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# Lesson 3.4

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### Lesson 3.4

1. 1. Transformation of Graphs Lesson 3.4
2. 2. Tools for Exploration <ul><li>Consider the function f(x) = 0.1(x 3 – 9x 2 ) </li></ul><ul><li>Enter this function into your calculator on the y= screen </li></ul><ul><li>Set the window to be  -10 < x < 10   and   -20 < y < 20 </li></ul><ul><li>Graph the function </li></ul>
3. 3. Shifting the Graph <ul><li>Enter the following function calls of our original function on the y= screen: </li></ul><ul><ul><li>y1=  0.1 (x 3 - 9x 2 )  </li></ul></ul><ul><ul><li>y2=   y1(x + 2)  </li></ul></ul><ul><ul><li>y3=   y1(x) + 2 </li></ul></ul><ul><li>Before you graph the other two lines, predict what you think will be the result. </li></ul>Use different styles for each of the functions
4. 4. Shifting the Graph <ul><li>How close were your predictions? </li></ul><ul><li>Try these functions – again, predict results </li></ul><ul><ul><li>y1=  0.1 (x 3 - 9x 2 )  </li></ul></ul><ul><ul><li>y2=   y1(x - 2)  </li></ul></ul><ul><ul><li>y3=   y1(x) - 2 </li></ul></ul>
5. 5. Which Way Will You Shift?  Matching -- match the letter of the list on the right with the function on the left. <ul><li>f(x) + a </li></ul><ul><li>f(x - a) </li></ul><ul><li>f(x)*a </li></ul><ul><li>f(x + a) </li></ul><ul><li>f(x) - a </li></ul>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. 6. Which Way Will It Shift?  <ul><li>It is possible to combine more than one of the transformations in one function: </li></ul><ul><li>What is the result of graphing this transformation of our function, f(x)? </li></ul><ul><li>f(x - 3) + 5 </li></ul>
7. 7. Numerical Results <ul><li>Given the function defined by a table </li></ul><ul><li>Determine the value of the following transformations </li></ul>x -3 -2 -1 0 1 2 3 f(x) 7 4 9 3 12 5 6 (x) + 3               f(x + 1)                f(x - 2)
8. 8. Sound Waves <ul><li>Consider a sound wave </li></ul><ul><ul><li>Represented by the function y = sin x)  </li></ul></ul><ul><li>Place the function in your Y= screen </li></ul><ul><ul><li>Make sure the mode is set to radians </li></ul></ul><ul><ul><li>Use the ZoomTrig option </li></ul></ul>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. 9. Sound Waves <ul><li>To model making the sound LOUDER we increase the maximum and minimum values (above and below the x-axis) </li></ul><ul><li>We increase the amplitude of the function </li></ul><ul><li>We seek to &quot;stretch&quot; the function vertically </li></ul><ul><li>Try graphing the following functions.  Place them in your Y= screen </li></ul>Predict what you think will happen before you actually graph the functions Function Style <ul><li>y1=sin x </li></ul><ul><li>y2=(1/2)*sin(x) </li></ul><ul><li>y3=3*sin(x) </li></ul><ul><li>dotted </li></ul><ul><li>thick </li></ul><ul><li>normal </li></ul>
10. 10. Sound Waves <ul><li>Note the results of graphing the three functions. </li></ul><ul><li>The coefficient 3  in  3 sin(x)  stretches the function vertically </li></ul><ul><li>The coefficient 1/2  in  (1/2) sin (x) compresses the function vertically </li></ul>
11. 11. Compression <ul><li>The graph of f(x) = (x - 2)(x + 3)(x - 7) with a standard zoom graphs as shown to the right. </li></ul><ul><ul><li>Enter the function in for y1=(x - 2)(x + 3)(x - 7) in your Y= screen. </li></ul></ul><ul><li>Graph it to verify you have the right function.   </li></ul>
12. 12. Compression <ul><li>What can we do (without changing the zoom) to force the graph to be within the standard zoom? </li></ul><ul><ul><li>We wish to compress the graph by a factor of 0.1 </li></ul></ul><ul><li>Enter the altered form of your y1(x) function into y2=  your Y= screen which will  do this. </li></ul>
13. 13. Compression <ul><li>When we multiply the function by a positive fraction less than 1, </li></ul><ul><ul><li>We compress the function </li></ul></ul><ul><ul><li>The local max and min are within the bounds of the standard zoom window. </li></ul></ul>
14. 14. Flipping the Graph of a Function <ul><li>Given the function below </li></ul><ul><ul><li>We wish to manipulate it by reflecting it across one of the axes </li></ul></ul>Across the x-axis Across the y-axis
15. 15. Flipping the Graph of a Function <ul><li>Consider the function </li></ul><ul><ul><li>f(x) = 0.1*(x 3 - 9x 2 + 5) : place it in y1(x) </li></ul></ul><ul><ul><li>graphed on the window   -10 < x < 10  and  -20 < y < 20 </li></ul></ul>
16. 16. Flipping the Graph of a Function <ul><li>specify the following functions on the Y= screen: </li></ul><ul><ul><li>y2(x) = y1(-x)                dotted style </li></ul></ul><ul><ul><li>y3(x) = -y1(x)                thick style </li></ul></ul><ul><li>Predict which of these will rotate the function </li></ul><ul><ul><li>about the x-axis </li></ul></ul><ul><ul><li>about the y-axis </li></ul></ul>
17. 17. Flipping the Graph of a Function <ul><li>Results </li></ul><ul><li>To reflect f(x) in the x-axis       or rotate about   </li></ul><ul><li>To reflect f(x) in the y-axis         or rotate about </li></ul>use -f(x) use f(-x)
18. 18. Assignment <ul><li>Lesson 3.4A </li></ul><ul><li>Page 209 </li></ul><ul><li>Exercises 1 – 35 odd </li></ul><ul><li>Lesson 3.4B </li></ul><ul><li>Page 210 </li></ul><ul><li>Exercises 37 – 51 odd </li></ul>