5.1 Graphing Sine and Cosine Functions When Function values repeat at regular intervals the function could be referred to as a cyclic function or periodic function. You can model these types of behaviour with sine or cosine functions. Math 30-1

Math 30-1

Functions that repeat themselves over a particular interval Periodic Functions Functions that repeat themselves over a particular interval of their domain are periodic functions. The length of the interval of repeat is called the period of the function. One cycle Math 30-1

Characteristics of a Periodic Function Graph y = sin x , radians The amplitude of a periodic function is one half the distance between the maximum and minimum values. 1 Period: 2p Domain: all real numbers Range: {y| -1 ≤ y ≤ 1} Amplitude: 1 y-intercept: 0 zeros : 0, ±, ±2 , ... General expression for zeros Math 30-1

Graphing a Periodic Function Graph y = cos x, radians 1 Period: 2p Domain: all real numbers Range: {y| -1 ≤ y ≤ 1} Amplitude: 1 y-intercept: 1 zeros: General expression Math 30-1

y = af(b(x - c)) + d y = asin[b(x - c)] + d Parameters that affect the graphs of sine and cosine. Vertical stretch factor |a| 5.1 Transformations abcd y = af(b(x - c)) + d Horizontal and vertical translation y = asin[b(x - c)] + d amplitude |a| phase shift displacement Math 30-1

Effect of parameter a in y = a sin x Graph y = 2sin x and y = 0.5sin x. y = 2sin x y = sin x y = sin x y = 0.5sin x Math 30-1

Comparing the Graphs of y = a sin x y = sin x y = 2sin x y = 0.5sin x 2  2  2  Period Amplitude Domain Range Zeros 1 2 0.5 All real numbers All real numbers All real numbers {-1 ≤ y ≤ 1} {-2 ≤ y ≤ 2} {-0.5 ≤ y ≤ 0.5} The amplitude of the graph of y = a sin x is | a |. When a > 0, there is a vertical stretch by a factor of |a|. When a < 0, there is a vertical stretch by a factor of |a| and a reflection in the x-axis. Math 30-1

Determining the Period for y = sin bx, b > 0 Graph y = sin 2x y = sin x y = sin 2x y = sin x transformed graph Length of period of parent Math 30-1

Determining the Period for y = sin bx, b > 0 Graph y = sin x y = sin x transformed graph Length of period of parent Math 30-1

Comparing the Graphs of y = sin bx y = sin x y = sin 2 x y = sin 0.5 x 2   4  Period Amplitude Domain Range Zeros 1 1 1 All real numbers All real numbers All real numbers {-1 ≤ y ≤ 1} {-1 ≤ y ≤ 1} {-1 ≤ y ≤ 1} The period for y = sin bx is When b > 0, there is a horizontal stretch by a factor of 1/|b|. When b < 0, there is a reflection and a horizontal stretch of 1/|b|. Math 30-1

Determining the Period and Amplitude of y = a sin bx Given the function y = 3sin 4x, determine the period and the amplitude. The period of the function is . Therefore, the period is . The amplitude of the function is | a |. Therefore, the amplitude is 3. y = 3sin 4x Math 30-1

Determining the Period and Amplitude of y = a sin bx Determine the characteristics of y = -3sin 3x. The period is . The amplitude is 3. Math 30-1

Writing the Equation of the Periodic Function Amplitude Period  = 2 b = 2 Therefore, the equation as a function of sine is y = 2sin 2x. Math 30-1

Writing the Equation of the Periodic Function Amplitude Period 4  = 3 b = 0.5 Therefore, the equation as a function of cosine is y = 3cos 0.5x. Math 30-1

Assignment Suggested Questions: Pages 233 1, 2, 4, 5, 6, 8, 10, 11b, c 14, 17a, 20, C4 Math 30-1