Vertical Stretches and Compressions Lesson 5.3

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?

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

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

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.  

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.

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.

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

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

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

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

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

Assignment Lesson 5.3 Page 216 Exercises 1 – 35 odd