x=45‹and x=-45‹are asymptote.

@@@ƒConsideration 3„
@@@@@@@@In general , w hat can be said about the period of th graph of y=tanax ?

@y‚Q‚P‚Pz Draw a graph of the trigonometric functions y=tanx and y=2tanx.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.
@@@@@@@ƒReference„
@@@@@@@@@@Please refer to y‚Q‚O‚Xz how to complete the worksheet.

column B column C column D
row 11 x y=tanx y=2tanx
row 12 -180 1.2251ĽĽĽ 2.4503ĽĽĽ
row 13 -178 0.0349ĽĽĽ 0.0698ĽĽĽ
row 14 -176 0.0699ĽĽĽ 0.1398ĽĽĽ
Ľ Ľ Ľ Ľ
row 54 -96 9.5143ĽĽĽ 19.0287ĽĽĽ
row 55 -94 14.3006ĽĽĽ 28.6013ĽĽĽ
row 56 -92 28.6362ĽĽĽ 57.2725ĽĽĽ
row 57 blank blank blank
row 58 -88 -28.6362ĽĽĽ -57.2725ĽĽĽ
row 59 -86 -14.3006ĽĽĽ -28.6013ĽĽĽ
row 60 -84 -9.5143ĽĽĽ -19.0287ĽĽĽ
Ľ Ľ Ľ Ľ
row 144 84 9.5143ĽĽĽ 19.0287ĽĽĽ
row 145 86 14.3006ĽĽĽ 28.6013ĽĽĽ
row 146 88 28.6362ĽĽĽ 57.2725ĽĽĽ
row 147 blank blank blank
row 148 92 -28.6362ĽĽĽ -57.2725ĽĽĽ
row 149 94 -14.3006ĽĽĽ -28.6013ĽĽĽ
row 150 96 -9.5143ĽĽĽ -19.0287ĽĽĽ
Ľ Ľ Ľ Ľ
row 190 176 -0.0699ĽĽĽ -0.1398ĽĽĽ
row 191 178 -0.0349ĽĽĽ -0.0698ĽĽĽ
row 192 180 -1.2251ĽĽĽ -2.4503ĽĽĽ

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cellrange B11:D192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area vertically and horizontally by dragging the top and bottom edges and the
@@@@@@@right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the number on the x-axis ¨ Axis formating ¨ Scale interval Fixed 15.0
@@@@@@¨ Close ¨ Right-click on the number on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the number on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 5.0
@@@@@@¨ Close ¨ Right-click on the number on the x-axis ¨ Add auxiliary gridlines
@@@@@ ‡B Right-click on the number on the y-axis ¨ Axis-formating ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the number on the y-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@@Find the period of the graph of y=2tanx from the graph.

@@@ƒConsideration 2„
@@@@@@@@State the positional relationship between the graphs of ‚™‚”‚‚Ž‚˜ and ‚™‚Q‚”‚‚Ž‚˜.

@@@ƒConsideration 3„
@@@@@@@@In general , what is the positional relationship between the graphs of ‚™‚”‚‚Ž‚˜ and ‚™‚‚”‚‚Ž‚˜.

@y‚Q‚P‚Qz Draw a graph of the trigonometric function y=2tan{(1/2))x-30‹}.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.
@@@@@@@ƒReference„
@@@@@@@@@@Please refer to y‚Q‚O‚Xz how to complete the worksheet.

column B column C
row 11 x y=2tan(x/2-30‹)
row 12 -360 -1.1547ĽĽĽ
row 13 -358 -1.1086ĽĽĽ
row 14 -356 -1.0634ĽĽĽ
Ľ Ľ Ľ
row 129 -126 38.1622ĽĽ
|row 130 -124 57.2725ĽĽĽ
row 131 -122 114.5799ĽĽĽ
row 132 blank blank
row 133 -118 -114.5799ĽĽĽ
row 134 -116 -57.2725ĽĽĽ
row 135 -114 -38.1622ĽĽĽ
Ľ Ľ Ľ
row 309 234 38.1622ĽĽĽ
row 310 236 57.2725ĽĽĽ
row 311 238 114.5799ĽĽĽ
row 312 blank blank
row 313 242 -114.5799ĽĽĽ
row 314 244 -57.2725ĽĽĽ
row 315 246 -38.1622ĽĽĽ
Ľ Ľ Ľ
row 370 356 -1.2497ĽĽ
row 371 358 -1.2017ĽĽĽ
row 372 360 -1.1547ĽĽĽ

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C372 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linesn.
@@@@@@‡B Expand the graph area vertically and horizontally by dragging the right and reft edges and the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Make sure that the straight lines x=240‹ and x=-120‹ are asymptote of the graph@of y=2tan{(1/2)x-30‹}

@@@ƒConsideration 2„
@@@@@@@Make sure that the graph oh y=2tan(1/2)x is translated by +60 in the x-axis ditrction.

@y‚Q‚P‚Rz When 0‹…‚˜…360‹
@@@@@@Draw a graph of the functions y=cos2x+2sinx+2 and y=-2(sinx)^2+2sinx+3.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 x y=cos2x+2sinx+2 x y=-2(sinx)^2+2sinx+3
row 12 0 3 0 3
row 13 2 3.0673ĽĽĽ 2 3.0673ĽĽĽ
row 14 4 3.1297ĽĽĽ 4 3.1297ĽĽĽ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 190 356 2.8507ĽĽĽ 356 2.8507ĽĽĽ
row 191 358 2.9277ĽĽĽ 358 2.9277ĽĽĽ
row 192 360 3 360 3

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell rangeB14:B192 , right-click , and select Paste.
@@@@@@@Enter 0 in half-width in cell D12.
@@@@@@@Enter =D12+2 in half-width in cell D13.
@@@@@@@Right-click on cell D13 and select Copy.
@@@@@@@Drag the cell rangeD14:D192 , right-click , and select Paste.
@@@@@@@Enter =COS(RADIANS(2*B12))+2*SIN(RADIANS(B12))+2@in half-width in cell C12.
@@@@@@@Enter =-2SIN(RADIANS(B12))^2+2*SIN(RADIANS(B12))+3@in half-width in cell E12.
@@@@@@@Right-click on cell D12 and select Copy.
@@@@@@@Drag the cell rangeC13:C192 , right-click , and select Paste.
@@@@@@@Right-click on cell E12 and select Copy.
@@@@@@@Drag the cell rangeE13:E192 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) avove.
@@@@@@‡@ Drag and select the cell range B11:C192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨msmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.
@@@@@@‡C Drag and select the cell range D11:E192 in the correspondence table of (1) above.
@@@@@@‡D Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡E Expand the graph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@
@@@@@@

@@@ƒlŽ@„
@@@@@@@Make sure that the graphs for y=cos2x+2sinx+2 and y=-2(sinx)^2+2sinx+3 are the same.

@y‚Q‚P‚Sz When 0‹…‚˜…360‹
@@@@@@Draw a graph of the functions of y=sinx+cosx and y=ă2sin(x+45‹).
@ƒProcedure„
@@@(1) Create the following correspondence tabe on the worksheet.

column B column C column D column E
row 11 x y=sinx+cosx x y=ă‚Qsin(x+45‹)
row 12 0 1 0 1
row 13 2 1.0342ĽĽĽ 2 1.0342ĽĽĽ
row 14 4 1.0673ĽĽĽ 4 1.0673ĽĽĽ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 190 356 0.9278ĽĽĽ 356 0.9278ĽĽĽ
row 191 358 0.9644ĽĽĽ 358 0.9644ĽĽĽ
row 192 360 1 360 1

@@@ƒReference„sHow to complete the worksheet abovet
@@@@@@Enter 0 in half-width in cell B12.
@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@Right-click cell B13 amd select Copy.
@@@@@@Drag the cell range B14:B192 , Right-click , and select Paste.
@@@@@@Enter 0 in half-width in cell D12.
@@@@@@Enter =D12+2 in half-width in cell D13.
@@@@@@Right-click cell D13 amd select Copy.
@@@@@@Drag the cell range D14:D192 , Right-click , and select Paste.
@@@@@@Enter =SIN(RADIANS(B12))+COS(RADIANS(B12)) in half-width in cell C12.
@@@@@@Enter =SQRT(2)*SIN(RADIANS(B12+45)) in half-width in cell E12.
@@@@@@Right-click cell C12 amd select Copy.
@@@@@@Drag the cell range C13:C192 , Right-click , and select Paste.
@@@@@@Right-click cell E12 amd select Copy.
@@@@@@Drag the cell range E13:E192 , Right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Expand the graph area horizontally by dragging the right and left edges of the graph area.
@@@@@@‡C Drag and select the cell range D11:E192 in the correspondence table of (1) above.
@@@@@@‡D Left-click mInsertn¨mScatter plotn¨mSmooth linen
@@@@@@‡E Expand the gtaph area horizontally by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 30.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 10.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines

@@@@@@
@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that the gtaphs for ‚™‚“‚‰‚Ž‚˜{‚ƒ‚‚“‚˜ and y=ă2sin(x+45‹)@are the same.




@To table of contents@
@y‚R‚O‚Pz Draw a graph of the exponential functions of y=2^x , y=3^x , and y=4^x.
@
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 x y=2^x y=3^x y=4^x
row 12 -2 0.25 0.111111 0.0625
row 13 -1 0.5 0.333333 0.25
row 14 0 1 1 1
row 15 1 2 3 4
row 16 2 4 9 16

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -2 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click B13 and select Copy.
@@@@@@@Drag the cell range B14:B16, right-click , and select Paste.
@@@@@@@Enter =POWER(2,B12) in half-width in cell C12.
@@@@@@@Enter =POWER(3,B12) in half-width in cell D12.
@@@@@@@Enter =POWER(4,B12) in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12, right-click , and select Copy
@@@@@@@Drag the cell range C13:C16 , right-click , and select Paste

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E16 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth lines and Marker]
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ Close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines
@@@(4) Delete the points(markers) in the correspondence table from yje graph.
@@@@@@‡@ Left-click on the garph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Find the coordinates of the point through which the graphs pf y=2^x , y=3^x , and y=4^x commonly pass.

@@@ƒConsideration 2„
@@@@@@@Find the asymptote that the graphs of y=2^x , y=3^x , and y=4^x have in common.

@@@ƒConsideration 3„
@@@@@@@State the difference between graphs y=2^x , y=3^x , and y=4^x.

@y‚R‚O‚Qz Draw a graph of the exponential functions of y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x.
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Refer to y‚R‚O‚Pz how to complete the worksheet.

column B column C column D column E
row 11 x y=(1/2)^x y=(1/3)^x y=(1/4)^x
row 12 -2 4 9 16
row 13 -1 2 3 4
row 14 0 1 1 1
row 15 1 0.5 0.333333 0.25
row 16 2 0.25 0.111111 0.0625

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E16 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Maekern.
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the bgraph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ Close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delite the points(markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen.

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Find the coordinates of the point through which the graphs of y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x commonly pass.

@@@ƒConsideration 2„
@@@@@@@Find the asymptote that the graphs of y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x have in common.
@@@@@@‚Č‚ł‚˘B

@@@ƒConsideration 3„
@@@@@@@State the difference between the graphs y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x.

@y‚R‚O‚Rz Draw a graph of the exponential functions of y=2^x and y=(1/2)^x .

@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚R‚O‚Pz how to complete the worksheet

column B column C column D
row 11 x y=2^x y=(1/2)^x
row 12 -3 0.125 8
row 13 -2 0.25 4
row 14 -1 0.5 2
row 15 0 1 1
row 16 1 2 0.5
row 17 2 4 0.25
row 18 3 8 0.125

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D18 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line snd markern.
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ Close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete the points(markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen.

@@@@@@

@@@ƒConsideration 1„
@@@@@@@State the positional relationship of the graphs of y=2^x and y=(1/2)^x.

@@@ƒConsideration 2„
@@@@@@@In general , what is the positional relationship between graphs y=a^x and y=(1/a)^x ?

@y‚R‚O‚Sz Draw a graph of the exponential functions of y=2^x , y=2^(x-2) , and y=2^(x+2).
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet..

@@@ƒReference„
@@@@@Please refer to y‚R‚O‚Pz how to complete the worksheet.

column B column C column D column E
row 11 x y=2^x y=2^(x-2) y=2^(x+2)
row 12 -3 0.125 0.03125 0.5
row 13 -2 0.25 0.0625 1
row 14 -1 0.5 0.125 2
row 15 0 1 0.25 4
row 16 1 2 0.5 8
row 17 2 4 1 16
row 18 3 8 2 32

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E18 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Markern.
@@@@@@‡B Expand the graph area vertically by dragging the right and left edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formating ¨ Scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formating ¨ Scale interval Fixed 5.0
@@@@@@¨ Close
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formating ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ Close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete the points(markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen.

@@@@@@

@@@ƒConsideration 1„
@@@@@@@state the positional relationship of the graph of y=2^x , y=2^(x-2) , and y=2^(x+2).

@@@ƒConsideration 2„
@@@@@@@In general , what is the positional relationship between graphs y=a^x and y=a^(x-k).
@y‚R‚O‚Tz Solve the equation 4^x@-@2^(x+2)@=@32 using a graph
@ƒProcedure„
@@@@@@@@
@@@@@@@@Draw the graphs of y=4^x and y=2^(x+2) +32.

@@@(1) Create the following correspondence table on a worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚R‚O‚Pz how to complete the worksheet.

column B column C column D
row 11 x y=4^x y=2^(x+2)+32
row 12 -4 0.003906 32.25
row 13 -3 0.015625 32.5
row 14 -2 0.0625 33
row 15 -1 0.25 34
row 16 0 1 36
row 17 1 4 40
row 18 2 16 48
row 19 3 64 64
row 20 4 256 96

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Merkern
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 10.0
@@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete the points(markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click the graph area to activare.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration„
@@@@@@Solve the equation 4^x-2^(x+2)=32 from the intersection of the graphs y=4^x and y=2^(x+2)+32

@y‚R‚O‚Uz Solve the inequality 9^x@<@2Ľ3^x@+@3 using a graph.
@ƒProcedure„
@@@@@@@@Draw the graphs y=9^x and y=2Ľ3^x+3

@@@(1) Create the following correspondence table on a worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚R‚O‚Pz how to complete the worksheet.

column B column C column D
row 11 x y=9^x y=2Ľ3^x+3
row 12 -2 0.012346 3.222222222
row 13 -1 0.111111 3.666666667
row 14 0 1 5
row 15 1 9 9
row 16 2 81 21

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D16 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line and Marker]
@@@@@@‡B Expand the graph area vertially by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is draw.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Auxoliary add gridlines

@@@(4) Delete the points(markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click the graph area to activate it..
@@@@@@‡A Left-click mInsetn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration„
@@@@@@Solve the inequality 9^x@<@2Ľ3^x@+@3 from the intersection of the graphs of y=9^x and y=2Ľ3^x+3.

@
@y‚R‚O‚Vz Compare the size of 8^(1/4) , 16^(1/3) , 64^(1/5) using a graph.
@ƒŽč‡„
@@@@@@@@

@@@(1) Create the following correspondence tablre on a worksheet.

columnB column C
row 12 ‡@ @‚W^(1/4) 1.681793
row 13 ‡A ‚P‚U^(1/3) 2.519842
row 14 ‡B ‚U‚S^(1/5) 2.297397

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet@@@@@@@
@@@@@@@Enter the information in cells B12 , B13 , and B14 as shown in the worksheet avbove.
@@@@@@@Enter =POWER(8,1/4) in Cell C12.
@@@@@@@Enter =POWER(16,1/3) in Cell C13.
@@@@@@@Enter =POWER(64,1/5) in Cell C14

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B12:C14 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mVertical barn¨m2-D vertical bar clustern

@@@@@@

@@@ƒConsideration„
@@@@@@Compare the sizes of 4^ă8@3^ă16@5^ă64 from the bar graphs of 8^(1/4) , 16^(1/3) , and 64^(1/5)
@@@@@@




@To table of contents@
@y‚S‚O‚Pz Draw a graph of the logarithmic functions of y=log(2)x , y=log(3)x , y=log(4)x
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 x y=log(x,2) y=log(x,3) y=log(x,4)
row 12 1/16 -4 -2.52372 -2
row 13 1/8 -3 -1.89279 -1.5
row 14 1/4 -2 -1.26186 -1
row 15 1/2 -1 -0.63093 -0.5
row 16 1 0 0 0
row 17 2 1 0.63093 0.5
row 18 4 2 1.26186 1
row 19 8 3 1.892789 1.5
row 20 16 4 2.523719 2

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Drag the cell range B12:B20 and right-click , and select Cell formatting.
@@@@@@@Select a number from the classification and set the number of decimal places to 5.
@@@@@@@Enter 1/16 in half-width in cell B12 (Displayed as a decimal).
@@@@@@@Enter data in the cell range B13:B20 using half-width characters as shown in the worksheet above.
@@@@@@@Enter =LOG(B12,2) in half-width in cell C12.
@@@@@@@Enter =LOG(B12,3) in half-width in cell D12.
@@@@@@@Enter =LOG(B12,4) in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C20 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line and Markern
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points(markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@Find the coordinates of point which the graphs of y=log(2)x , y=log(3)x , and y=log(4)x commonly pass through.

@@@ƒConsideration 2„
@@@@@@Find the common asymptote for the graphs of y=log(2)x , y=log(3)x , and y=log(4)x.

@@@ƒConsideration 3„
@@@@@@@
@@@@@@State the difference between the graphs of y=log(2)x , y=log(3)x , and y=log(4)x.

@
@y‚S‚O‚Qz Draw a graph of the logarithmic functions of y=log(1/2)x , y=log(1/3)x , and y=log(1/4)x.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

@@@ƒRefer„
@@@@@@Refer to y‚S‚O‚Pz how to complete the wprksheet.

column B column C column D column E
row 11 x y=log(x,1/2) y=log(x,1/3) y=log(x,1/4)
row 12 1/16 4 2.52371ĽĽĽ 2
row 13 1/8 3 1.892789ĽĽĽ 1.5
row 14 1/4 2 1.261859ĽĽĽ 1
row 15 1/2 1 0.630929ĽĽĽ 0.5
row 16 1 0 0 0
row 17 2 -1 -0.63092ĽĽĽ -0.5
row 18 4 -2 -1.26185ĽĽĽ -1
row 19 8 -3 -1.89278ĽĽĽ -1.5
row 20 16 -4 -2.52371ĽĽĽ -2

@@@(2) Draw a graph from the correspondence table in (1) avobe.
@@@@@@‡@ Drag and select the cell range B11:E20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line and Markern
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom of graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@Find the coordinates of the point that the graphs of y=log(1/2)x , y=log(1/3)x and , y=log(1/4)x pass commonly through.

@@@ƒConsideration 2„
@@@@@@Find the common asymptote for the graphs of y=log(1/2)x , y=log(1/3)x and , y=log(1/4)x.

@@@ƒConsideration 3„
@@@@@@State the difference between thegraphs of y=log(1/2)x , y=log(1/3)x and , y=log(1/4)x.
@y‚S‚O‚Rz Draw a graph of the logarithmic functions of y=log(2)x and y=log(1/2)x.
@ƒProcedure„
@@@(1) Create the following corresspondence table on the worksheet.

@@@ƒReferrence„
@@@@@@Please refer to y‚S‚O‚Pz how to complete the worksheet.

column B column C column D
row 11 x y=log(x,2) y=log(x,1/2)
row 12 1/16 -4 4
row 13 1/8 -3 3
row 14 1/4 -2 2
row 15 1/2 -1 1
row 16 1 0 0
row 17 2 1 -1
row 18 4 2 -2
row 19 8 3 -3
row 20 16 4 -4

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mSmooth line and Markern
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@@State the positional relationship of the graphs of y=log(2)x and y=log(1/2)x.

@@@ƒConsideration 2„
@@@@@@@In general , what can be said about the positional relationship between the graphs of y=log(2)x and y=log(1/2)x.

@y‚S‚O‚Sz Draw a graph of the logarithmic functions of y=log(2)x and y=log(2)(1/x)
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReference„
@@@@@@Please refer to y‚S‚O‚Pz how to complete the worksheet.

column B column C column D
row 11 x y=log(x,2) y=log(1/x,2)
row 12 1/16 -4 4
row 13 1/8 -3 3
row 14 1/4 -2 2
row 15 1/2 -1 1
row 16 1 0 0
row 17 2 1 -1
row 18 4 2 -2
row 19 8 3 -3
row 20 16 4 -4

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correspndence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line and Markersn
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines
@@@@@@‡B Right-click on the numbers on the y-axis¸ ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration 1„
@@@@@@@State the positional relationship between the graphs of y=log(2)x and y=log(2)(1/x).

@@@ƒConsideration 2„
@@@@@@In general , what can be said about the positional relationship between the graphs of y=log(a)x and y=log(a)(1/x).

@
@y‚S‚O‚Tz Solve equation log(10)x+log(10)(x-3)=1 using graphs.
@ƒProcedure„
@@@@@@@@@@
@@@@@@@@@@Draw graphs of y=log(10)x and y=-log(10)(x-3)+1.

@@@(1) Create the following correspondence table on the worksheet.

@@@ƒReferation„
@@@@@@Please refer to y401z how to complete the worksheet.
column B column C column D
row 11 x y=log(x,10) y=-log(x-3,10)+1
row 12 3.5 0.54406ĽĽĽ 1.30102ĽĽĽ
row 13 4 0.60205ĽĽĽ 1
row 14 4.5 0.65321ĽĽĽ 0.82390ĽĽĽ
row 15 5 0.69897ĽĽĽ 0.69897ĽĽĽ
row 16 5.5 0.74036ĽĽĽ 0.60205ĽĽĽ
row 17 6 0.77815ĽĽĽ 0.52287ĽĽĽ
row 18 6.5 0.81291ĽĽĽ 0.45593ĽĽĽ
row 19 7 0.84509ĽĽĽ 0.39794ĽĽĽ
row 20 7.5 0.87506ĽĽĽ 0.34678ĽĽĽ

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D20 in the correpondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line and markern
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.1
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@(4) Delete points (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration„
@@@@@@@Solve equation log(10)x+log(10)(x-3)=1 from the intersection of the graphs of y=log(10)x and y=-log(10)(x-3)+1.
@y‚S‚O‚Uz Solve inequality log(3)(x-3)+log(3)(x-5)<1 using a graph.
@ƒŽč‡„
@@@@@@@@@@log(3)(x-3) < -log(x-5)+1
@@@@@@@@@@Draw graphs of y=log(3)(x-3) and y=-log(3)(x-5)+1.

@@@(1) Create the following correspondence table on a worksheet.

@@@ƒReference„
@@@@@Please refer to y401z how to complete the worksheet.

column B column C column D
row 11 x y=log(x-3,3) y=-log(x-5,3)+1
row 12 5.5 0.83404ĽĽĽ 1.63092ĽĽĽ
row 13 6 1 1
row 14 6.5 1.14031ĽĽĽ 0.63092ĽĽĽ
row 15 7 1.26185ĽĽĽ 0.36907ĽĽĽ
row 16 7.5 1.36907ĽĽĽ 0.16595ĽĽĽ
row 17 8 1.46497ĽĽĽ 0
ro 18 8.5 1.55172ĽĽĽ -0.14031ĽĽĽ

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D18 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line and Markern
@@@@@@‡B Expand the graph area vertically by dragging the top and bottom edges of the graph area.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on te x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.00
@@@@@@¨ close ¨ Right-click on the numbers on te x-axis ¨ Add gridline
@@@@@@‡A Right-click on the numbers on te y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.1
@@@@@@¨ close ¨ Right-click on the numbers on te y-axis ¨ Add auxiliary gridline

@@@(4) Delete poinfs (markers) in the correspondence table from the graph.
@@@@@@‡@ Left-click on the graph area to activate it.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen

@@@@@@

@@@ƒConsideration„
@@@@@@@Solve inequality log(3)(x-3)+log(3)(x-5) < 1 from the intersection of the graphs of y=log(3)(x-3) and y=-log(3)(x-5)+1
@@@@@@
@y‚S‚O‚Vz Compare the size of 1/2, -log(2)(1/3) , and log(1/2)7 using the graphs.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 12 @‡@ 1/2 0.50000
row 13 @‡A -log(1/3,2) 1.58496
row 14 @‡B log(7,1/2) -2.80735

@@@ƒReference„
@@@@@sHow to complete a worksheet abovet
@@@@@@@Enter data into cells B12 , B13 , and B14 as shown in the worksheet above.
@@@@@@@Drag the cell range C12:C14 , right-click , and select Cell Formatting.
@@@@@ @Select a number from the crassification and set the number pf decimal places to 5.
@@@@@@@Enter 1/2 in half-width in cell C12.
@@@@@@@Enter =-1*LOG(1/3,2) in half-width in cell C13.
@@@@@@@Enter =LOG(7,1/2) in half-width in cell C14.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B12:C14 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mVertical barn¨m2-D vertical bar clustern

@@@@@@

@@@ƒConsideratin„
@@@@@@From the bar graphs of 1/2 , -log(2)(1/3) , and log(1/2)7 , compare the sizes of 1/2 , -log(2)(1/2) , and log(1/2)7.




@To table of contents@
@y‚T‚O‚Pz Draw a ciircle x^2+y^2+6x-8y-11=0
@ƒProcedure„@
@@@@@@@@@@x^2+y^2+6x-8y-11=0
@@@@@@@@@@x^2+6x+9+y^2-8y+16=11+9+16
@@@@@@@@@@(x+3)^2+(y-4)^2=36 , center(-3,4 ) , radius 6
@@@@@@@@@@(y-4)^2=36-(x+3)^2
@@@@@@@@@@y-4=+ă{36-(x+3)^2} , -ă{36-(x+3)^2}
@@@@@@@@@@y=4+ă{36-(x+3)^2} , 4-ă{36-(x+3)^2}

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 x y=4+SQRT(36-(x+3)^2) y=4-SQRT(36-(x+3)^2)
row 12 -9 4 4
row 13 -8.9 5.09087ĽĽĽ 2.90912ĽĽĽ
row 14 -8.8 5.53622ĽĽĽ 2.46377ĽĽĽ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 130 2.8 5.53622ĽĽĽ 2.46377ĽĽĽ
row 131 2.9 5.09087ĽĽĽ 2.90912ĽĽĽ
row 132 3 4 4

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -9 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B132 , Right-click , and select Paste.
@@@@@@@Enter =4+SQRT(36-(B12+3)^2) in half-width in cell C12.
@@@@@@@Enter =4-SQRT(36-(B12+3)^2) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , Right-click , and select Copy.
@@@@@@@Drag the cell range C13:C132 , Right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D132 in the correspondence table of (1) above.
@@@@@@‡A mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area to stretch and adjust.

@@@(3) Arrange the xy plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that it is a circle with center(-3,4) amd radius 6.

@
y‚T‚O‚Qz Find the coordinates of the common point between circle x^2+y^2=5 and straight line y=-x+1 by drawing a graph.
@ƒProcedure„
@@@@@@@@@@
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 x y=SQRT(5-x^2) y=-SQRT(5-x^2) y=-x+1
row 12 -2.236 0.01743ĽĽĽ -0.01743ĽĽĽ 3.236
row 13 -2.2 0.4 -0.4 3.2
row 14 -2.1 0.76811ĽĽĽ -0.76811ĽĽĽ 3.1
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 56 2.1 0.76811ĽĽĽ -0.76811 -1.1
row 57 2.2 0.4 -0.4 -1.2
row 58 2.236 0.01743ĽĽĽ -0.01743ĽĽĽ -1.236

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -2.236 in half-width in cell B12.
@@@@@@@Enter -2.2 in half-width in cell B13.
@@@@@@@Enter =B13+0.1 in half-width in cell B14.
@@@@@@@Right-click on cell B14 and select Copy.
@@@@@@@Drag the cell range B15:B57 , right-click , and select Paste.
@@@@@@@Enter 2.236 in half-width in cell C12.
@@@@@@@Enter =SQRT(5-B12^2) in half-width in cell C12.
@@@@@@@Enter =-SQRT(5-B12^2) in half-width in cell D12.
@@@@@@@Enter =-B12+1 in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy..
@@@@@@@Drag the cell range C13:C58 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E58 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@Find the coordinates of the common point from the graphs circle x^2+y^2=5 and straight line y=-x+1.

@
y‚T‚O‚Rz Draw an ellipse whose sum of distances from two fixed points (3,0) and (-3,0) is 10.
@ƒŽč‡„
@@@@@@@@@@Let x^2/a^2+y^2/b^2=1 be the ellipse to seek..
@@@@@@@@@@‚ƒ‚R
@@@@@@@@@@Since ‚Q‚‚P‚O , ‚‚T
@@@@@@@@@@Since c=ă(a^2-b^2) , 3=ă(5^2-b^2)
@@@@@@@@@@‚‚‚S
@@@@@@@@@@Therefore ,
@@@@@@@@@@@@x^2/5^2 + y^2/4^2 = 1 , y=4ă(1 - x^2/25) , y=-4ă(1 - x^2/25)

@@@(1) Create the following correspondence table in a worksheet.

column B columnC columnD
row 11 x y=4SQRT(1-x^2/25) y=-4SQRT(1-x^2/25)
row 12 -5 0 0
row 13 -4.9 0.79598ĽĽĽ -0.79598ĽĽĽ
row 14 -4.8 1.12 -1.12
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 110 4.8 1.12 -1.12
row 111 4.9 0.79598ĽĽĽ -0.79598ĽĽĽ
row 112 5 0 0

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -5 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B112 , right-click , and select Paste.
@@@@@@@Enter =4*SQRT(1-B12^2/25) in half-width in cell C12.
@@@@@@@Enter =-4*SQRT(1-B12^2/25) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C112 , right-click , and select Paste.

@@@(2) Drag a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D112 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the right and left edges of the graph areas holizontally to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that ellipse x^2/5^2 + y^2/4^2 = 1 is drawn.

y‚T‚O‚Sz Draw a parabola whose distance from rhe fixed point (2,0) is equal to the distance from the straight line x=-2.
@ƒProcedure„
@@@@@@@@@@Let y^2=4px be the parabola to seek.
@@@@@@@@@@Since p=2 , y^2=8x
@@@@@@@@@@y=2ă(2x) , y=-2ă(2x)

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 x y=2SQRT(2x) y=-2SQRT(2x)
row 12 0 0 0
row 13 0.1 0.89442ĽĽĽ -0.89442ĽĽĽ
row 14 0.2 1.26491ĽĽĽ -1.26491ĽĽĽ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 50 3.8 5.51361ĽĽĽ -5.51361ĽĽĽ
row 51 3.9 5.58569ĽĽĽ -5.58569ĽĽĽ
row 52 4 5.65685ĽĽĽ -5.65685ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B12.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B52 , right-click , and select Paste.
@@@@@@@Enter =2*SQRT(2*B12) in half-width in cell C12.
@@@@@@@Enter =-2*SQRT(2*B12) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C52 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D52 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of graph area vertically to stretch and adjust.

@@@(3) Arrange the coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that parabola y^2=8x is drawn.

@y‚T‚O‚Tz Draw a hyperbola whose dtfference between distances from two fixed points (5,0) and (-5,0) is 6.
@ƒProcedure„
@@@@@@@@@@Let x^2/a^2 - y^2/b^2 = 1 be the hyperbola to seek.
@@@@@@@@@@Since c=5 , 2a=6 , c=ă(a^2+b^2) , a=3 , b=4
@@@@@@@@@@Therefore
@@@@@@@@@@@@@x^2/3^2 - y^2/4^2 = 1 , y=4ă(x^2/9 - 1) , y=-4ă(x^2/9 - 1)

@@@(1) Create the following correspondence table in a worksheet.

column B column C column D
row 11 x y=4SQRT(x^2/9-1) y=-4SQRT(x^2/9-1)
row 12 10 12.71918ĽĽĽ -12.71918ĽĽĽ
row 13 9.9 12.57934ĽĽĽ -12.57934ĽĽĽ
row 14 9.8 12.43936ĽĽĽ -12.43936ĽĽĽ
Ľ Ľ Ľ Ľ
row 80 3.2 1.48473ĽĽĽ -1.48473ĽĽĽ
row 81 3.1 1.04136ĽĽĽ -1.04136ĽĽĽ
row 82 3 0 0
row 83 blank Blank blank
row 84 -3 0 0
row 85 -3.1 1.04136ĽĽĽ -1.04136ĽĽĽ
row 86 -3.2 1.48473ĽĽĽ -1.48473ĽĽĽ
Ľ Ľ Ľ Ľ
row 152 -9.8 12.43936ĽĽĽ -12.43936ĽĽĽ
row 153 -9.9 12.57934ĽĽĽ -12.57934ĽĽĽ
row 154 -10 12.71918ĽĽĽ -12.71918ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 10 in half-width in cell B12.
@@@@@@@Enter =B12-0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B82 , right-click , and select Paste.
@@@@@@@ƒZƒ‹B84‚É |‚R ‚𔟊p‚Ĺ“ü—Í‚ˇ‚éBEnter -3 in half-width in cell B84.
@@@@@@@Enter =B84-0.1 in half-width in cell B85.
@@@@@@@Right-click on cell B85 and select Copy.
@@@@@@@Drag the cell range B86:B154 , right-click , and select Paste.
@@@@@@@Enter =4*SQRT(B12^2/9-1) in half-width in cell C12.
@@@@@@@Enter =-4*SQRT(B12^2/9-1) in half-width in cell D12
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C154 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@@See y504z

@@@(3) Arrange the xy coordinate plane on whic the graph is drawn.
@@@@@@‡@ Right-click on the numbers on x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on x-axis ¨ Add auxiliary gridlines
@@@@@@‡A Right-click on the numbers on y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on y-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Make sure that hyperbola x^2/3^2 - y^2/4^2 = 1 is drawn.




@To table of contents@
@y‚U‚O‚Pz Draw graphs of function y=-2x^3+6x+1 and y'=-6x^2+6.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 x y=-2x^3+6x+1 y'=-6x^2+6
row 12 -3 37 -48
row 13 -2.9 32.378 -44.46
row 14 -2.8 28.104 -41.04
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 70 2.8 -26.104 -41.04
row 71 2.9 -30.378 -44.46
row 72 3 -35 -48

@@@ƒReference„
@@@@@sHow to complete the worksheet avove.t
@@@@@@@Enter -3 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B72 , right-click and , select Paste.
@@@@@@@Enter =-2*B12^3+6*B12+1 in half-width in cell C12.
@@@@@@@Enter =-6*B12^2+6 in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C72 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:D72 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen.
@@@@@@‡B Drag the top and bottom area of the graph area vertically to strech and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@When y'<0 , what does the graph of y=-2x^2+6x+1 look like ?

@@@ƒConsideration 2„
@@@@@@@When y'>0 , what does the graph of y=-2x^2+6x+1 look like ?

@@@ƒConsideration 3„
@@@@@@@When y'=0 , what does the graph of y=-2x^2+6x+1 look like ?

@
y‚U‚O‚Qz Find the maximam and minimum values by drawing a graph of function y=2x^3-3x^2-12x-6 (-2…x…4)
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 x y=2x^3-3x^2-12x-6
row 12 -2 -1
row 13 -1.9 -7.748
row 14 -1.8 -5.784
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 70 3.8 14.824
row 71 3.9 20.208
row 72 4 26

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -2 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B72 , right-click , and select Paste.
@@@@@@@Enter =2*B12^3-3*B12^2-12*B12-6 in half-width in cell C12.
@@@@@@@Right-click on cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C72 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C72 in the orrespondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines
@@@@@@‡B Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Scale interval Fixed 5.0
@@@@@@¨ 3close

@@@@@@

@@@ƒConsideration 1„
@@@@@@@ Find the maximum value when -2…x…4 from the graph.

@@@ƒConsideration 2„
@@@@@@@Find the minimum value when -2…x…4 from the graph.

@
y‚U‚O‚Rz Find the number of different real solutions for cubic equation x^3-3x^2-a=0 using a graph.
@ƒProcedure„
@@@@@@@@@@x^3-3x^2=a
@@@@@@@@@@Draw graphs of y=x^3-3x^2 and y=a.

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 8 a= 1 ‹ó”’
row 9 blank blank blank
row 10 blank blank blank
row 11 x y=x^3-3x^2 y=a
row 12 -2 -20 1
row 13 -1.9 -17.689 1
row 14 -1.8 -15.552 1
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 70 3.8 11.552 1
row 71 3.9 13.689 1
row 72 4 16 1

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter a= in cell B81.
@@@@@@@Enter 1 in half-width in cell C81.
@@@@@@@Enter -2 in half-width in cell B12.o stretch and
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cellrangeB14:B72 , right-click , and select Paste.
@@@@@@@Enter =B12^3-3*B12^2 in half-width in cell C12.
@@@@@@@Enter =$C$8 in half-width in cell D12.
@@@@@@@Drag the cellrange C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cellrange C13:C72 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) avove.
@@@@@@‡@ Drag and select the cell range B11:D72 in the correspondence table of (1) above..
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 1.0
@@@@@@¨ close ¨Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@When a=1 , a=-5 , find the number of different real solutions.
@@@ƒConsideration 2„
@@@@@@@When a=0 , a=-4 , find the number of different real solutions.
@@@ƒConsideration 3„
@@@@@@@When a=-2 , find the number of different real solutions.
@@@ƒConsideration 4„
@@@@@@@How does the number of different real solutions change depending on the value of the constant a ?

y‚U‚O‚Sz When x†0 , draw a graph to confirm that the inequality x^3-3x^2+4†0.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 x y=x^3-3x^2+4
row 12 -3.2 -59.488
row 13 -3.1 -54.621
row 14 -3.0 -50.0
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 42 3.8 15.552
row 43 3.9 17.689
row 44 4.0 20.0

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B44 , right-click , and select Paste.
@@@@@@@Enter =B12^3-3*B12^2+4 in half-width in cell C12.
@@@@@@@Right-click on cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C44 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C44 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@When x†0 ,
@@@@@@@look at the graph and confirm that the inequality x^3-3x^2+4†0 holds true.
@y‚U‚O‚Tz Draw a graph of the curve y=logx and the tangent and normal at point(1,0) on it.
@ƒProcedure„
@@@@@@@@@@
@@@@@@@@@@Tangent F ‚™|‚O‚Pi‚˜|‚Pj@@@@ˆ@‚™‚˜|‚P
@@@@@@@@@@Normal F ‚™|‚O|‚Pi‚˜|‚Pj@@ ˆ@‚™|‚˜{‚P

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 x y=logx y=x-1 y=-x+1
row 12 0.05 -2.99573ĽĽĽ -0.95 0.95
row 13 0.1 -2.30258ĽĽĽ -0.9 0.9
row 14 0.15 -1.89711ĽĽĽ -0.85 0.85
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 109 4.9 1.58923ĽĽĽ 3.9 -3.9
row 110 4.95 1.59938ĽĽĽ 3.95 -3.95
row 111 5 1.60943ĽĽĽ 4 -4

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0.05 in half-width in cell B12.
@@@@@@@Enter =B12+0.05 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell rsange B14:B111 and right-click , and select Paste.
@@@@@@@Enter =LN(B12) in half-width in cell C12.
@@@@@@@Enter =B12-1 in half-width in cell D12.
@@@@@@@Enter =-B12+1 in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 and right-click , and select Copy.
@@@@@@@Drag the cell range C13:C111 and right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E111 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Arrange the top and bottom of the graph area vertically to sttetch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Make sure that y=x-1 is tangent to point (1,0) on y=logx.

@@@ƒConsideration 2„
@@@@@@@Make sure that y=-x+1 is normal to point (1,0) on y=logx.

@y‚U‚O‚Uz Draw the graph of ellipse x^2/4 + y^2 = 1 and the tangent at the point (6/5,4/5) on it.
@ƒProcedure„
@@@@@@@@@@ Since y^2 = 1 - x^2/4 , y=}ă(1-x^2/4)
@@@@@@@@@@Since the tangent formula , x/4 ~ 6/5 + y ~ 4/5 = 1
@@@@@@@@@@Therefore , tangent : y=-(3/8)x + 5/4
@@@@@@@@@@Draw the graphs of y = ă(1-x^2/4) , y = -ă(1-x^2/4) , and y = -3/8 + 5/4

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 x y=SQRT(1-x^2/4) y=-SQRT(1-x^2/4) y=(-3/8)x+5/4
row 12 -2 0 0 2
row 13 -1.95 0.22220ĽĽĽ -0.22220ĽĽĽ 1.98125
row 14 -1.9 0.31224ĽĽĽ -0.31224ĽĽĽ 1.9625
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 90 1.9 0.31224ĽĽĽ -0.31224ĽĽĽ 0.5375
row 91 1.95 0.2222ĽĽĽ -0.22220ĽĽĽ 0.51875
row 92 2 0 0 0.5

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -2 in half-width in cell B12.
@@@@@@@Enter =B12+0.05 in half-width in cell B13.
@@@@@@@Right-click on cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B32 , right-click , and select Paste.
@@@@@@@Enter =SQRT(1-B12^2/4) in half-width in cell C12.
@@@@@@@Enter =-SQRT(1-B12^2/4) in half-width in cell D12.
@@@@@@@Enter =-3*B12/8+5/4 in half-width in cell E12.
@@@@@@@Drag the cell range C12:E12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C92 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:E92 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth line]
@@@@@@‡B Drag the left and right edges of graph area horizontally and the top and bottom edges of graph area vertically to strech and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0 ¨ close
@@@@@@¨ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.2¨ close
@@@@@@¨ Right-click on the numbers on the x-axis ¨ Add auxiliary gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Axis formatting
@@@@@@¨ Auxiliary scale interval Fixed 0.2 ¨ close ¨ Right-click on the numbers on the y-axis
@@@@@@¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@Make sure that y=(-3/8)x+5/4 is the tangent to the point (6/5,4/5) on the ellipse x^2/4+y^2=1.




@To table of contents@
@y‚V‚O‚Pz Draw a graph of function y=xă(9-x^2).
@ƒProcetdure„
@@@(1) Create the forrowing correspondence table on a worksheet.

column B column C
row 11 x y=xSQRT(9-x^2)
row 12 -3 0
row 13 -2.9 -2.2753ĽĽĽ
row 14 -2.8 -3.01569ĽĽĽ
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 70 2.8 3.01569ĽĽĽ
row 71 2.9 2.22753ĽĽĽ
row 72 3 0

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -3 in half-width in the cell B12.
@@@@@@@Enter =B12+0.1 in half-width in the cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B72 , right-click , and select Paste.
@@@@@@@Enter =B12*SQRT(9-B12^2) in half-width in the cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C72 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C72 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges og the graph area vertically to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 1.0
@@@@@@¨ close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@@@@‡A Right-click on the numbers on the y-axis ¨ Axis formatting ¨ Auxiliary scale interval Fixed 0.5
@@@@@@¨ close ¨ Right-click on the numbers on the y-axis ¨ Add auxiliary gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Find the local maximum from the graph.

@@@ƒConsideration 2„
@@@@@@@Find the local minimum from the graph.

@@@ƒConsideration 3„
@@@@@@@Find the maximum value from the graph.

@@@ƒConsideration 4„
@@@@@@@Find the minimum value from the graph.

@
y‚V‚O‚Qz For 0…x…2ƒÎ , draw the graph of the function y=x+2sinx.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 x y=x+2sinx
row 12 0 0
row 13 2 0.10470ĽĽĽ
row 14 4 0.20932ĽĽĽ
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 190 356 6.07385ĽĽĽ
row 191 358 6.17847ĽĽĽ
row 192 360 6.28318ĽĽĽ

@@@ƒReference„
@@@@@sHw to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B12.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B192 , right-click , and select Paste.
@@@@@@@Enter =RADIANS(B12)+2*SIN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C192 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Axis formatting ¨ Scale interval Fixed 60.0
@@@@@@¨ Close ¨ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Use a graph to find the value of x that takes the local maximum value when 0…x…2ƒÎ.@Then caluculate the local maximum value.

@@@ƒConsideration 2„@
@@@@@@@Use a graph to find the value of x that takes the local miniimum value when 0…x…2ƒÎ.@Then caluculate the local minimum value.

@@@ƒConsideration 3„
@@@@@@@Use a graph to find the value of x that takes the maximum value when 0…x…2ƒÎ.@Then caluculate the maximum value.

@@@ƒConsideration 4„
@@@@@@@Use a graph to find the value of x that takes the miniimum value when 0…x…2ƒÎ.@Then caluculate the minimum value.

@
y‚V‚O‚Rz Draw a graph of yhe function y=x+ă(4-x^2)
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 x y=x+SQRT(4-x^2)
row 12 -2 -2
row 13 -1.9 -1.27550ĽĽĽ
row 14 -1.8 -0.92822ĽĽĽ
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 50 1.8 2.67177ĽĽĽ
row 51 1.9 2.52449ĽĽĽ
row 52 2 2

@@@ƒReference„
@@@@@sHow to complete thw worksheet abovet
@@@@@@@Enter -2 in half-width in cell B12.
@@@@@@@Enter =B12+0.1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B52 and right-click , and select Paste.
@@@@@@@Enter =B12+SQRT(4-B12^2) in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C52 and right-click , and select Paste.
@@@@@@ Enter 2 in half-width in cell C52.

@@@(2) Draw a graph from the correspondence in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C52 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration„
@@@@@@@Find the minimum value and th value of x at that time from the graph.

y‚V‚O‚Sz When -1…x…1 , draw a graph of the function y=e^(2x)-2e^x+1
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 x y=e^2x-2e^x+1
row 12 -1 0.39957ĽĽĽ
row 13 -0.95 0.37608ĽĽĽ
row 14 -0.9 0.35215ĽĽĽ
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 50 0.9 2.13044ĽĽĽ
row 51 0.95 2.51447ĽĽĽ
row 52 1 2.95249ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter -1 in half-width in cell B12.
@@@@@@@Enter =B12+0.05 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B52 , right-click , and select Paste.
@@@@@@@Enter =EXP(2*B12)-2*EXP(B12)+1 in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C52 , right-click , and select Paste.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B11:C52 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Riht-click on the numbers on the x-axis ¨ Add gridlines

@@@@@@

@@@ƒConsideration 1„
@@@@@@@When -1…x…1 , find the maximum value and the value of x at the time using the graph.

@@@ƒConsideration 2„
@@@@@@@When -1…x…1 , find the minimum value and the value of x at the time using the graph.

@



@To table of contents@
@y‚W‚O‚Pz Find the area enclosed by the curve y=x^2 , the x-axis , and the straight line x=1 using the piecewise method.
@ƒProcedure„
@@@(1) Create the following correspondence method on a worksheet.

column B column C column D
row 11 x y=x^2 0.01~x^2
row 12 0.01 0.0001 0.000001
row 13 0.02 0.0004 0.000004
row 14 0.03 0.0009 0.000009
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 109 0.98 0.9604 0.009604
row 110 0.99 0.9801 0.009801
row 111 1 1 0.01
row 112s blank blank 0.33835

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0.01 in halh-width in cell B12.
@@@@@@@Enter =B12+0.01 in halh-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B111 , right-click , and select Paste.
@@@@@@@Enter =B12^2 in halh-width in cell C12.
@@@@@@@Enter =0.01*C12 in halh-width in cell D12.
@@@@@@@Right-click the cell range C12:D12 , and select Copy.
@@@@@@@Drag the cell range C13:C111 , right-click , and select Paste.

@@@(2) Find the piecewisw from the correspondence table in (1) abobe.
@@@@@@‡@ Find the sum of the cell range D12:D111 in the correspondence table in (1) above.
@@ @@@@@@Enter =SUM(D12:D111) in half-width in cell D112 .of the worksheet.

@@@ƒConsideration 1„
@@@@@@@Calcurate the following definite integral and find the area enclosed by the curvey=x^2 , the x-axis , and the straight line x=1.

@@@@@@@

@@@ƒConsideration 2„
@@@@@@@Compare the value of the definite integral obtained in consideration 1 with the value of D112 , which is the sum of the cell range D12:D111 in the correspondence table of the worksheet.

@
@y‚W‚O‚Qz Draw a graph of cycloid ‚˜‚”|‚“‚‰‚Ž‚”A‚™‚P|‚ƒ‚‚“‚” (‚O…‚”…‚QƒÎ). Also , find its length.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E column F column G
row 11 x x=t-sint y=1-cost C(n)-C(n-1) D(n)-D(n-1) SQRT(E()^2+F()^2)
row 12 0 0 0 ‹ó”’ ‹ó”’ ‹ó”’
row 13 2 7.088ĽĽ 0.000ĽĽ 7.088ĽĽ 0.000ĽĽ 0.000ĽĽ
row 14 4 5.669ĽĽ 0.002ĽĽ 4.960ĽĽ 0.001ĽĽ 0.001ĽĽ
Ľ Ľ Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ Ľ Ľ
row 190 356 6.283ĽĽ 0.002ĽĽ 0.000ĽĽ -0.003ĽĽ 0.003ĽĽ
row 191 358 6.283ĽĽ 0.000ĽĽ 4.960ĽĽ -0.001ĽĽ 0.001ĽĽ
row 192 360 6.283ĽĽ 0 7.088ĽĽ -0.000ĽĽ 0.000ĽĽ
row 193 blank blank blank blank blank 7.99989ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 0 in half-width in cell B1.
@@@@@@@Enter =B12+2 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B192 , right-click , and select Paste.
@@@@@@@Enter =RADIANS(B12)-SIN(RADIANS(B12)) in half-width in cell C12.
@@@@@@@Enyer =1-COS(RADIANS(B12)) in half-width in cell D12
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:D192 , right-click , and select Paste.
@@@@@@@Enter =C13-C12 in half-width in cell E13.
@@@@@@@Enter =D13-D12 in half-width in cell F13.
@@@@@@@Enter =SQRT(E13^2+F13^2) in half-width in cell G13.
@@@@@@@Drag the cell range E13:G13 , right-click , and select Copy.
@@@@@@@Drag the cell range E14:E192 , right-click , and select Paste.
@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range C11:D192 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mSmooth linen
@@@@@@‡B Drag the top and bottom edges of the graph area vertically to stretch and adjust.
@@@(3) Arrange the xy-coordinate plane on which the graph is drawn.
@@@@@@‡@ Right-click on the numbers on the x-axis ¨ Add gridlines
@@@(4) Find the length of the cycloid from the correspondence table in (1) above.
@@@@@@‡@ Find the sum of the cell range G13:G192 in the correspondence table of (1) above.
@@ @@@@@@Enter =SUM(G13:G192) in half-width in cell G193 of the worksheet.

@@@@@@

@@@ƒConsideration 1„
@@@@@@@Calculate the following definite integral and find the length of the cycloid ‚˜‚”|‚“‚‰‚Ž‚”A‚™‚P|‚ƒ‚‚“‚” (‚O…‚”…‚QƒÎ)
@@@@@@@

@@@ƒConsideration 2„
@@@@@@@Compare the value of the definite integral obtained in Consideration 1 with the value of cell G193 , which is the sum of the cell range G13:G192 in the correspondence table of the worksheet.

@



@To table of contents@
@y‚X‚O‚Pz Find the general terms of the arithmetic progression where the 3rd term is 5 and the 12th term is 59.
@ƒProcedure„
@@@@@@@Let@
@@@@@@@@@@
@@@@@@@@@@
@@@@@@@@@@If you solve ‡@ and ‡A simultaneously , ‚|‚VA‚„‚U
@@@@@@@@@@Therefore ,
@@@@@@@@@@@ @ @ @ @

@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 Section number 6n-13
row 12 1 -7
row 13 2 -1
row 14 3 5
Ľ Ľ Ľ
Ľ Ľ Ľ
Ľ Ľ Ľ
row 109 98 575
row 110 99 581
row 111 100 587

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B111 , right-click , and select Paste.
@@@@@@@Enter =6*B12-13 in half-width in cell C12.
@@@@@@@Right-click cell C12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C111 , right-click , and select Paste.

@@@ƒConsideration 1„
@@@@@@@Solving for a(n)=6n-13=143 gives n=26.
@@@@@@@Check on the worksheet that 143 is the 26th term.

@@@ƒConsideration 2„
@@@@@@@
@@@@@@ Check on the worksheet that the 96th term is 563.

@
@y‚X‚O‚Qz Find the general terms of the geometric progression where the 3rd term is 12 and the 7th term is 192.
@ƒProgression„
@@@@@@@@@ @Let
@@@@@@@@@@
@@@@@@@@@@
@@@@@@@@@@Solving ‡@ and ‡A simultaneously gives ‚‚RA‚’}‚Q
@@@@@@@@@@Therefore ,

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 Section number 3*2^(n-1) 3*(-2)^(n-1)
row 12 1 3 3
row 13 2 6 -6
row 14 3 12 12
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 44 33 12884ĽĽĽ 12884ĽĽĽ
row 45 34 25769ĽĽĽ -25369ĽĽĽ
row 46 35 51539ĽĽĽ 51539ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter the information in the cell range B11:D111 as shown in the table above.
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 , right-click , and select Paste.
@@@@@@@Enter =3*2^(B12-1) in half-width in cell C12.
@@@@@@@Enter =3*(-2)^(B12-1) in half-width in cell D12.
@@@@@@@Drag the cell range C12:D12 , right-click , and select Copy.
@@@@@@@Drag the cell range C13:C46 , right-click , and select Paste.

@@@ƒConsideration 1„
@@@@@@@
@@@@@@@
@@@@@@@Check on the worksheet that the 8th term is 384 and -384.

@@@ƒConsideration 2„
@@@@@@@Solving a(n)=3*2^(n-1)=786432 gives n=19.
@@@@@@@Solving a(n)=3*(-2)^(n-1)=786432 gives n=19.
@@@@@@@Check on the worksheet that 786432 is the 19th term.

@
@y‚X‚O‚RzFind the sum from the first term to the nth term of an arithmetic progression where the first term is 50 and the commom difference is -6.
@ƒProcedure„
@@@@@@@@@@
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 section number -6n+56 sum -3n^2+53n
row 12 1 50 50 50
row 13 2 44 94 94
row 14 3 38 132 132
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 59 48 -232 -4368 -4368
row 60 49 -238 -4606 -4606
row 61 50 -244 -4850 -4850

@@@ƒReference„
@@@@@sHow to complette the worksheet abovet
@@@@@@@Drag the cell range B11:E11 , right-click and select Cell formatting.
@@@@@@ @Select Tablar format ¨ Classification string¨ OK
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B61 , right-click , and select Paste.
@@@@@@@Enter =-6*B12+56 in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C61 , right-click , and select Paste.
@@@@@@@Enter 50 in half-width in cell D12.
@@@@@@@Enter =D12+C13 in half-width in cell D13.
@@@@@@@Right-click on the cell D13 and select Copy.
@@@@@@@Drag the cell range D14:D61 , right-click , and select Paste.
@@@@@@@Enter =-3*B12^2+53*B12 in half-width in cell E12.
@@@@@@@Right-click on the cell E12 and select Copy.
@@@@@@@Drag the cell range E13:E61 , right-click , and select Paste.

@@@ƒConsideration 1„
@@@@@@@Make sure that the sum in column D of the worksheet matches -3n^2+53n in column E.

@@@ƒConsideration 2„
@@@@@@@
@@@@@@@
@@@@@@@Therefore , when n=9 , it is maximum.
@@@@@@@Check on the worksheet that when n=9 , it is the maximum.

@y‚X‚O‚SzFind the sum from the first term to the nth term of a geometric progression with an initial term of 1/2and a common ratio of 2.
@ƒŽč‡„
@@@@@@@@@@
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 Section number 2^(n-2) Sum (2^n-1)/2
row 12 1 0.5 0.5 0.5
row 13 2 1 1.5 1.5
row 14 3 2 3.5 3.5
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 44 33 21474ĽĽĽ 42949ĽĽĽ 42949ĽĽĽ
row 45 34 42949ĽĽĽ 85899ĽĽĽ 85899ĽĽĽ
row 46 35 85899ĽĽĽ 17179ĽĽĽ 17179ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the woksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 , right-click , and select Paste.
@@@@@@@Enter =2^(B12-2) in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range B13:C46 , right-click , and select Paste.
@@@@@@@Enter 0.5 in half-width in cell D12.
@@@@@@@Enter =D12+C13 in half-width in cell D13.
@@@@@@@Right-click on the cell D13 and select Copy.
@@@@@@@Drag the cell range D14:D46 , right-click , and select Paste.
@@@@@@@Enter =(2^B12-1)/2 in half-width in cell E12.
@@@@@@@Right-click on the cell E12 and select Copy
@@@@@@@Drag the cell range E13:E46 , right-click , and select Paste.

@@@ƒConsideration 1„
@@@@@@@Make sure that the sum in column D of the worksheet matches (2^n-1)/2 in column E.

@@@ƒConsideration 2„
@@@@@@@
@@@@@@@Solving this gives n=11.
@@@@@@@Check on the sheet that the fist term the sum exceedds 1000 is the sum up to the 11th term.

@
@y‚X‚O‚Tz Find the sum from 1^2 to n^2.
@ƒProcedure„
@@@@@@@@@@
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 Section number n^2 Sum n(n+1)(2n+1)/6
row 12 1 1 1 1
row 13 2 4 5 5
row 14 3 9 14 14
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 44 33 1089 12529 12529
row 45 34 1156 13685 13685
row 46 35 1225 14910 14910

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 ,right-click . and select Paste.
@@@@@@@Enter =B12^2 in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C46 ,right-click . and select Paste.
@@@@@@@Enter 1 in half-width in cell D12.
@@@@@@@Enter =D12+C13 in half-width in cell D13.
@@@@@@@Right-click on the cell D13 and select Copy.
@@@@@@@Drag the cell range D14:D46 ,right-click . and select Paste.
@@@@@@@Enter =B12*(B12+1)*(2*B12+1)/6 in half-width in cell E12.
@@@@@@@Right-click on the cell E12 and select Copy.
@@@@@@@Drag the cell range E13:E46 ,right-click . and select Paste.

@@@ƒConsideration„
@@@@@@@Make sure that the sum in column D of the worksheet natches n(n+1)(2n+1)/6 in column E.

@y‚X‚O‚Uz Find the sum from 1^3 to n^3.
@ƒProcedure„
@@@@@@@@@@
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E
row 11 Section number n^3 Sum {n(n+1)/2}^2
row 12 1 1 1 1
row 13 2 8 9 9
row 14 3 27 36 36
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 44 33 35937 314721 314721
row 45 34 39304 354025 354025
row 46 35 42875 396900 396900

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 , right-click , and select Paste.
@@@@@@@Enter =B12^3 in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C46 , right-click , and select Paste.
@@@@@@@Enter 1 in half-width in cell D12.
@@@@@@@Enter =D12+C13 in half-width in cell D13.
@@@@@@@Right-click on the cell D13 and select Copy.
@@@@@@@Drag the cell range D14:D46 , right-click , and select Paste.
@@@@@@@Enter =(B12*(B12+1)/2)^2 in half-width in cell E12..
@@@@@@@Right-click on the cell E12 and select Copy.
@@@@@@@Drag the cell range E13:E46 , right-click , and select Paste.

@@@ƒConsideration„
@@@@@@@Make sure that the sum in column D of the worksheet matches {n(n+1)/2}^2 in column E.

@y‚X‚O‚VzFind the general term of the following sequence.
@@@@@@‚RC‚SC‚UC‚P‚OC‚P‚WC‚R‚SCEEE
@ƒProcedure„
@@@(1) Create the following correspondence table on the worksheet.

column B column C column D column E
row 11 Section number Difference Sum 2^(n-1)+2
row 12 1 blank 3 3
row 13 2 1 4 4
row 14 3 2 6 6
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ Ľ
row 44 33 21474ĽĽĽ 42949ĽĽĽ 42949ĽĽĽ
row 45 34 42949ĽĽĽ 8599ĽĽĽ 85899ĽĽĽ
row 46 35 85899ĽĽĽ 17179ĽĽĽ 17179ĽĽĽ

@@@ƒReferrence„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cel B12.
@@@@@@@Enter =B12+1 in half-width in cel B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 , right-click , and select Paste.
@@@@@@@Enter =2^(B13-2) in half-width in cel C13.
@@@@@@@Right-click on the cell C13 and select Copy.
@@@@@@@Drag the cell range C14:C46 , right-click , and select Paste.
@@@@@@@Enter 3 in half-width in cel D12.
@@@@@@@Enter =D12+C13 in half-width in cel D13.
@@@@@@@Right-click on the cell D13 and select Copy.
@@@@@@@Drag the cell range D14:D46 , right-click , and select Paste.
@@@@@@@Enter =2^(B12-1)+2 in half-width in cel E12.
@@@@@@@Right-click on the cell E12 and select Copy.
@@@@@@@Drag the cell range E13:E46 , right-click , and select Paste.

@@@ƒConsideration„
@@@@@@@Make sure that the sum in column D of the worksheet matches 2^(n-1)+2 in column E.

@y‚X‚O‚WzFind the general term of the sequence {a(n)} defined by the following recurrence formula.
@@@@@@
@@@@@@
@ƒProcedure„
@@@@@@@@@@When a(n+1) =2a(n)+3 is transformed , a(n+1)+3=2{a(n)+3}
@@@@@@@@@@Sequence {a(n)+3} is a geometric progression where the first term is 4 and the common ratio is 2.
@@@@@@@@@@Therefore , a(n)+3=4~2^(n-1)=2^(n+1)
@@@@@@@@@@a(n)=2^(n+1)-3

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 Section number Recurrence formula 2^(n+1)-3
row 12 1 1 1
row 13 2 5 5
row 14 3 13 13
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 44 33 17179ĽĽĽ 17179ĽĽĽ
row 45 34 34359ĽĽĽ 34359ĽĽĽ
row 46 35 68719ĽĽĽ 68719ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 , right-click , annd select Paste.
@@@@@@@Enter 1 in half-width in cell C12.
@@@@@@@Enter =2*C12+3 in half-width in cell C13.
@@@@@@@Right-click on the cell C13 and select Copy.
@@@@@@@Drag the cell range C14:C46 , right-click , annd select Paste.
@@@@@@@Enter =2^(B12+1)-3 in half-width in cell D12.
@@@@@@@Right-click on the cell D12 and select Copy.
@@@@@@@Drag the cell range D13:D46 , right-click , annd select Paste.

@@@ƒReference„
@@@@@@@Make sure that the recurrence formula in column C of the worksheet matches 2^(n+1)-3 in column D.

@y‚X‚O‚Xz Find the general term of the sequence {a(n)} defined by the following recurrence formula.
@@@@@@@@
@@@@@@@@
@@@@@@@@
@ƒProcedure„
@@@@@@@@@@Solve this recurrence formula to find the general term.
@@@@@@@@@@@
@@@@@@@@@@
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 Section number Recurrence formula General term
row 12 1 0 0
row 13 2 1 1
row 14 3 1 1
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 44 33 12477.ĽĽĽ 12477.ĽĽĽ
row 45 34 17044.ĽĽĽ 17044.ĽĽĽ
row 46 35 23282.ĽĽĽ 23282.ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the cell B13 and select Copy.
@@@@@@@Drag the cell range B14:B46 , right-click , and select Paste.
@@@@@@@Enter 0 in half-width in cell C12.
@@@@@@@Enter 1 in half-width in cell C13.
@@@@@@@Enter =(2*C13+C12)/2 in half-width in cell C14
@@@@@@@Right-click on the cell C14 and select Copy.
@@@@@@@Drag the cell range C15:C46 , right-click , and select Paste.
@@@@@@@Enter =(((1+SQRT(3))/2)^(B12-1)-((1-SQRT(3))/2)^(B12-1))/SQRT(3) in half-width in cell D12.
@@@@@@@Right-click on the cell D12 and select Copy.
@@@@@@@Drag the cell range D13:D46 , right-click , and select Paste.

@@@ƒConsideration„
@@@@@@@Make sure that the recurrence formula in column C of the worksheet matches the general term in column D.

@y‚X‚P‚Oz Consider the Febonacci sequence determined by the following recurrence formula.
@@@@@@@
@@@@@@@
@@@@@@@
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 Section number Recurrence formula nth term/ n-1th term
row 12 1 1 blank
row 13 2 1 1
row 14 3 2 2
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
Ľ Ľ Ľ Ľ
row 59 48 48075ĽĽĽ 1.618033989
row 60 49 77787ĽĽĽ 1.618033989
row 61 50 12586ĽĽĽ 1.618033989

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter 1 in half-width in cell B12.
@@@@@@@Enter =B12+1 in half-width in cell B13.
@@@@@@@Right-click on the B13 and select Copy.
@@@@@@@Drag the cell range B14:B61 , right-click , and select Paste.
@@@@@@@Enter 1 in half-width in cell C12.
@@@@@@@Enter 1 in half-width in cell C13.
@@@@@@@Enter =C12+C13 in half-width in cell C14.
@@@@@@@Right-click on the C14 and select Copy.
@@@@@@@Drag the cell range C15:C61 , right-click , and select Paste.
@@@@@@@Enter =C13/C12 in half-width in cell D13.
@@@@@@@Right-click on the D13 and select Copy.
@@@@@@@Drag the cell range D14:D61 , right-click , and select Paste.


@@@ƒConsideration„
@@@@@@@Make sure that the value of (the 50th term/ the 49th term) in column D of the worksheet is close to the golden ratio (1+ă5)/2ŕ1.618033989.





@To table of contens@
@y1001z Inverse trigonometric functions and Conversion from arc degree method to degree method
@@@@@@i‚`‚r‚h‚m functionA‚`‚b‚n‚r functionA‚`‚s‚`‚m functionA‚c‚d‚f‚q‚d‚d‚r functionj
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E column F column G
row 11 SIN ASIN COS ACOS TAN ATAN
row 12 -1 -90 1 0 -SQRT(3) -60
row 13 -SQRT(3)/2 -60 SQRT(3)/2 30 -1 -45
row 14 -SQRT(2)/2 -45 SQRT(2)/2 45 -1/SQRT(3) -30
row 15 -1/2 -30 1/2 60 0 0
row 16 0 0 0 90 1/SQRT(3) 30
row 17 1/2 30 -1/2 120 1 45
row 18 SQRT(2)/2 45 -SQRT(2)/2 135 SQRT(3) 60
row 19 SQRT(3)/2 60 -SQRT(3)/2 150 ‹ó”’ ‹ó”’
row 20 1 90 -1 180 ‹ó”’ ‹ó”’

@@@ƒReference„
@@@@@sHow to complete the worksheet above.t
@@@@@@@Drag the cell range B12:B20 , right-click and select Cell firmatting.
@@@@@@@Select Display format ¨ Classification string ¨ OK
@@@@@@@Enter data in the cell range B12:B20 as shown in the table avobe
@@@@@@@Drag the cell range D12:D20 , right-click ,and select Cell formatting
@@@@@@@Select Display format ¨ Classification string ¨ OK
@@@@@@@Enter data in the cell range D12:D20 as shown in the table avobe
@@@@@@@Drag the cell range F12:F20 , right-click ,and select Cell formatting
@@@@@@@Select Display format ¨ Classification string ¨ OK
@@@@@@@Enter data in the cell range F12:F18 as shown in the table avobe
@@@@@@@Enter =DEGREES(ASIN(-1)) in cell C12.
@@@@@@@Similarly , enter =DEGREES(ASIN(value in the same row of column B)) in half-width from C13 to C20
@@@@@@@Enter =DEGREES(ACOS(1)) in cell E12.
@@@@@@@Similarly , enter =DEGREES(ACOS(value in the same row of column D)) in half-width from E13 to E20
@@@@@@@Enter =DEGREES(ATAN(-SQRT(3))) in cell G12.
@@@@@@@Similarly , enter =DEGREES(ACOS(value in the same row of column F)) in half-width from G13 to G18

@@@ƒAbout functions„
@@@@@@@‚`‚r‚h‚m functionĽĽĽ ASIN(real number 1)
@@@@@@@@@@Return the angle of SIN that has a real number 1 in radians.

@@@@@@@‚`‚b‚n‚r functionĽĽĽ ACOS(real number 1)
@@@@@@@@@@Return the angle of COS that has a real number 1 in radians.

@@@@@@@‚`‚s‚`‚m functionĽĽĽ ATAN(real number 1)
@@@@@@@@@@Return the angle of TAN that has a real number 1 in radians.

@@@@@@@‚c‚d‚f‚q‚d‚d‚r functionĽĽĽ DEGREES(ƒ‰ƒWƒAƒ“)
@@@@@@@@@@Converts an angle in radians to an angle in degrees.

@
@y1002z Sum of squares of complex numbers
@@@@@@i‚r‚t‚l‚r‚p functionj
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 String Numerical value SUMSQ
row 12 0 0.00000 blank
row 13 1 1.00000 1
row 14 SQRT(2) 1.41421 3
row 15 SQRT(3) 1.73205 5
row 16 2 2.00000 7
row 17 SQRT(5) 2.23607 9
row 18 SQRT(6) 2.44949 11
row 19 SQRT(7) 2.64575 13
row 20 SQRT(8) 2.82843 15

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Drag the cell range B11:B20 , right-click , and select Cell formatting.
@@@@@@@Select Display format ¨ Classification string ¨ OK.
@@@@@@@Enter data in the cell range B11:B20 as shown in the table above.
@@@@@@@Enter =SQRT(0) in half-width in cell C12.
@@@@@@@Enter =SQRT(1) in half-width in cell C13.
@@@@@@@Enter =SQRT(2) in half-width in cell C14.
@@@@@@@Enter =SQRT(3) in half-width in cell C15.
@@@@@@@Enter =SQRT(4) in half-width in cell C16.
@@@@@@@Enter =SQRT(5) in half-width in cell C17.
@@@@@@@Enter =SQRT(6) in half-width in cell C18.
@@@@@@@Enter =SQRT(7) in half-width in cell C19.
@@@@@@@Enter =SQRT(8) in half-width in cell C20.
@@@@@@@Enter =SUMSQ(C12,C13) in half-width in cell D13.
@@@@@@@Right-click on the cell D13 and select Copy.
@@@@@@@Drag the cell range D14:D20 , right-click , and select Paste.

@@@ƒAbout function„
@@@@@@@‚r‚t‚l‚r‚p functionĽĽĽ SUMSQ(real number 1Creal number 2)
@@@@@@@@@@ Returns the value of (real number 1)^2 + (real number 2)^2
@
@y1003z Four arithmetic calculations of complex numbers <sum , difference , product , quotient> and display in complex number format
@@@@@@i‚h‚l‚r‚t‚l function A‚h‚l‚r‚t‚a function A‚h‚l‚o‚q‚n‚c‚t‚b‚s function A‚h‚l‚c‚h‚u function A‚b‚n‚l‚o‚k‚d‚w functionj
@ƒProcedure„
@@@(1) Create the correspondence table on a worksheet.

column B column C column D column E column F column G
row 11 blank A= 5-3i blank C= 2+i
row 12 blank B= -2+7i blank D= 2-i
row 13 blank blank blank blank blank blank
row 14 sum A+B= 3+4i blank blank blank
row 15 difference A-B= 7-10i blank blank blank
row 16 product A*B= 11+41i blank blank blank
row 17 quotient C/D= 0.6+0.8i blank blank blank

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter A= in cell C11.
@@@@@@@Enter B= in cell C12.
@@@@@@@Enter =COMPLEX(5,-3) in half-width in cell D11.
@@@@@@@Enter =COMPLEX(-2,7) in half-width in cell D12.
@@@@@@@Enter C= in half-width in cell F11.
@@@@@@@Enter D= in half-width in cell F12.
@@@@@@@Enter =COMPLEX(2,1) in half-width in cell G11.
@@@@@@@Enter =COMPLEX(2,-1) in half-width in cell G12.
@@@@@@@Enter data in the cell range B14:B17 as shown in the table above.
@@@@@@@Enter data in the cell range C14:C17 as shown in the table above.
@@@@@@@Enter =IMSUM(D11,D12) in half-width in cell D14..
@@@@@@@Enter =IMSUB(D11,D12) in half-width in cell D15.
@@@@@@@Enter =IMPRODUCT(D11,D12) in half-width in cell D16.
@@@@@@@Enter =IMDIV(G11,G12) in half-width in cell D17.

@@@ƒAbout functions„
@@@@@@@‚h‚l‚r‚t‚l functionĽĽĽ IMSUM(complex number 1Ccomplex number 2)
@@@@@@@@@@Returns the value of (complex number 1 + complex number 2)

@@@@@@@‚h‚l‚r‚t‚a functionĽĽĽ IMSUB(complex number 1Ccomplex number 2)
@@@@@@@@@@Returns the value of (comlex number 1 - complex number 2)

@@@@@@@‚h‚l‚o‚q‚n‚c‚t‚b‚s functionĽĽĽ IMPRODUCT(comlex number 1Ccomplex number 2)
@@@@@@@@@@Returns the value of (comlex number 1 ~ complex number 2)

@@@@@@@‚h‚l‚c‚h‚u functionĽĽĽ IMDIV(comlex number 1Ccomplex number 2)
@@@@@@@@@@Returns the value of (comlex number 1 € complex number 2)

@@@@@@@‚b‚n‚l‚o‚k‚d‚w functionĽĽĽ COMPLEX(real number 1Creal number 2)
@@@@@@@@@@Returns the complex number form (real number 1) + (real number 2) i

@y1004zConjugate complex number and Abloluete values
@@@@@@i‚h‚l‚b‚n‚m‚i‚t‚f‚`‚s‚d functionA‚h‚l‚`‚a‚r functionj
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 Complex number Conjugate complex number Absolute value
row 12 3+4i 3-4i 5
row 13 3 3 3
row 14 4i -4i 4

@@@ƒReference„
@@@@@sHow to compleete the worksheet abovet
@@@@@@@Enter 3+4i in half-width in cell B12.
@@@@@@@Enter 3 in half-width in cell B13.
@@@@@@@Enter 3i in half-width in cell B14.
@@@@@@@Enter =IMCONJUGATE(B12) in half-width in cell C12.
@@@@@@@Enter =IMCONJUGATE(B13) in half-width in cell C13.
@@@@@@@Enter =IMCONJUGATE(B14) in half-width in cell C14.
@@@@@@@Enter =IMABS(B12) in half-width in cell D12.
@@@@@@@Enter =IMABS(B13) in half-width in cell D13.
@@@@@@@Enter =IMABS(B14) in half-width in cell D14.

@@@ƒAbout functions„
@@@@@@@‚h‚l‚b‚n‚m‚i‚t‚f‚`‚s‚d functionĽĽĽ IMCONJUGATE(complex number 1)
@@@@@@@@@@Returns complex number that is the conjugate of complex number 1.

@@@@@@@‚h‚l‚`‚a‚r functionĽĽĽ IMABS(complex number 1)
@@@@@@@@@@Returns the absolute value of a complex number 1.

@
@y1005zargument , real part , imaginary part of complex number
@@@@@@i‚h‚l‚`‚q‚f‚t‚l‚d‚m‚s functionA‚h‚l‚q‚d‚`‚k functionA‚h‚l‚`‚f‚h‚m‚`‚q‚x functionj
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C column D column E column F
rows 11 String Value Argument Real part Imaginary part
rows 12 1 1 0 1 0
rows 13 SQRT(3)+i 1.73205ĽĽ+i 30 1.73205ĽĽĽ 1
rows 14 1+i 1+i 45 1 1
rows 15 1+SQRT(3)i 1+1.73205ĽĽi 60 1 1.73205ĽĽĽ
rows 16 i i 90 0 1
rows 17 -1+SQRT(3)i -1+1.73205ĽĽi 120 -1 1.73205ĽĽĽ
rows 18 -1+i -1+i 135 -1 1
rows 19 -SQRT(3)+i -1.73205ĽĽ+i 150 -1.73205ĽĽĽ 1
rows 20 -1 -1 180 -1 0

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Drag the cell range B11:B20 , right-click and select Cell formatting.
@@@@@@@Select Display format ¨ Classification string ¨ OK
@@@@@@@Enter data in the ccll range B11:B20 as shown in the table above.
@@@@@@@Enter =COMPLEX(1,0) in half-width in cell C12.
@@@@@@@Enter =COMPLEX(SQRT(3),1) in half-width in cell C13.
@@@@@@@Enter =COMPLEX(1,1) in half-width in cell C14.
@@@@@@@Enter =COMPLEX(1,SQRT(3)) in half-width in cell C15.
@@@@@@@Enter =COMPLEX(0,1) in half-width in cell C16.
@@@@@@@Enter =COMPLEX(-1,SQRT(3)) in half-width in cell C17.
@@@@@@@Enter =COMPLEX(SQRT(-1,1) in half-width in cell C18.
@@@@@@@Enter =COMPLEX(-SQRT(3),1) in half-width in cell C19.
@@@@@@@Enter =COMPLEX(-1,0) in half-width in cell C20.
@@@@@@@Enter =DEGREES(IMARGUMENT(C12)) in half-width in cell D12.
@@@@@@@Right-click on the cell D12 and select Copy.
@@@@@@@Drag the cell range D13:D20 , right-click and select Paste.
@@@@@@@Enter =IMREAL(C12) in half-width in cell E12.
@@@@@@@Right-click on the cell E12 and select Copy.
@@@@@@@Drag the cell range E13:E20 , right-click and select Paste.
@@@@@@@Enter =IMAGINARY(C12) in half-width in cell F12.
@@@@@@@Right-click on the cell F12 and select Copy.
@@@@@@@Drag the cell range F13:F20 , right-click and select Paste.

@@@ƒAbout functions„
@@@@@@@‚h‚l‚`‚q‚f‚t‚l‚d‚m‚s functionĽĽĽ IMARGUMENT(complex number 1)
@@@@@@@@@@Returns theargument of a complex number 1 in radians. unit

@@@@@@@‚h‚l‚q‚d‚`‚k functionĽĽĽ IMREAL(complex number 1)
@@@@@@@@@@Returns the real part of a complex number 1.

@@@@@@@‚h‚l‚`‚f‚h‚m‚`‚q‚x functionĽĽĽ IMAGINARY(complex number 1)
@@@@@@@@@@Returns the imaginary part of a complex number 1.

@y1006zThink of
@@@@@@@@@@de Moivle's theorem
@@@@@@@@@@@@
@@@@@@@@@@@@i‚h‚l‚o‚n‚v‚d‚q functionj
@ƒProcedure„

@@@(1) Create the following correspondence table on a worksheet.

@@@ƒReference„
@@@@@Refer to y401z how to complete the worksheet.

column B column C column D column E
row 9 z= 1+1.73205ĽĽĽi blank blank
row 10 blank blank blank blank
row 11 n (1+SQRT(3)i)^n argument absolute value
row 12 1 1+1.73205ĽĽĽi 60 2
row 13 2 -2.00000ĽĽĽ+3.46410ĽĽĽi 120 4
row 14 3 -8.00000ĽĽĽ-2.03361ĽĽĽi -180 8
row 15 4 -8-13.85640ĽĽĽi -120 16
row 16 5 16.00000ĽĽĽ-27.71281ĽĽĽi -60 32
row 17 6 64.00000ĽĽĽ+3.25378ĽĽĽi 2.91294E-13 64
row 18 7 64+110.85125ĽĽĽi 60 128
row 19 8 -128.00000ĽĽĽ+221.70250ĽĽĽi 120 256
row 20 9 -512.00000ĽĽĽ-3.449796ĽĽĽi -180 512
row 21 10 -512-886.810013ĽĽĽi -120 1024
row 22 11 1024.00000ĽĽĽ-1773.62002ĽĽĽi -60 2048
row 23 12 4096.00000ĽĽĽ+4.164846ĽĽĽi 5.82588E-13 4096

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter data in the cell range B9:B23 as shown in the table above.
@@@@@@@Enter =COMPLEX(1,SQRT(3)) in half-width in cell C9.
@@@@@@@Enter data in the cell C11 as shown in the table above.
@@@@@@@Enter =IMPOWER($C$9,B12) in half-width in cell C12.
@@@@@@@Right-click on the cell C12 and select Copy.
@@@@@@@Drag the cell range C13:C23 , right-click and select Paste.
@@@@@@@Enter =DEGREES(IMAGUMENT(C12)) in half-width in cell D12.
@@@@@@@Right-click on the cell D12 and select Copy.
@@@@@@@Drag the cell range D13:D23 , right-click and select Paste.
@@@@@@@Enter =IMABS(C12) in half-width in cell E12.
@@@@@@@Right-click on the cell E12 and select Copy.
@@@@@@@Drag the cell range E13:E23 , right-click and select Paste.

@@@ƒAbout functions„
@@@@@@@‚h‚l‚o‚n‚v‚d‚q functionĽĽĽ IMPOWER(complex number 1C‚Ž)
@@@@@@@@@@Returns (complex number 1)^n.

@y1007z@When the points represented by ƒż,ƒŔ, and ƒÁare A , B ,and C respectively , find the magnitude of Ú‚`‚a‚b.@@@@@@@
@ƒProcedure„
@@@@@@@@@@For points ‚`(ƒż)A‚a(ƒŔ)Aand ‚b(ƒÁ)
@@@@@@@@@@

@@@(1) Create the following correspondence table on a worksheet.

column B column C column D
row 11 A(ƒż) 2.73205ĽĽĽ-0.26794ĽĽĽi blank
row 12 B(ƒŔ) -i blank
row 13 C(ƒÁ) 1-2i blank
row 14 blank blank blank
row 15 ƒż|ƒŔ 1.73205ĽĽĽ+1.73205ĽĽĽi blank
row 16 ƒŔ|ƒÁ -1+i blank
row 17 blank blank blank
row 18 Ú‚`‚b‚a arg(ƒŔ|ƒÁ)/(ƒż|ƒÁ) 90

@@@ƒReference„
@@@@@sHow to compllete the worksheet abovet
@@@@@@@Enter data in the cell range B11:B18 as shown in the table above.
@@@@@@@Enter =COMPLEX(1+SQRT(3),-2+SQRT(3)) in half-width in cell C11.
@@@@@@@Enter =COMPLEX(0,-1) in half-width in cell C12.
@@@@@@@Enter =COMPLEX(1,-2) in half-width in cell C13.
@@@@@@@Enter =IMSUB(C11,C13) in half-width in cell C15.
@@@@@@@Enter =IMSUB(C12,C13) in half-width in cell C16.
@@@@@@@Enter data in the cell C18 as shown in the table above.
@@@@@@@Enter =DEGREES(IMARGUMENT(IMDIV(C16,C15))) in half-width in cell D18.

@@@ƒlŽ@„
@@@@@@@
@@@@@@@ƒŔ|ƒÁ|‚‰|(‚P|‚Q‚‰)|‚P{‚‰
@@@@@@@Calculate and find arg{(ƒŔ|ƒÁ)/(ƒż|ƒÁ)}.




@To table of contents@
@y1101z@When the integer part of ă5 is a and the decimal part of ă5 is b , find the value of (a^2-b^2).
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet above.

column B column C
row 11 integer part of ă‚T ‚ 2
row 12 decimal part of ă‚T ‚‚ 0.23606ĽĽĽ
row 13 blank blank
row 14 ‚^2|‚‚^2 3.94427ĽĽĽ
row 15 blank blank
row 16 ‚Să‚T|‚T 3.94427ĽĽĽ

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter data in the cell range B11:B16 as shown in the table above.
@@@@@@@Enter =TRUNC(SQRT(5)) in half-width in cell C11.
@@@@@@@Enter =SQRT(5)-C11 in half-width in cell C12.
@@@@@@@Enter =POWER(C11,2)-POWER(C12,2) in half-width in cell C14.
@@@@@@@Enter =4*SQRT(5)-5 in half-width in cell C16.

@@@ƒAbout functions„
@@@@@@@‚o‚n‚v‚d‚q functionĽĽĽ POWER(real number 1C‚Ž)
@@@@@@@@@@Returns (real number 1)^n

@@@@@@@‚s‚q‚t‚m‚b functionĽĽĽ TRUNC(real number 1)
@@@@@@@@@@Returns a real number 1 with the decimal part truncated.

@@@ƒConsideration„
@@@@@@@Since a=2 , b=ă5 - 2
@@@@@@@
@@@@@@Make sure that the value of a^2-b^2 on the worksheet matches the value of 4ă5 - 5.

@
@y1102z@Find the value of a^2+8a when the decimal part of ă23 is a.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

column B column C
row 11 integer part of ă23  4
row 12 decimal part of ă23 ‚ 0.79583ĽĽĽ
row 13 blank blank
row 14 ‚^2{‚W‚ 7

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter data in the cell range B11:B14 as shown in the table above.
@@@@@@@Enter =TRUNC(SQRT(23)) in half-width in cell C11.
@@@@@@@Enter =SQRT(23)-C11 in half-width in cell C12.
@@@@@@@Enter =POWER(C12,2)+8*C12 in half-width in cell C14.

@@@ƒAbout functions„
@@@@@@@‚o‚n‚v‚d‚q functionĽĽĽ POWER(real number 1Cn)
@@@@@@@@@@Returns (real nimber 1)^n

@@@@@@@‚s‚q‚t‚m‚b function ĽĽĽ TRUNC(real number 1)
@@@@@@@@@@Returns a real number 1 with the decimal part truncated.

@@@ƒConsideration„
@@@@@@@Since a=ă‚Q‚R@|@‚S
@@@@@@@
@@@@@@@Make sure that this matches the value of a^2+8a on the woksheet.

@

@To table of contens@
@y1201z@Regarding the tangent at point (t , t^3+t^2-2t) on the cubic function y=x^3+x^2-2x , observe the state of the tangent when t moves within the range of -2.8 to 2.1.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.
_ column A column B column C column E column F column G column H
row 1 x y=x^3+x^2-2x y=(3t^2+2t-2)x-2t^3-t^2 _ _ _ _
row 2 -3 -12 -32.249 contacti _ _ j
row 3 -2.9 -10.179 -31.474 _ _ _ _
row 4 -2.8 -8.512 -30.699 _ _ _ _
_ _ _ _ _ _ _ _
row 60 2.8 24.192 12.699 _ _ _ _
row 61 2.9 26.999 13.474 _ _ _ _
row 62 3.0 30 14.249 _ _ _ _
@@@@@However , _in the cells in the table above presents a blank.

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter data in the cell range A2:A62 in increments of 0.1 as shown in the table above.
@@@@@@@Enter A2^3+A2^2-2*A2 in half-widyh in cell B2.
@@@@@@@Copy cell B2 to the cell range B3:B62.
@@@@@@@Enter (3*$F$2^2+2*$F$2-2)*A2-2*$F$2^3-$F$2^2 in half-width in cell C2.
@@@@@@@Copy cell C2 to the cell range C3:C62.
@@@@@@@Enter F2^3+F2^2-2*F2 in half-width in cell G2.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range A1:C62 in the correspondence table of (1) above.
@@@@@@‡A Left-click mInsertn¨mScatter plotn¨mScatter linen
@@@@@@‡B Delete the legend

@@@(3) Arrange the xy coordinate plane on which the graph is drawn
@@@@@@‡@ Set the x-axis scale to -3 or more and 3 or less , and set the scale interval to 1.
@@@@@@‡A Set the y-axis scale to -10 or more and 10 or less , and set the scale interval to 1
@@@@@@‡B Set the scale line style to dotted lines for both the x-axis and y-axis.

@@@@@

@@@ƒAbout Macro„
@@@@@@@
@@@@@Sub Sessen()

@@@@@@@@Dim t As Single @@@@@ @'Declare it as single-precision floating point type

@@@@@@@@DoEvents @@@@@@@@@@ 'Allow you to force exit from ForNext repetition.

@@@@@@@@For t = -2.8 To 2.1 Step 0.01
@@@@@@@@@@@Cells(2, "F") = t @@@'Assign the the value of t to row 2 and column F.
@@@@@@@@@@@Calculate @@@@@@@ 'Recalculate
@@@@@@@@Next

@@@@@End Sub


@@@ƒConsideration„
@@@@@@@Let's run the macro and observe the movement of the tangent line.
@@@@@@@You can observe the state of the tangent line as the x-coordinate of the contact point changes from -2.8 to 2.1.
@
@y1202z@Observe that the graph of the quadratic function y=a(x-p)^2+q is obtained by translaing the graph of y=ax^2 by +p in the x-axis direction and +q in the y-axis direction.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

_ column B column C column D column F column G column H column I column J
row 2 x y=ax^2 y=a(x-p)^2+q a= -1 _ _ _
row 3 -3 -9 -8 p= 0 _ _ _
row 4 -2.9 -8.41 -7.21 q= 0 _ _ _
row 5 -2.8 -7.84 -6.44 _ _ _ x y
row 6 -2.7 -7.29 -5.69 _ _ origin 0 0
row 7 -2.6 -6.76 -4.96 _ _ perpendicular foot 1 0
row 8 -2.5 -6.25 -4.25 _ _ vertex 1 8
_ _ _ _ _ _ _ _ _
row 61 2.8 -7.84 4.76 _ _ _ _ _
row 62 2.9 -8.41 4.39 _ _ _ _ _
row 63 3.0 -9 4 _ _ _ _ _
@@@@@@However , "_" in the cells in the table above represents a blank.

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter data in the cell range B3:B63 in increments of 0.1 as shown in the table above.
@@@@@@@Enter $G$2*(B3-$G$3)^2+$G$4 in half-width in cell C3
@@@@@@@Copy the cell C3 to the cell range C4:C63.
@@@@@@@Enter $G$2*(B3-$I$8)^2+$J$8 in half-width in cell D3.
@@@@@@@Copy the cell D3 to the cell range D4:D63.
@@@@@@@Enter =I8 in half-width in cell I7.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range B2:D63 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mScatter linen
@@@@@@‡B Delete the legend.

@@@(3) Arrange the xy coordinate plane on which the grapf is drawn.
@@@@@@‡@ Set the x-axis scale to -3 or more and 3 or less , and set the scale interval to 1.
@@@@@@‡A Set the y-axis scale to -10 or more and 10 or less , and set the scale interval to 1.
@@@@@@‡B Set the scale line style to dotted lines for the x-axis and y-axis.

@@@@@

@@@ƒAbout macro„
@@@@@@@
@@@Option Explicit@@@@@'Force bariable declaration

@@@Sub apq()

@@@@@@@Dim a, p, q As Single@@@@@'Declare as simple-precision floating point type.
@@@@@@@Dim sp, sq As Single@@@@@'Declare as simple-precision floating point type

@@@@@@@sp = Cells(8, "I")@@@@@'Assign the value of row 8 and column I to sp.
@@@@@@@sq = Cells(8, "J")@@@@@'Assign the value of row 8 and column J to sq.

@@@@@@@Cells(3, "G") = 0@@@@@'Write 0 in row 3 and couimn G.
@@@@@@@Cells(4, "G") = 0@@@@@'Write 0 in row 4 and column G.

@@@@@@@DoEvents@@@@@'Allow forcible exit from ForNext statement.

@@@@@@@If sp > 0 And sq > 0 Then@@@@@'If the vertex is in the first quadrant

@@@@@@@@@@Cells(3, "G") = 0@@@@@'Write 0 in row 3 and column G.
@@@@@@@@@@Cells(4, "G") = 0@@@@@'Write 0 in row 4 and column G.

@@@@@@@@@@For p = 0 To sp Step 0.05
@@@@@@@@@@@@@Cells(3, "G") = p@@@@@'Write the value of p in row 3 and column G.
@@@@@@@@@@@@@Calculate@@@@@'Recaluculate
@@@@@@@@@@Next@p

@@@@@@@@@@For q = 0 To sq Step 0.05
@@@@@@@@@@@@@Cells(4, "G") = q@@@@@'Write the value of q in row 4 and column G.
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next@q

@@@@@@@End If

@@@@@@@If sp > 0 And sq < 0 Then@@@@@'If the vertex is in the 4th quadrant

@@@@@@@@@@Cells(3, "G") = 0@@@@@'Write 0 in row 3 and column G.
@@@@@@@@@@Cells(4, "G") = 0@@@@@'Write 0 in row 4 and column G

@@@@@@@@@@For p = 0 To sp Step 0.05
@@@@@@@@@@@@@Cells(3, "G") = p@@@@@'Write the value of p in row 3 and column G.
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next p

@@@@@@@@@@For q = 0 To sq Step -0.05
@@@@@@@@@@@@@Cells(4, "G") = q@@@@@'Write the value of q in row 4 and column G
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next q

@@@@@@@End If

@@@@@@@If sp < 0 And sq > 0 Then@@@@@'If the vertex is in the 2nd quadrant

@@@@@@@@@@Cells(3, "G") = 0@@@@@'Write 0 in row 3 and column G
@@@@@@@@@@Cells(4, "G") = 0@@@@@'Write 0 in row 4 and column G

@@@@@@@@@@For p = 0 To sp Step -0.05
@@@@@@@@@@@@@Cells(3, "G") = p@@@@@'Write the value of p in row 3 and column G
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next p

@@@@@@@@@@For q = 0 To sq Step 0.05
@@@@@@@@@@@@@Cells(4, "G") = q@@@@@'Write the value of q in row 4 and column G
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next q

@@@@@@@End If

@@@@@@@If sp < 0 And sq < 0 Then@@@@@''If the vertex is in the 3rd quadrant

@@@@@@@@@@Cells(3, "G") = 0@@@@@'Write 0 in row 3 and column G
@@@@@@@@@@Cells(4, "G") = 0@@@@@'Write 0 in row 4 and column G

@@@@@@@@@@For p = 0 To sp Step -0.05
@@@@@@@@@@@@@Cells(3, "G") = p@@@@@'Write the value of p in row 3 and column G
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next p

@@@@@@@@@@For q = 0 To sq Step -0.05
@@@@@@@@@@@@@Cells(4, "G") = q@@@@@'Write the value of q in row 4 and column G
@@@@@@@@@@@@@Calculate@@@@@'Recalculate
@@@@@@@@@@Next q

@@@@@@@End If

@@@End Sub


@@@ƒConsideration„
@@@@@@@Let's run the macro and observe that the quadratic function y=a(x-p)^2+q is obtained by translating y=ax^2 by +p in the x-axis direction and +q in the y-axis direction.
@



@To table of contents@
@y1301z Find prime numbers less than or equal to 20 using a macro.@@@@@@@@@@@

@@@ƒAbout prime number„

@@@@@@@A prime number is a natural number greater than or equal to 2 that has no positive divisors other than 1 and itself.
@@@@@@@The following macro determines whether a number is prime by dividing it one by one until 2 or more and 1 smaller than itself and checking whether it is divisible.

@@@ƒAbout macro„
@@@@@@@
@@@Sub Sosu2()

@@@@@@For n = 2 To 20
@@@@@@@@@d = 0 @@@@@'Initialize d
@@@@@@@@@For k = 2 To n - 1
@@@@@@@@@@@@If n Mod k = 0 Then @@@@@'If k is a divisor of n
@@@@@@@@@@@@@@@d = d + 1@@@@@ 'Add 1 to d.
@@@@@@@@@@@@End If
@@@@@@@@@Next k

@@@@@@@@@If d = 0 Then @@@@@'If d=0
@@@@@@@@@@@@MsgBox (n & "‚Í‘f”‚Ĺ‚ˇI") @@@@@'Display the message that d is a prime number.
@@@@@@@@@End If
@@@@@@Next n

@@@@@@MsgBox ((n - 1) & " ˆČ‰ş‚Ě‘f”‚Ě’Tő‚đI—š‚ľ‚Ü‚ľ‚˝B") @'Display the "End" message.

@@@End Sub


@@@ƒConsideration„
@@@@@@@Let's run the macro and find prime numbers less than or equal to 20.
@@@@@@@Next , let's rewrite the macro and find prime numbers less than or equal to 30.
@
@y1302z@Find perfect numbers using a method other than brute force.@
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

_ column A column B column C column D column E
row 1 n 2^n-1 prime numberH (2^n-1)Ľ2^(n-1) perfect numberH
row 2 _ _ _ _ _
row 3 _ _ _ _ _
_ _ _ _ _ _
row 29 _ _ _ _ _
row 30 _ _ _ _ _
row 31 _ _ _ _ _
@@@@@@ However , "_" in the cells in the table above represents a blank.

@@@@@@@Ś Follow the steps below so that column D does not display "ĽĽĽĽĽE-12".
@@@@@@@@@Select column D ¨ Right+click ¨ Format cells ¨ Display format ¨ Numerical value ¨ ‚n‚j


@@@ƒAbout perfect number„

@@@@@@@A perfect number is a natural number whose sum of positive divisors , excluding itself , is equal to itself. For example , the divisors of 6 , excluding itself , are 1,2,3 , so 1+2+3=6 , thefore , 6 is a perfect number.

@@@@@@When n is a natural number greater than or equal to 2 and is a prime number , is a perfect number. (This can be easily proven using the formula for the sum of geometric progressions.)

@@@@@@So , are all perfect numbers expressed as when is a prime number ?
@@@@@@It has been proven that all even perfect numbers are denoted by when is prime.
@@@@@@However , the odd perfect number has not yet been discovered. It has not been proven that the odd perfect numbers don't exist.

@@@@@@The following macro uses the fact that "When n is a natural number greater than or equal to 2 and is a prime number , is a perfect number" to find a perfect number.


@@@ƒAbout macro„

@@@Sub Kanzensusp()

@@@@@@Dim n As Long @@@@@'Declare as long integer type
@@@@@@Dim k As Long @@@@@'Declare as long integer type
@@@@@@Dim d As Long @@@@@'Declare as long integer type

@@@@@@'Cells.Clear
@@@@@@DoEvents @@@@@'Make it possible to stop even in the middle of a ForLoop

@@@@@@For n = 2 To 31

@@@@@@@@@Cells(n, "A") = n @@@'Assign the value of n to row n and column A
@@@@@@@@@Cells(n, "B") = 2 ^ n - 1@@@ 'Assign the value of 2^n-1 to row n and column B.
@@@@@@@@@d = 0@@@@@ 'nitialize d.

@@@@@@@@@For k = 2 To 2 ^ n - 1 - 1
@@@@@@@@@@@@If (2 ^ n - 1) Mod k = 0 Then @@@'If k is a divisor of 2^n-1 , then
@@@@@@@@@@@@@@@d = d + 1@@@ 'Add 1 to d
@@@@@@@@@@@@End If
@@@@@@@@@Next k

@@@@@@@@@If d = 0 Then @@@@@'If d=0 then , then
@@@@@@@@@@@@Cells(n, "C") = "Prime" @@@'Assign "Prime number " to row n and column C.
@@@@@@@@@End If

@@@@@@@@@Cells(n, "D") = (2 ^ (n - 1)) * (2 ^ n - 1) 'Assign the value of 2^(n-1)*(2^n-1) to row n and column D
@@@@@@@@@If Cells(n, "C") = "Prime" Then @@@@@'If row n and column C is "Prime number" , then
@@@@@@@@@@@@Cells(n, "E") = "Perfect" @@@@@'Assign "Perfect number" to row n and column E
@@@@@@@@@End If

@@@@@@Next n

@@@@@@MsgBox ("The search ofor perfect numbers has ended...") 'Display a completion message.

@@@End Sub


@@@ƒConsideration„
@@@@@@ Let's run the macro and find the perfect numbers.
@@@@@@ It will take some time to find 8 perfect numbers.
@



@To table of contents@
@y1401z@Draw a figure (positive leaf curve) repesented by the polar equetion ‚’‚“‚‰‚Ž‚ƒĆ.
@ƒProcedure„
@@@(1) Create the following correspondence table on a warksheet.

_ column A column B column C column D column E column F
row 1 t(degree) r=sinaƒĆ x=rcosƒĆ y=rsinƒĆ _ _
row 2 0 0 0 0 a= 2
row 3 1 0.172 0.172 0.003 _ _
row 4 2 0.340 0.340 0.011 _ _
row 5 3 0.497 0.497 0.026 _ _
_ _ _ _ _ _ _
row 360 358 -0.614 -0.613 0.021 _ _
row 361 359 -0.468 -0.468 0.008 _ _
row 362 360 -0.308 -0.308 7.56893E-17 _ _
@@@@@@However , "_" in the cells in the table above represents a blank..

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@ Enter data in the cell range A2:A362 in 1 complements as shown in the the table above.
@@@@@@ Enter SIN($F$2*RADIANS(A2)) in half-width in cell B2.
@@@@@@ Copy cell B2 to the cell range B3:B362.
@@@@@@ Enter B2*COS(RADIANS(A2)) in half-width i n cell C21.
@@@@@@ Copy cell C2 to cell range C3:D362.
@@@@@@ Enter B2*SIN(RADIANS(A2)) in half-width in cell D2.
@@@@@@ Copy cell D2 to the cell range D3:D362.
@@@@@@ Enter "a=" in cell E2.
@@@@@@ Enter "2" in cell F2.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range C2:D362 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mScatter plot (smooth line)n
@@@@@@‡B Delete the legend.

@@@(3) Arrange the xy coordinate plane on which the graph is drawn.
@@@@@@‡@ Set the scale interval of the x-axis to 0.1 between -1 and 1.
@@@@@@‡A Set the scale interval of the y-axis to 0.1 between -1 and 1.
@@@@@@‡B Set the scale line style to dotted lines for both the x-axis and y-axis.




@@@ƒAbout functions„
@@@@@@@‚q‚`‚c‚h‚`‚m‚r function ĽĽĽ ‚q‚`‚c‚h‚`‚m‚r(Angle in degrees)
@@@@@@@@@@Convert an angle in degrees to arc degrees.


@@@ƒAbout macro„

@@@@@@@Sub Seiyou()

@@@@@@@@@@Dim a As Single@@@'Declare a as a single-precision floating point type.

@@@@@@@@@@For a = 0 To 10.05 Step 0.05
@@@@@@@@@@@@@Cells(2, "F") = a@@@'Assign the value of a to row 2 and column F.
@@@@@@@@@@@@@Calculate@@@'Recalculate
@@@@@@@@@@Next a

@@@@@@@End Sub


@@@ƒConsideration„
@@@@@@@Let's sequentially substitute 2 , 3 , 4 , ĽĽĽ,10 into cell F2 (value of a) and observe the drawn figure (positive leaf curve).
@@@@@@@Also , let's run the macro and observe the figures drawn as the value of a changes from 0 to 10 in 0.05 incements .

@
@y1402z@Draw the figure (Archimedes spiral line) repesented by the polar equation r=aƒĆ.
@ƒProcedure„
@@@(1) Create the following correspondence table on a worksheet.

_ column A column B column C column D column E column F
row 1 t(“x) r=aƒĆ x=rcosƒĆ y=rsinƒĆ _ _
row 2 0 0 0 0 a= 2
row 3 1 0.0349 0.0349 0.0006 _ _
row 4 2 0.0698 0.0697 0.0024 _ _
row 5 3 0.1047 0.1045 0.0054 _ _
_ _ _ _ _ _ _
row 720 718 25.0629 25.0476 -0.8746 _ _
row 721 719 25.0978 25.0940 -0.4380 _ _
row 722 720 25.1327 25.1327 -1.23165E-14 _ _
@@@@@@However , "_" in cells in the table above represents a blank.

@@@ƒReference„
@@@@@sHow to complete the worksheet abovet
@@@@@@@Enter data in the cell range A2:A722 in increments of 1 as shown in the table above.
@@@@@@@Enter $F$2*RADIANS(A2) in half-width in cell B2.
@@@@@@@Copy cell B2 to the cell range B3:B722.
@@@@@@ Enter B2*COS(RADIANS(A2)) in half-width in cell C2.
@@@@@@@Copy cell C2 to the cell range C3:C722.
@@@@@@ Enter B2*SIN(RADIANS(A2)) in half-width in cell D2.
@@@@@@@Copy cell D2 to the cell range D3:D722.
@@@@@@@Enter "a=" in cell E2.
@@@@@@@Enter "2" in cell F2.

@@@(2) Draw a graph from the correspondence table in (1) above.
@@@@@@‡@ Drag and select the cell range C2:D722 in the correspondence table of (1) above.
@@@@@@‡A Left-click mIncertn¨mScatter plotn¨mScatter plotiSmooth linejn
@@@@@@‡B Delete legend.

@@@(3) Arrange the xy-coordinate plane on which the graph is drawn.
@@@@@@‡@ Set the x-axis scale to -150 or more and 150 or less , and set the scale interval to 30.
@@@@@@‡A Set the y-axis scale to -150 or more and 150 or less , and set the scale interval to 30.
@@@@@@‡B Set the scale line style to the dotted lines for both the x-axis and y-axis.



@@@ƒAbout functions„
@@@@@@@‚q‚`‚c‚h‚`‚m‚r functionĽĽĽ ‚q‚`‚c‚h‚`‚m‚r(Angle in degrees)
@@@@@@@@@@Convert angles in degrees to arc degrees.


@@@ƒAbout macro„

@@@@@@@Sub Aruki()

@@@@@@@@@@Dim a As Single@@@'Declare a as a singe-precision floating point type

@@@@@@@@@@For a = 0 To 10.05 Step 0.05
@@@@@@@@@@@@@Cells(2, "F") = a@@@'Assign the value of a to row 2 and column F.
@@@@@@@@@@@@@Calculate@@@'Recalculate
@@@@@@@@@@Next a

@@@@@@@End Sub


@@@ƒConsideration„
@@@@@@@Let's sequentially assign 2 , 3 , 4 , ĽĽĽ, 10 to cell F2 (value of a) and obseve the drawn figure (Archmedean spiral line).
@@@@@@@Also , let's run the macro and observe the figures drawn as the value of a changes from 0 to 10 in 0.05 increments.

@


Ś@You can down load the above sample data files "`.xls" below.


@To table contents@
@
@When you downloaded the data below for how to use Excel , compressed file file "sample_excel.lzh" will be downloaded.
@When the downloaded compressed file "sample_excel.lzh" is decompressed , a folder "Sample data for thinking about how to use "Excel" is created.

@When you download the sample data for "simulation" , "Prime and Perfect Numbers" , and "Polar Equation"below , copmressed file "ExcelSample2.lzh" will be downloaded.
@When the downloaded compressed file "ExcelSample2.lzh" is downloaded , the sample data "Simulation" , "Prime and Perfect Number" , and "Polor Equation" are created.

@‚d‚˜‚ƒ‚…‚Œ sample data @Sample data for thinking about how to use "Excel"
@‚d‚˜‚ƒ‚…‚Œ sample data 2@ @"Simulation" , "Prime and Perfect Number" , "Polar Equation"sample data



@To table of contents@
@There are 11 folders "[0100]Quadratic function" , "[0200]Trigonometric function" , "[0300]Exponential function" , "[0400]Logarithmic function" , "[0500]Various curves" , "[0600]Differentiation(Part 1)" , "[0700]Differentiation (Part 2)" , "[0800]Integral" , "[0900]Number sequence" , "[1000]Complex number" , "[1100]Application of square roots" in the folder "Sample data for thinking about how to use Excel". Open the files "`.xlsm" in each folder in Excel.
@There are six files in the folder "Simulation " , "Prime and Perfect Numbers" , "Polor Equations" sample data. Open each file "`.xlsm" in Excel.


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yReferencesz
Easy math to learn with Excel (from trigonometric functions to differential and integral calculus) Yukihisa Takahashi / Yaichi Watanabe Ohmsha
Handmade math simulation using Excel - Freely manipulate graph functions and VBA programs. Written by Haruhiko Tanuma Bluebacks
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