# Graph Transformations: Discovering Manipulating Functions

- Author:
- Adam Kelly

- Topic:
- Functions

Graph Transformations: Discovering How to Manipulate Functions
Parent function: a basic function used as the building block for more complicated functions. Some examples of parent functions are:
Quadratics (parabolas): f(x)=x^2 (typed x^2)
Cubics: f(x)=x^3 (typed x^3)
Absolute value: f(x)=|x| (typed abs(x))
Square root: f(x)=√x (typed sqrt(x))
Use the Geogebra worksheet to explore the four functions listed above. Type the parent function you wish to investigate into the input box. Click each of the buttons (yellow, green, blue, and red) one at a time and move the slider left and right to see what different values of the variable do to transform the function. Make sure to check positive and negative integers and rational numbers (decimals).
Each colored box will display the transformation function on the graph. The parent function will be visible in blue dotted font and the transformation function will be visible in red font.
Complete the exercises below, making conjectures about how the values of each of the variables (a, h, k, and z) transform the parent function.

Graph Transformations
Part 1
Type x^2 into the input box and press enter.
Click the yellow button to explore the graph of g(x)=a*f(x).
Move the slider to change the value of a, or enter values into the input box.
Make a conjecture about how the value of a transforms the graph of f(x) to the graph of g(x).
Repeat the same work with the following functions and test your conjecture.
x3 (type x^3)
|x| (type abs(x))
√x (sqrt(x))
Does your conjecture hold? If not change it.
Does your conjecture hold if 0<a<1? If not, modify it.
Does your conjecture still hold if a < 0? If not, modify it.
Does your conjecture still hold if a = 0? If not, modify it.
Part 2.
Type x^2 into the input box and press enter.
Click the green button to explore the graph of g(x)=f(x-h).
Move the slider to change the value of h.
Your task consists of making a conjecture about how the value of h transforms the parent function.
Observe the transformations of the graph with the changes of the value h.
Do the same using the following functions: x3 (type x^3) ; |x| (type abs(x)); and √x (sqrt(x))
Based upon your observations, make a conjecture on how the value of “h” affects the graph of the equation g(x)=f(x-h).
Does your conjecture hold for h>0? If not, modify your conjecture.
Does your conjecture hold for h<0? If not, modify your conjecture.
Part 3.
Type x^2 into the input box and press enter.
Click the blue button to explore the graph of g(x)=f(x)+k.
Move the slider to change the value of k.
Your task consists of making a conjecture about how the value of k transforms the parent function.
Observe the transformations of the graph with the changes of the value k.
Do the same using the following functions: x3 (type x^3) ; |x| (type abs(x)); and √x (sqrt(x))
Based upon your observations, make a conjecture on how the value of “k” affects the graph of the function g(x)=f(x)+k.
Does your conjecture hold for k>0? If not, modify your conjecture.
Does your conjecture hold for k<0? If not, modify your conjecture.
Part 4
Type x^2 into the input box and press enter.
Click the blue button to explore the graph of g(x)=f(z*x).
Move the slider to change the value of z.
Your task consists of making a conjecture about how the value of z transforms the parent function.
Observe the transformations of the graph with the changes of the value z.
Do the same using the following functions: x3 (type x^3) ; |x| (type abs(x)); and √x (sqrt(x))
Based upon your observations, make a conjecture on how the value of “z” affects the graph of the function g(x)=f(z*x).
Does your conjecture hold if 0<z<1? If not, modify it.
Does your conjecture still hold if z < 0? If not, modify it.
Does your conjecture still hold if z = 0? If not, modify it.