# MEI Conference Active learning with GeoGebra

```MEI Conference 2015
Active learning with
GeoGebra
Tom Button
[email protected]
www.mei.org.uk/geogebra
Differentiation 1 – Exploring the gradient on a curve
1. In the input bar enter a cubic function: e.g. f(x)=x^3-2x^2-2x+2
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 gradient 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 for discussion

How is the gradient of the tangent (as the point moves) related to the shape of the
Problem
Change your function in GeoGebra so that is has the following gradient functions:
Find the point on the function f( x)  x3  6 x 2  9 x  1 where the tangent has its maximum
downwards slope. Investigate the point with maximum downward slope for other cubic
functions.
1
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Differentiation 2 – Stationary points
1. In the input bar enter a cubic function: e.g. f(x)=x^3-2x^2-2x+1
2. Find the turning points of the function: in the input bar enter TurningPoint[f]
3. 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 for discussion

How can you use the graph of the gradient function to explain why the function has a
local maximum at A and a local minimum at B?
Problem (Try this on paper first then check your answer on GeoGebra)
Find the values of x for which the following functions have turning points and determine
whether they are maxima or minima:

f( x)  x3  3 x 2  9 x  3

g( x)  x 4  4 x3  36 x 2  8
Use GeoGebra to find the gradient function of f( x)  x3  6 x 2  12 x  5 . Explain why the
function has a stationary point that is neither a maximum nor a minimum (a stationary point
of inflection).
Find some other functions that have stationary points of inflection.
2
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Integration: Area under a curve
It is essential that this is
entered as a function f(x).
1. In the Input bar enter: f(x)=x^2
to create a slider for a.
3. In the Input bar enter: A=Integral[f, 0, a]
Create the slider with
minimum value 0.
Question for discussion

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 (Try this on paper first then check your answer on GeoGebra)
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
3
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Functions: Transformations
to create sliders for a and b.
It is essential that this is
entered as a function f(x).
2. In the Input bar enter: f(x)=x^2
3. In the Input bar enter: g(x)=f(x+a)+b
Question for discussion

What transformation maps f(x) onto g(x)?

Does this work if other functions are entered for f(x)?
Problem (Try this on paper first then check your answer on GeoGebra)
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(d*x).
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)
4
Changing f(x) to
f(x)=x&sup3;–x might help
make it clearer.
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AS Mathematics Construction Problems
Example
0.
Create two points A and B on the x axis.
Construct a quadratic graph that passes
through A and B.
A possible solution (there are other ways to do this)
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)
Problems
1.
2.
Create points A, B and C fixed to the
x-axis and D fixed to the yaxis. Construct a cubic that passes
through A, B, C and D.
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.
3.
4.
Plot the curve f(x)=x&sup2; and add a point
A to the curve. Construct the point B
such that the tangents at A and B are
perpendicular.
Create two points A and B. Construct
a cubic that has stationary points at A
and B.
5
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```