Further Transformations Chapter 4 Trigonometric Functions 4.4 Further Transformations of Sine and Cosine Functions MATHPOWERTM 12, WESTERN EDITION 4.4.1

Transformations of Functions The principles of transformations of functions apply to trigonometric functions and can be summarized as follows: Vertical Stretch y = af(x) y = a sin x changes the amplitude to | a | Horizontal Stretch y = f(bx) y = sin bx changes the period Vertical Translation y = f(x) + k y = sin x + k shifts the curve vertically k units upward when k > 0 and k units downward when k < 0 Horizontal Translation y = f(x + h) y = sin (x + h) shifts the curve horizontally h units to the left when h > 0 and h units to the right when h < 0 4.4.2

Transforming a Trigonometric Function Graph y = sin x + 2 and y = sin x - 3. y = sin x + 2 y = sin x - 3 The range for y = sin x + 2 is 1 ≤ y ≤ 3. The range for y = sin x - 3 is -4 ≤ y ≤ -2. 4.4.3

Transforming a Trigonometric Function y = sin x A horizontal translation of a trigonometric function is called a phase shift. 4.4.4

Transforming a Trigonometric Function Sketch the graph of y = sin x y = 3sin 2x y = 3sin 2x 4.4.5

Analyzing a Sine Function p 2p y- intercept: x = 0 Domain: Range: Amplitude: Vertical Displacement: Period: Phase Shift: the set of all real numbers -5 ≤ y ≤ 1 3 2 units down p units to the left 4.4.6

Analyzing a Sine Function In the equation of y = asin[b(x + c)] + d: a = 4, b = 3, d = -3, and Compare the graph of this function to the graph of y = sin x with respect to the following: a) domain and range b) amplitude Domain: Amplitude: 4 Range: -7 ≤ y ≤ 1 c) period d) x- and y-intercepts x-intercepts: Period: 0.02, 0.5, 2.12, 2.80 y-intercept: e) phase shift f) vertical displacement right down 3 units g) equation 4.4.7

Determining an Equation From a Graph A partial graph of a sine function is shown. Determine the equation as a function of sine. a = 2 d = 1 b = 2 Therefore, the equation is . 4.4.8

Determining an Equation From a Graph A partial graph of a cosine function is shown. Determine the equation as a function of cosine. a = 2 d = -1 b = 2 Therefore, the equation is . 4.4.9

Determining an Equation From a Graph A partial graph of a sine function is shown. Determine the equation as a function of sine. Amplitude: Vertical Displacement: Period: The equation as a function of sine is 3 2 p 4.4.10

Graphing Sine as a Function of Time The motion of a weight on a spring can be described by the equation Sketch this function. y = sin t The period is 2. The amplitude is 2. The phase shift is indicating a shift to the right. 4.4.11

Assignment Suggested Questions: Pages 218 and 219 1-23 odd, 25-33, 34 (graphing calculator) 4.4.12