20.3 Period and Phase Shift 13. For each of the following, draw a graph of the indicated transformation of the primary period of y = sin(I). Hint: Your graphs for parts (a) and (b) should not be the same! (a) First compress the period to n, then shift right by ~. (b) First shift right by 5, then compress the period to . Hint: Remember that the compression happens towards the y-axis. 14. The equations for the transformations in #13 are formed like this: (a) y = sin(x) compress, y = sin(2x) shift y = sin (2(x - 7) ) (b) y = sin(r) shift y = sin(x - compress, y = sin (22) - 7) For each graph, take the final phase shift (after both transformations) and plug it in for a in that graph's equation. What do you notice? Summary: The wave given by y = Asin( Br + C) + D or y = Acos( Bx + C) + D has the following properties: . The Midline is y = D. . The Amplitude is |A|. . The Period is . The Phase Shift is the solution to Be + C =0, which is a =- 16320.4 Drawing a Graph Use th i e procedure below to help you draw the primary cycle of the graph of y = 251'11(;'5 + ';_) ca 15. 16. 17. 18. 19. 20. 21. 22. 23. . Draw this as a dashed line on the graph. Identify the equation of the midline: y = Compute the maximum and minimum values of the wave, Hint: What is the wave's amplitude\" maximum: =__ minimum: = Compute the phase shift: 5 = idline. h its z-coordinate at s and its y-coordinate on the m tead of the midline. Plot the first point of our cycle wit, Note: If we had cosine instead of sine, we would start our cycle at the maximum y-value ins Compute the period: p = We need to divide the period into four equal parts, so compute %p = : 1 Add ip to s to get the next important z-coordinate: s+ 3P = Plat the second major point of our cycle with its z-coordinate at s + 7P and its y-coordinate at the maximurm. Note: 1f we had a negative coefficient in front of our function, we would plot this point at the minimum y-value instead of the maximum. p-coordinates. Plot the third point on the midline, the fourth at the Repeatedly add ip to get the three remaining aw & smooth wave through the five points. minimum, and the fifth on the midline. Then dr 164