# MEI Conference Using GeoGebra in A level Core ```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)
2. Use New Point (2nd 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 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.

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.
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MEI GeoGebra Tasks for AS Core
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 x&sup2; – 2x – 8 = 0 by both factorising and completing the square.
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?
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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?
Problem
Can you find functions that have the following gradient functions:
 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
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.
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)=x&sup3; + 3x&sup2; + 4x +1
www.mei.org.uk/geogebra
MEI GeoGebra Tasks for AS Core
Functions: Transformations
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 – 8x&sup3; + 24x&sup2; – 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).
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)=x&sup3;–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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