MEI Conference Using GeoGebra in A level Core

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MEI Conference 2014
Using GeoGebra in
A level Core
Tom Button
[email protected]
MEI GeoGebra Tasks for AS Core
Coordinate Geometry: Perpendicular lines
1. Use New Point (2nd menu)
to add a point, A.
2. Use New Point (2nd menu)
to add a point, B.
3. Use Line (3rd menu)
to create the line through A and B.
4. Use New Point (2nd menu)
5. Use Perpendicular Line (4th menu)
perpendicular to the line AB.
Click on the point C
and then the line.
to add a point, C.
to create the line through C and
You can display
the gridlines by
clicking the
gridlines icon in
Graphics style bar.
Questions

What is the relationship between the equations of the lines?

What is the relationship between the equations of the lines then they are written in
the form y = mx + c?
Problem
Show that the line perpendicular to the line through (5,1) and (1,3) that passes through the
point (3,4) has equation y = 2x – 2.
Further Tasks

For two points A and B what are the possible positions for C so that the line through
C is a perpendicular bisector?

For three points A, B and C find the point of intersection of the two lines.
www.mei.org.uk/geogebra
MEI GeoGebra Tasks for AS Core
Algebra: Quadratic equations
1. Use New Point (2nd menu)
2. In the input bar enter:
a=x(A)
b=x(B)
to add two new points on the x-axis, A and B.
Enter these separately and
press enter after each one.
3. In the input bar enter: y=(x-a)(x-b)
4. Use New Point (2nd menu)
to add a new point (not on either axis), C.
5. In the input bar enter:
p=x(C)
q=y(C)
Enter these separately and
press enter after each one.
6. In the input bar enter: y=(x-p)^2+q
You can display
the gridlines by
clicking the
gridlines icon in
Graphics style bar.
Questions

Can you find positions for A, B and C so that the two graphs are the same?

What is the relationship between the values of a, b, p and q when the graphs are the
same?
Problem
Solve the equation x² – 2x – 8 = 0 by both factorising and completing the square.
Further Tasks
Add a Slider (11th menu)
and set its name to k.
Change the equation in step 7 to y=k(x-p)^2+q


Where does this curve cross the x-axis?
Can you change the equation in step 4 so the curves are the same?
www.mei.org.uk/geogebra
MEI GeoGebra Tasks for AS Core
Differentiation: Exploring the gradient on a curve
1. In the input bar enter a cubic function: e.g. f(x)=x^3-2x^2-2x+1
2. Use New Point (2nd menu) to add a point on the curve.
3. Use Tangent (4th menu) to create a tangent to the curve at point A.
4. Use Slope (8th menu) to measure the slope of the tangent.
5. Plot the gradient function by entering g(x)=f '(x) in the input bar.
You might find it easier to see if you change the gradient function to a red dotted line using
the Graphics Styling bar.
Question

How is the gradient of the slope (as the point moves) related to shape of the gradient
graph?
Verify your comments by trying some other functions for f(x).
Problem
Can you find functions that have the following gradient functions:
Further Tasks
 Describe the gradient graph for cubics that have 0, 1 and 2 stationary points.
 Investigate the minimum (or maximum) point on the gradient graph for a cubic.
www.mei.org.uk/geogebra
MEI GeoGebra Tasks for AS Core
Integration: Area under a curve
1. In the Input bar enter: f(x)=x^2
2. Use Slider (11th menu)
It is essential that this is
entered as a function f(x).
to create a slider for a.
3. In the Input bar enter: A=Integral[f, 0, a]
Create the slider with
minimum value 0.
Questions

What is the relationship between the area and the value of a?

What is the relationship if f(x) is changed to a different power of x?
Problem
Find the area under f(x) = x5 between x = 0 and x = 3.
Further Tasks
Add a Slider (11th menu)
for b.

Investigate the area under f(x)= xn between x = a and x = b.

Investigate the areas under functions that are the sums of powers of x:
e.g. f(x)=x³ + 3x² + 4x +1
www.mei.org.uk/geogebra
MEI GeoGebra Tasks for AS Core
Functions: Transformations
1. Use Slider (11th menu)
to create sliders for a and b.
2. In the Input bar enter: f(x)=x^2
It is essential that this is
entered as a function f(x).
3. In the Input bar enter: g(x)=f(x+a)+b
Questions

What transformation maps f(x) onto g(x)?

Does this work if other functions are entered for f(x)?
Problem
Show that f(x) = x4 – 8x³ + 24x² – 32x +13 can be written in the form (x+a)4 + b and hence
find the coordinates of the minimum point on the graph of y = f(x).
Further Tasks
Use Slider (11th menu)
to create sliders for c and d.

In the Input bar enter: h(x)=c*f(x*d).
What transformation maps f(x) onto h(x)?

Investigate g(x) and h(x) for f(x)=log10x.
NB this is entered as: f(x)=log10(x)
Changing f(x) to
f(x)=x³–x might help
make it clearer.
www.mei.org.uk/geogebra
MEI GeoGebra Tasks for AS Core
Constructing objects in GeoGebra
Testing students’ understanding of ideas and reinforcing generalisation
Example
0. Create a two points A and B on the x axis.
Construct a quadratic graph that passes
through A and B.
Use the New Point button to add
points A and B fixed to the x-axis.
In the Input bar define two variables:
a = x(A) and b = x(B).
Define a new curve on the Input bar:
y = (x–a)(x–b)
Ideas for AS Core Mathematics
1. Create two points A and B. Construct a third point C which lies on the line
perpendicular to AB passing through A and is twice as far away from A as B is.
2. Create points A, B and C fixed to the x-axis and D fixed to the y-axis. Construct a
cubic that passes through A, B, C and D.
3. Create a triangle with one point on the origin and one point on the x-axis.
Construct circles centred on each vertex such that all three circles touch each
other.
4. Create a graph of a quadratic equation that can be moved by dragging the vertex.
a. Construct the tangent to the curve with gradient 2 (that works for the vertex
in any position).
b. Construct the tangent to the curve with gradient b (that works for the vertex
in any position).
5. Draw the graph of a straight line through the origin (NB this must be defined as a
function, e.g. f(x)= x or f(x)=2x). Add a point A on the positive x-axis.
a. Construct a point B such that the integral of f(x) between A and B is 8.
b
The GeoGebra function for
 f ( x)dx
is: Integral[f, a, b]
a
b. Construct a point B such that the integral of f(x) between A and B is d.
c. Construct a point B such that the integral of f(x)=mx between A and B is d
for any value of m or d.
6. Construct a triangle with sides a and b and angle A that demonstrates the
ambiguous case of the Sine rule.
7. Draw the graph of y = ax and add the point A on the curve. Construct a point B
based on A that you can use with Trace function to obtain the shape of y = logax.
8. (A challenge!)
Create two points A and B. Construct a cubic that has stationary points at A and
B. (Hint – the midpoint of A and B may help).
www.mei.org.uk/geogebra
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