Section 8.4: The Equations of Sinusoidal Functions

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1 Section 8.4: The Equations of Sinusoidal Functions In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation. Transformed sinusoidal equations are written in the following forms: y = a sin b(x c) + d y = a cos b(x c) + d 1

2 Examining the Impact of the "a" Value on the Graph of a Sinusoidal Function 1. Use your calculator to graph each function. A) y = sin x B) y = 2 sin x 2

3 C) y = ½ sin x D) y = 2 sin x 3

4 Questions: 1. What happens to the amplitude if a > 0? 2. Is the shape of the graph affected by the parameter a? 3. How is the range affected by the parameter a? 4. Will the value of a affect the cosine graph in the same way that it affects the sine graph? Why or why not? 4

5 Summary of the "a" Value The "a" value stretches or compresses a graph vertically. It equals the amplitude of the function. Making a negative will cause a reflection in the x-axis. Maximums will become Minimums and vice versa. However points on the original midline (sinusoidal axis) will not change (yet). A = amplitude Example: Determine the "a" value and state the amplitude for each equation. (A) y = 2 sin 3(x 90 o ) + 1 (B) y = 0.75 cos 2(x + 45 o ) +3 5

6 Examining the Impact of the "d" Value on the Graph of a Sinusoidal Function Use your calculator to graph each function. (A) y = sin x + 2 (B) y = sin x 3 6

7 Questions 1. How does each graph change when compared to y = sinx? 2. How is the value of d related to the equation of the midline? 3. Is the shape of the graph or the location of the graph affected by the parameter d? 4. Is the period affected by changing the value of d? 5. Will the value of d affect the cosine graph in the same way that it affects the sine graph? Why or why not? 7

8 Summary of the "d" Value The "d" value gives us the equation of the midline of a sinusoidal function. Equation of Midline: y = d Examples: Identify the equation of the midline for each equation. (A) y = 2 sin 3(x 90E) + 1 (B) y = 0.75 cos 2(x + 45E)!3 8

9 Examining the Impact of the "b" Value on the Graph of a Sinusoidal Function (A) y = sin ½x (B) y = sin 2x 9

Write an equation that relates period to the b value.

10 QUESTIONS 1. How does each graph change when compared to y = sinx? 2. How is the value of b related to the period? Write an equation that relates period to the b value. Summary of the "b" Value The "b" value gives us the period of a sinusoidal function. In degrees, In radians, 10

11 Examples: Determine the period for each equation. (A) y = 2 sin 3(x 90E) + 1 (B) y = 0.75 cos 2(x + 45E)!3 (C) (D) 11

12 Examining the Impact of the "c" Value on the Graph of a Sinusoidal Function Use your calculator to graph each function. (A) y = sin (x 60E) (B) y = sin (x + 30) 12

13 (C) y = cos(x 60E) (D) y = cos (x + 30) 13

14 QUESTIONS 1. How does each graph change when compared to y = sinx? y = cos x? 2. How is the value of c related to starting point? 3. How can we determine the "phase shift" from the equation of the sinusoidal function? More on the "c" Value... The "c" value shifts a graph horizontally. The shift is obtained by taking the opposite sign of the number after x in the equation. Horizontal Shift/Translation = c 14

Section 8.4 Equations of Sinusoidal Functions soln.notebook. May 17, Section 8.4: The Equations of Sinusoidal Functions.

Section 8.4 Equations of Sinusoidal Functions soln.notebook. May 17, Section 8.4: The Equations of Sinusoidal Functions. Section 8.4: The Equations of Sinusoidal Functions Stop Sine 1 In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation.