1. Transformation of Graphs
Lesson 3.4

2. Tools for Exploration Consider the function f(x) = 0.1(x3 – 9x2)
Enter this function into your calculator on the y= screen
Set the window to be  -10 < x < 10   and   -20 < y < 20
Graph the function

3. Use different styles for each of the functions
Shifting the Graph
Enter the following function calls of our original function on the y= screen:
y1=  0.1 (x3 - 9x2) 
y2=   y1(x + 2) 
y3=   y1(x) + 2
Before you graph the other two lines, predict what you think will be the result.
Use different styles for each of the functions

4. Shifting the Graph How close were your predictions?
Try these functions – again, predict results
y1=  0.1 (x3 - 9x2) 
y2=   y1(x - 2) 
y3=   y1(x) - 2

5. Which Way Will You Shift?
Matching -- match the letter of the list on the right with the function on the left.
f(x) + a
f(x - a)
f(x)*a
f(x + a)
f(x) - a
A) shift down a units  B) shift right a units  C) shift left a units  D) shift up a units  E) turn upside down  F) none of these

6. Which Way Will It Shift? 
It is possible to combine more than one of the transformations in one function:
What is the result of graphing this transformation of our function, f(x)?
f(x - 3) + 5

7. Numerical Results Given the function defined by a table
Determine the value of the following transformations x -3 -2 -1 1 2 3
f(x) 7 4 9 12 5 6
(x) + 3
f(x + 1)   
f(x - 2)
   

8. Sound Waves Consider a sound wave Place the function in your Y= screen
Represented by the function y = sin x) 
Place the function in your Y= screen
Make sure the mode is set to radians
Use the ZoomTrig option
The rise and fall of the graph model the vibration of the object creating or transmitting the sound.
What should be altered on the graph to show increased intensity or loudness?

9. Sound Waves
To model making the sound LOUDER we increase the maximum and minimum values (above and below the x-axis)
We increase the amplitude of the function
We seek to "stretch" the function vertically
Try graphing the following functions.  Place them in your Y= screen
Function
Style
y1=sin x
y2=(1/2)*sin(x)
y3=3*sin(x)
dotted
thick
normal
Predict what you think will happen before you actually graph the functions

10. Sound Waves Note the results of graphing the three functions.
The coefficient 3  in  3 sin(x)  stretches the function vertically
The coefficient 1/2  in  (1/2) sin (x) compresses the function vertically

11. Compression
The graph of f(x) = (x - 2)(x + 3)(x - 7) with a standard zoom graphs as shown to the right.
Enter the function in for y1=(x - 2)(x + 3)(x - 7) in your Y= screen.
Graph it to verify you have the right function.  

12. Compression
What can we do (without changing the zoom) to force the graph to be within the standard zoom?
We wish to compress the graph by a factor of 0.1
Enter the altered form of your y1(x) function into y2=  your Y= screen which will  do this.

13. Compression
When we multiply the function by a positive fraction less than 1,
We compress the function
The local max and min are within the bounds of the standard zoom window.

14. Flipping the Graph of a Function
Given the function below
We wish to manipulate it by reflecting it across one of the axes
Across the x-axis
Across the y-axis

15. Flipping the Graph of a Function
Consider the function
f(x) = 0.1*(x3 - 9x2 + 5) : place it in y1(x)
graphed on the window   -10 < x < 10  and  -20 < y < 20

16. Flipping the Graph of a Function
specify the following functions on the Y= screen:
y2(x) = y1(-x)                dotted style
y3(x) = -y1(x)                thick style
Predict which of these will rotate the function
about the x-axis
about the y-axis

17. Flipping the Graph of a Function
Results
To reflect f(x) in the x-axis       or rotate about  
To reflect f(x) in the y-axis         or rotate about
use -f(x)
use f(-x)

18. Assignment Lesson 3.4A Page 209 Exercises 1 – 35 odd Lesson 3.4B