1. 4.5 – Graphs of Sine and Cosine A function is periodic if f(x + np) = f(x) for every x in the domain of f, every integer n, and some positive number p (called the period).

2. 4.5 – Graphs of Sine and Cosine xy 00 1 π0 −1 2π2π0 y = sin x −2π −π π 2π 3π 4π

3. Characteristics of the Sine Function 1)The domain is. 2)The range is [-1, 1]. 3)The period is. 4)The sine function is an odd function. It is symmetric with respect to the origin. sin (x) = -sin (x)

4. 4.5 – Graphs of Sine and Cosine xy 01 0 π−1 0 2π2π1 y = cos x −2π −π π 2π 3π 4π

5. Characteristics of the Cosine Function 1)The domain is. 2)The range is [-1, 1]. 3)The period is. 4)The sine function is an even function. It is symmetric with respect to the y-axis. cos (x) = cos (-x)

6. 4.5 – Graphs of Sine and Cosine −2π −π π 2π 3π 4π Graphing y = a sin x y = 2 sin x y = sin x ½

7. 4.5 – Graphs of Sine and Cosine −2π −π π 2π 3π 4π Graphing y = sin bx y = sin 2x y = sin x ½

8. 4.5 – Graphs of Sine and Cosine −2π −π π 2π 3π 4π Graphing y = sin(x − c) y = sin ( x + 2 ) y = sin x y = sin(x − π) π

9. 4.5 – Graphs of Sine and Cosine Graphing y = sin(bx − c) 0  bx − c  2π c  bx  c + 2π starting pointending point

10. 4.5 – Graphs of Sine and Cosine Graphing y = a sin(bx − c) 1.Find amplitude = | a | 2.Find period = 3.Find phase shift = 4.Find the interval on the x-axis. 5.Divide the interval into fourths to plot “key points”. 6.Graph one period. Extend if necessary.

11. 4.5 – Graphs of Sine and Cosine Graph the equation y = 3 sin(2x − π) amplitude: period: phase shift: interval: 3 = π π2π2π

12. 4.5 – Graphs of Sine and Cosine Graph the equation y = amplitude: period: p. s.: interval: 4π4π2π2π = 2π(2)= 4π = −π(2)= −2π [−2π, 2π] −2π 1 −1

13. 4.5 – Graphs of Sine and Cosine −2π −π π 2π 3π 4π Graphing y = sin(x) + d y = sin ( x) + 2 y = sin x y = sin(x) − 1

14. 4.5 – Graphs of Sine and Cosine Graph the equation y = sin(2x − π) + 2 amplitude: period: phase shift: interval: vertical shift: 1 = π up 2 units π2π2π