Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine)

Transcription

1 Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine) Reflections Horizontal Translation (c) Vertical Translation (d) Remember: vertical stretch horizontal stretch 1

2 Part A: Reflections on the x and y axis Example 1: Graph the functions 2

3 Example 2: Graph the functions 3

4 Part B: Vertical Translation (vertical displacement) Example 3: Graph the functions How does changing the value of affect the graph? vertical translation: midline: maximum/minimum: vertical translation: midline: maximum/minimum: vertical translation: midline: maximum/minimum: Use amplitude and vertical translation to determine the maximum and minimum value 4

5 Part C: Horizontal Translation (Phase Shift) Cosine Function Sine Function the start of the first cycle of the cosine curve (0,1) the start of the first cycle of the cosine curve (0,0) Example 4: Graph the functions How does changing the value of affect the graph? Locate the start of the first cycle of the cosine curve (0,1) 5

6 Since the function is periodic, there are several equations that can correspond to a given graph where the phase shift is different. Think about the equations: The value that is chosen for the phase shift will determine whether the graph is perceived as having a reflection on the x axis or not. The x value of the maximum point The x value of the minimum point no reflection reflection on x axis 6

7 Example 5: Graph the functions Can you identify other equations that would produce the graph of? 7

8 Example 6: Graph the functions Locate the start of the first cycle of the sine curve (0,0) Think about other equations: The value that is chosen for the phase shift will determine whether the graph is perceived as having a reflection on the x axis or not. (i) where the x coordinate hits the sinusoidal axis going from a minimum to a maximum no reflection (ii) where the x coordinate hits the sinusoidal axis going from a maximum to a minimum reflection on x axis 8

9 Example 7: Sketch the graph of over two cycles. Method 1: Use Mapping Rule 9

10 Example 7 cont'd Method 2: Use Inequality 10

11 Example 8: Sketch the graph of over two cycles. 11

12 Example 9: Sketch the graph of over two cycles. Identify the vertical displacement, amplitude, period, phase shift, domain and range for the function. Method 1: Use Mapping Rule 12

13 Example 9 cont'd Method 2: Use Inequality 13

14 Example 10: Sketch the graph of over two cycles. 14

15 Your Turn Sketch the following functions over two cycles. Identify the vertical displacement, amplitude, period, phase shift, domain and range. 15

16 Write the Equation of the Sinusoidal Function Given the Graph. Example 1: Write the equation of the function in the form and Identify the key characteristics of the graph and then link them to the parameters in the equation. maximum value = Sinusoidal Axis = minimum value = amplitude = period = b = From graph there are cyles in Sine Function phase shift locate start of cycle Cosine Function phase shift locate start of cycle Equation: Equation: 16

17 Example 2: Write the equation of the function in the form and 17

18 Example 3: Write the equation of the function in the form and 18

19 Example 4: Word Problems 19

20 Questions p.253 #6 7, p.253 #15 16 p.253 #19 20

21 21

22 22

23 23

24 Word Problems Assign p #20, 24 24

25 25