MEI Conference 2015
Teaching Functions
with Graphing
Technology
Rob Butler
rob.butler@mei.org.uk
Investigating range and domain
Go to the graph plotting mode (Menu > 5)
𝐴𝑥+𝐵
Enter 𝐶𝑥+𝐷 in the top line and press EXE:
Press “Modify” (F5) and set the initial values as below:
What is the range and domain of this function?
Is it possible to change the values of A, B, C and D so that the domain of the function is all the real
numbers?
Is it possible to change the values of A, B, C and D so that the range of the function is all the real
numbers?
How would you have to change the function so that either the range or domain has more than one
value missing?
Starting with A, B, C and D = 1. Try the following and investigate how changing coefficients affect the
range and domain. (you may need a step size of 0.1)
1. 𝑓(𝑥) =
𝐴𝑥+𝐵
√𝐶𝑥+𝐷
2. 𝑓(𝑥) = 𝐴 sin 𝑥 + 𝐵 cos 𝑥
3. 𝑓(𝑥) = 𝐴 𝑥 2 + 𝐵 𝑥 + 𝐶
4. 𝑓(𝑥) = 𝐴 |𝐵𝑥 + 𝐶| + 𝐷
Multiple Transformations of graphs
1. 𝑓(𝑥) = √𝑥 to 𝑔(𝑥) = √2𝑥 + 1
Using the calculators go into the graph plotting mode (Menu > 5)
Enter √𝐴𝑥 + 𝐵 in the top line and press EXE:
Press “Modify” (F5) and set the initial value for A to be 1 and B to be 0:
Change A to 2 and B to 1 (in that order) and describe the transformations that take place after each
step.
After returning the graph to its original form; change B to 1 and A to 2 (in that order) and describe
the transformations that take place after each step. Which do you prefer?
In each case find the sequence of transformations which map the function 𝑓(𝑥) to the function 𝑔(𝑥)
2.
3.
4.
5.
𝑓(𝑥) = √𝑥 to 𝑔(𝑥) = 2√𝑥 + 1
𝑓(𝑥) = 𝑥 2 to 𝑔(𝑥) = −2𝑥 2 + 3
𝑓(𝑥) = 𝑥 3 to 𝑔(𝑥) = 8𝑥 3 − 5
𝑓(𝑥) = |𝑥| to 𝑔(𝑥) = |2(𝑥 + 3)|
6. The transformations R, S and T are given by:
R: reflection in the 𝑥 axis
S: Stretch in the 𝑥 direction with scale factor 2
T: Translation in the positive 𝑥 direction by 3 units
i)The curve 𝑦 = ln 𝑥 is transformed by R followed by T. Find the equation of the resulting curve.
1
3
ii) Find in terms of S and T, a sequence of transformations that transforms 𝑦 = 𝑥 3 to 𝑦 = (4 𝑥 − 3)
The inverse of a function
Using the algebra view in GeoGebra, type “function[x^3-4x,a,b]” into the input bar.
Create sliders when prompted.
You should be able to adjust the two sliders to give limits to
the domain of the function
Type “y = x” into the input bar. Then click on the reflection
tool
then the curve and then the line y = x to reflect
the function in the line.
You can change the style of any of the lines by clicking on
them with the pointer and then selecting options from the
style bar.
For which domains is the reflection (i.e. the inverse) not a function?
How could you find the limits of those domains algebraically?
What is significant about the point where the inverse meets the original graph?
Try this for other functions (it’s possible to change the function by double clicking on the equation of
the function in the algebra window).
Composite and inverse functions
All of these graphs are a composition two of the following functions and their inverses. Find the
equation for each of the functions.
𝑓(𝑥) = √𝑥,
𝑔(𝑥) = 𝑒 𝑥 ,
𝑝(𝑥) = |𝑥|,
ℎ(𝑥) = sin 𝑥 ,
1)
2)
3)
𝑞(𝑥) = 𝑥 cos(𝑥)
All of these graphs are a composition of the following functions and their inverses. Find the
equation for each of the graphs.
𝑓(𝑥) = √𝑥,
𝑔(𝑥) = 𝑒 𝑥 ,
ℎ(𝑥) = sin 𝑥 ,
4)
5)
6)
7)
𝑝(𝑥) = |𝑥|,
𝑞(𝑥) = 𝑥 cos 𝑥