Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.5 Part 2 Graphs of Sine and Cosine Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Understand the graph of y = sin x. Understand the graph of y = cos x. Graph variations of y = sin x. Graph variations of y = cos x. Use vertical shifts of sine and cosine curves. Model periodic behavior.

Period Because y = a sin x completes one cycle from x = 0 to x = 2, it follows that y = a sin bx completes one cycle from x = 0 to x = 2 /b, where b is a positive real number. For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period 2/b. The graph of y = cos bx will resemble that of y = cos x, with period 2/b.

Period Note; when b > 1, the period of y = a sin bx (or y = a cos bx) is less than 2 and represents a horizontal shrinking (smaller period) of the graph of y = a sin x (or y = a cos bx). when 0 < b < 1, the period of y = a sin bx (or y = a cos bx) is greater than 2 and represents a horizontal stretching (larger period) of the graph of y = a sin x (or y = a cos bx).

Period When b is negative, the identities sin(–x) = –sin x and cos(–x) = cos x are used to rewrite the function and make b positive.

Period The period is the distance between two peaks or valleys. y = sin x has period 2π y = sin(bx) has period 2π/b

Period The length of a Period (or simply Period) = (horizontal distance between adjacent peaks) = (horizontal distance between adjacent troughs) one period

If b > 1, the graph of the function is shrunk horizontally. Period The period of a function is the x interval needed for the function to complete one cycle. For b  0, the period of y = a sin bx is 2𝜋 𝑏 . For b  0, the period of y = a cos bx is also 2𝜋 𝑏 . If b > 1, the graph of the function is shrunk horizontally. y x period: π period: 2π

Period If 0 < b < 1, the graph of the function is stretched horizontally. y x period: 4π period: 2π

Example: Graph y = cos 2x/3 over one period For y = cos 2x/3, b = 2/3. Then, the period (2π/b = 2π/(2/3) = 3π) is 3. The graph will complete one period over the interval [0, 3]. The endpoints are 0 and 3, the three middle points are: 1 4 3𝜋 , 1 2 3𝜋 , 3 4 3𝜋 Make a table of the 5 key points, plot the points and join in a smooth curve.

Example: Graph y = cos 2x/3 over one period The period is 3. Divide the interval into four equal parts. Obtain key points for one period. 1 1 cos 2x/3 2 3/2  /2 2x/3 3 9/4 3/4 x

Example: Graph y = cos 2x/3 over one period continued The amplitude is 1. Join the points and connect with a smooth curve.

Your Turn: Graph y = sin 2x over one period

Graphing Variations of y = a sin bx and y = a cos bx Identify the amplitude (|a|) and the period (2π/b). Start with 0 on the x-axis, and lay off a distance of one period (2/b). Divide the interval on the x-axis into four equal parts. Find the x values for the 5 key points; the x-intercepts, maximum and minimum points. Start with the value of x where the period begins and add quarter-periods (that is, 𝑝𝑒𝑟𝑖𝑜𝑑 4 ) to find successive values of x. Find the values of y by evaluating the function for each of the five x-values resulting from Step 2. Make a table of values. Connect the 5 key points with a smooth curve and graph one complete period of the given function. Extend the graph in step 4 to the left or right as needed.

Example #1: Graph y = 2 sin 4x Amplitude: a = -2 → |-2| = 2 amplitude, a = -2 → negative = reflected over x-axis. Period: b = 4 → 2π/b = 2/4 = /2 period. The function will be graphed over the interval [0, /2]. Divide the interval into four equal parts: 0, ( 1 4 ∙ 𝜋 2 ), ( 1 2 ∙ 𝜋 2 ), ( 3 4 ∙ 𝜋 2 ), π/2 → 0, 𝜋 8 , 𝜋 4 , 3𝜋 8 , 𝜋 2 .

Example #1: (continued) y = 2 sin 4x Make a table of values 2 2 2 sin 4x 1 1 sin 4x 2 3/2  /2 4x 3/8 /4 /8 x

Example #1: (continued) y = 2 sin 4x Plot the 5 key points and join them with a sinusoidal curve with amplitude 2. /4 3/8 /2 /8 /2 /8 x y /4 3/8 -2 2 Extend the graph as needed.

Your Turn #1: Determine the amplitude and period of 𝑦=2 sin 1 2 𝑥 . Then graph the function for 0 ≤ x ≤ 8π.

Find the amplitude and the period. Example #2: © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Graphing y = A cos (Bx) Graph 𝑦=3 cos 1 2 𝑥 Solution Find the amplitude and the period.

Example #2: (continued) 𝑦=3 cos 1 2 𝑥 © 2010 Pearson Education, Inc. All rights reserved Example #2: (continued) 𝑦=3 cos 1 2 𝑥 2. Find the x-coordinates for the five key points. 1 4 (period) = 1 4 (4π) = π x1 = 0 x2 = 1 4 ∙ 4π = π x3 = 1 2 ∙ 4π = 2π x4 = 3 4 ∙ 4π = 3π x5 = 4π

Example #2: (continued) 𝑦=3 cos 1 2 𝑥 © 2010 Pearson Education, Inc. All rights reserved Example #2: (continued) 𝑦=3 cos 1 2 𝑥 3. Find the y-coordinates for the five key points.

Example #2: (continued) 𝑦=3 cos 1 2 𝑥 © 2010 Pearson Education, Inc. All rights reserved Example #2: (continued) 𝑦=3 cos 1 2 𝑥 4. Graph of 𝑦=3 cos 1 2 𝑥 , 0 ≤ x ≤ 4π

Example #2: (continued) 𝑦=3 cos 1 2 𝑥 © 2010 Pearson Education, Inc. All rights reserved Example #2: (continued) 𝑦=3 cos 1 2 𝑥 5. Extend Graph of 𝑦=3 cos 1 2 𝑥 , as needed.

Graph y = –3 cos x over one period. Your Turn #2: GRAPHING y = a cos bx FOR b THAT IS A MULTIPLE OF π Graph y = –3 cos x over one period.

Graph y = –2 sin 3x over one period. Your Turn #3: GRAPHING y = a sin bx Graph y = –2 sin 3x over one period.