1. Vertical Stretches and Compressions
Lesson 5.3

2. 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?

3. 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

4. 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

5. 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.  

6. 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.

7. 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.

8. View the different versions of the altered graphs
Changes to a Graph
What has changed?
What remains the same?
View the different versions of the altered graphs

9. Changes to a Graph
Classify the following properties as changed or not changed when the function f(x) is modified by a coefficient    a*f(x)
Property
Changed
Not Changed
Zeros of the function
Intervals where the function increases or decreases
X locations of the max and min
Y-locations of the max and min
Steepness of curves where function is increasing/decreasing

10. Changes to a Graph
Consider the function below.  What role to each of the modifiers play in transforming the graph?
Modifier
Result a b c d

11. Combining Transformations
y = a * f (b * (x + c)) + d
a => vertical stretch/compression
|a| > 1 causes stretch
-1 < a < 1 causes compression of the graph
a < 0 will "flip" the graph about the x-axis
b => horizontal stretch/compression
b > 1 causes compression
|b| < 1 causes stretching

12. Combining Transformations
y = a * f (b * (x + c)) + d
c => horizontal shift of the graph
c < 0 causes shift to the right
c > 0 causes shift to the left
d => vertical shift of the graph
d > 0 causes upward shift
d < 0 causes downward shift

13. Assignment
Lesson 5.3
Page 216
Exercises 1 – 35 odd