Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
Task 1: Graphs of Trigonometric Functions
To create a graph of y = a sin x with adjustable coefficient a.
1.
In the input bar enter:
sin(x°)
Enter “°” by clicking
in the input bar, or press “Alt + o”.
2.
Right click on the empty space of the Graphics view. Choose “Graphics” and set values in the
tabs according to the following figures.
3.
Drag the curve to see that it is movable (and its equation changes accordingly). To fix it, press
Ctrl+Z to undo the action, then right-click the curve and choose “Properties”. In the “Basic” tab
check the “Fix Object” box.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
4.
To confine the curve in the range 0° ≤ x ≤ 360°,
double-click the curve and redefine it as:
Function[sin(x°), 0, 360]
5.
Enter in the input bar the command
a = 1
In the Algebra view, check the button of a to show the slider controlling
its value. Drag the slider to the right hand side of the Graphics view.
6.
Double-click the slider. Adjust the range of
values of a as shown in the figure.
7.
Double-click the curve and redefine it as:
Function[a sin(x°), 0, 360]
8.
Use the “Insert Text”
tool. Click on an appropriate position.
In the Text window type the content as shown in the figure. The
boxed variable a is obtained by choosing “a” in the “Objects”.
9.
To obtain square grid, enter the command
SetAxesRatio[30,0.25]
Use the slider to change the value of a and see how the
graph changes accordingly.
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Buttons to
show/hide objects
Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
Task 2 Locus of a Point at a Fixed Distance from a Line
To trace the locus of a point that is moving at a fixed distance from a line.
1.
Use the “Line through Two Points” tool
to draw a line (named a) passing through two
arbitrary points A and B. Hide the two points A and B. Fixed the line and hide the axes.
2.
Use the point tool to create a movable point in the plane and rename it as P.
3.
Create the foot of perpendicular C from P to the line by entering
Intersect[a,PerpendicularLine[P,a]]
4.
Use the segment tool “Segment” tool
in the Line toolbox
to draw a dashed-line joining P and C.
5.
Label the dashed-line by its value. Use the point tool
to create a movable point D on the line.
Use the angle tool
to measure the right angle∠DCP.
Right click it. Hide its label and set its value between 0° and
180°. Hide D.
6.
Use the “Slider” tool
to create a slider named Distance varies
from 1 to 5 with increment 0.5. Set the value of Distance as 2.
7.
Create the point Q by entering
Q = C + Distance UnitVector[Vector[C, P]]
Change its colour to red and hide its label.
8.
Choose in the Menu bar “Options | Rounding | 1 Decimal Place”.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
9.
Right click Q and choose “Object Properties”.
In the “Advanced” tab enter
Distance[Q, P] < 0.05
in the field “Condition to Show Object”.
10. Right click P and choose “Object Properties”.
In the “Advanced” tab enter
If[Distance[Q, P] < 0.05, 1, 0]
0
0
in the fields “Red”, “Green” and “Blue” of
“Dynamic Colours” respectively.
11. In the Algebra View, right click Q and choose “Trace On” to show the trace of P (i.e. Q) when
its distance from the line is 2 units.
12. Right click the slider of Distance and choose
“Object Properties”. In the “Scripting” tab enter
ZoomIn[1]
in the “On Update” tab so that the trace of P
would be erased when Distance is changed.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
Task 3 Tangent and Derivative of a Function
To investigate the value of the derivative at turning points and at points of inflection. .
1.
Input the polynomial
x^3 – 6x^2 + 9x + 1
Label the curve as “y = x3 – 6x2 + 9x + 1”.
2.
Use the “New Point” tool
, click on the curve to create a movable point A on the curve.
3.
Use the “Tangents” tool
in the Special Lines toolbox
, select A and then the curve to
construct the tangent to the curve at A.
4.
Use the “Slope” tool
in the Measurements toolbox
, select the tangent to show its
slope (named a). Label it by its value.
5.
Drag A to see how the slope varies, especially at the turning points and the point of inflection.
6.
Enter
(x(A), b)
to create the point on y = f’(x) (b is the name of the slope). Label this point B by its value.
Change its colour to red.
7.
Use the “Check Box”
tool in the Action Object toolbox
. Click on a position. Enter
“Trace Derivative” in the “Caption” field. Click B and click “Apply” afterwards.
8.
Right click the check box and choose “Object Properties”. In
the “Scripting” tab enter: ZoomIn[1] in the “On Update” tab.
9.
Right-click the red point and choose “Trace On”.
10. Drag A to trace the derivative function. Discuss with students
how to use the derivative to find the turning points and the
point of inflection.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
Task 4 Riemann Sums, Trapezoidal Sum and Definite Integral
To demonstrate how the Riemann sums and the trapezoidal sum tend to the definite integral.
1.
Input the following:
4x / (x^2 + 1)
n = 8
Show the slider for n. Change its “Properties” as shown
in the figure. Label it by its name and value.
2.
Input the Lower and Upper Riemann Sums, the Trapezoidal Sum and the definite integral in the
interval [0, 4] by the following commands.
LowerSum[f(x), 0, 4, n]
UpperSum[f(x), 0, 4, n]
TrapezoidalSum[f(x), 0, 4, n]
Integral[f(x), 0, 4]
3.
Change the colour of the LowerSum (a) to blue; the
UpperSum (b) to green, the Trapezoidal Sum (c) to magenta
and the definite integral (d) to red.
Label a, b and c by their values.
4.
Put the Lower Sum to layer 3, the definite integral to layer 2,
the Trapezoidal Sum to layer 1 in the “Advanced” tab.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
5.
Use the “Insert Text”
tool. Check the “LaTex formula” box. Select
∫
b
a
from the “Sums
and Integrals” list (Fig. 1).
Replace “a” and “b” by “0” and “4”.
a
In the third braces select “ ” from the “Roots and Fractions” list (Fig. 2). Replace “a” and “b”
b
by “4x” and “x^2+1”, and outside the “}” type “dx = ”, then choose “d” from the “Objects” list
(Fig. 3). Click OK afterward.
6.
Use the “Check Box to Show / Hide”
tool in the
toolbox. Click a position in the Geometry view. In the “Caption”
field enter “Lower Riemann Sum” and select “Number a:
LowerSum[f(x), 0, 4, n]” from the list. Click “Apply” afterwards.
Similarly, create the checkboxes for “Upper Riemann Sum”, the
“Trapezoidal Rule” and “Definite Integral”.
7.
Click the check boxes to show or hide the different sums and the definite integral. Drag n to see
how the sums approach the definite integral.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
Task 5 Matrix and Linear Transformation
To visualize how a matrix represents a linear transformation.
1.
Create three points A, B and O by entering:
A = (1,0)
B = (0,1)
O = (0,0)
Fix O. Change the colours of A and B to red and blue respectively. Hide their labels.
2.
Create vectors u and v by entering:
u = Vector[O, A]
v = Vector[O, B]
Change their thickness to Point 5. Change the colours of u and v to red and blue respectively.
Label them by S(i) and S(j) respectively.
3.
Set the distance of ticks of the axes be 1 and hide the numbers of the axes. Set the distance of
grid be 1.
4.
Use the “Insert Image”
tool in the Special Object toolbox
given file “obama.png” to insert the photo of Barack Obama.
8
. Click on O, and select the
Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
5.
Right-click the image and choose “Object Properties”. In the
“Position” tab, enter the positions of Corner 2 and Corner 4 of
the photo as shown in the figure.
Hide O. Drag the red point and blue point to see how Obama’s
photo changes accordingly.
6.
Enter the transformation matrix M by entering in the input field:
M = {{x(A), x(B)}, {y(A), y(B)}}
The matrix M appears in the Algebra view.
7.
Choose “View | Graphics 2
axes in this view.
Ctrl+Shift+2” to activate the second graphics view. Hide the
8.
In Graphics 2 view, use the “Insert Text”
tool and check the “LaTex Formula” box. Enter
the texts as shown. Note that the objects in the boxes are the dynamic objects u, v and M, and
have to be chosen from the “Objects” list in the Text window.
u
v
u
v
M
9.
Drag the red point and blue point to change the vectors S(i) and S(j) and see how the matrix and
the photo changes accordingly. Explain to students how the linear transformation is represented
by a matrix.
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Use of GeoGebra in Trigonometry, Locus, Calculus and Linear Algebra (Advanced Level)
Task 6 Basic CAS
Turn on the CAS view. Input the following commands:
1.
sqrt(12)+sqrt(80)-sqrt(75)
2.
3x^2+6x-4-(4x+2-x^2)+(x-1)*(x-9)
3.
Solve[x^2-5x-8=0]
4.
Solve[x^2-5x+6>=0]
5.
Solve[(5y+x)/(1+x)=3,y]
6.
Solve[(5y+x)/(1+x)=3,x]
7.
Solve[{3x-4y=10,5x-2y=8}]
8.
Solve[{x - a y + z = 2, 2x + (1-2a)y + (2-b)z = a + 4, 3x + (1-3a)y
+ (3-a b)z = 4}]
9.
Solve[{x^2+y^2-2x+y-8=0,x+y=3}]
10. Expand[(3x-4y)^2]
11. Factor[2014]
12. Factor[x^6-1]
13. Derivative[x^2/sqrt(x+1)]
14. Integral[(3x+4)/sqrt(2x-1)]
15. Simplify[$]
($ means the previous output, $n means the output of row n)
16. {{2,5},{1,2}}
⎛ 2 5⎞
⎟⎟ .)
(Input the matrix ⎜⎜
⎝ 1 2⎠
17. {{2,5},{1,2}}^2
(You can click the previous output to get the list of the matrix.)
18. {{1, -a, 1},{2, 1 - 2a, 2 - b}, {3, 1 - 3a, -a b + 3}}
19. Determinant[{{1, -a, 1},{2, 1 - 2a, 2 - b}, {3, 1 - 3a, -a b + 3}}]
20. {{1, -a, 1},{2, 1 - 2a, 2 - b},{3, 1 - 3a, -a b + 3}}^-1 {{2},{a+4},{4}}
(Compare the result with that of No.8)
Reference: http://wiki.geogebra.org/en/CAS_View
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