2 2 Graphs of Other Trigonometric Functions Copyright

2. 2 Graphs of Other Trigonometric Functions Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 1

Objectives • Understand the graph of y = tan x. • Graph variations of y = tan x. • Understand the graph of y = cot x. • Graph variations of y = cot x. • Understand the graphs of y = csc x and y = sec x. • Graph variations of y = csc x and y = sec x. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 2

The Graph of y = tan x Period: The tangent function is an odd function. The tangent function is undefined at odd multiples of Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 3

The Tangent Curve: The Graph of y = tan x and Its Characteristics (1 of 2) Characteristics • Period: • Domain: All real numbers except odd multiples of • Range: All real numbers Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 4

The Tangent Curve: The Graph of y = tan x and Its Characteristics (2 of 2) • Vertical asymptotes at odd multiples of • An x-intercept occurs midway between each pair of consecutive asymptotes • Odd function with origin symmetry • Points on the graph of the way between consecutive asymptotes have y-coordinates of − 1 and 1, respectively. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 5

Graphing Variations of y = tan x (1 of 2) Graphing y = Atan (Bx − C), B > 0 1. Find two consecutive asymptotes by finding an interval containing one period A pair of consecutive asymptotes occurs at 2. Identify an x-intercept, midway between the consecutive asymptotes of the way 3. Find the points on the graph between the consecutive asymptotes. These points have y-coordinates of −A and A, respectively. 4. Use steps 1 -3 to graph one full period of the function. Add additional cycles to the left or right as needed Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 6

Graphing Variations of y = tan x (2 of 2) Graphing y = Atan (Bx − C), B > 0 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 7

Example 1: Graphing a Tangent Function (1 of 4) Graph y = 3 tan 2 x for Solution: A = 3, B = 2, C = 0 Step 1 Find two consecutive asymptotes. An interval containing one period is Thus, two consecutive asymptotes occur at Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 8

Example 1: Graphing a Tangent Function (2 of 4) Graph y = 3 tan 2 x for Step 2 Identify an x-intercept, midway between the consecutive asymptotes. x = 0 is midway between The graph passes through (0, 0). Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 9

Example 1: Graphing a Tangent Function (3 of 4) Graph y = 3 tan 2 x for Step 3 Find points on the graph of the way between the consecutive asymptotes. These points have y-coordinates of –A and A. The graph passes through Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 10

Example 1: Graphing a Tangent Function (4 of 4) Graph y = 3 tan 2 x for Step 4 Use steps 1 -3 to graph one full period of the function. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 11

The Cotangent Curve: The Graph of y = cot x and Its Characteristics (1 of 2) • Period: • Domain: All real numbers except integral multiples of • Range: all real numbers Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 12

The Cotangent Curve: The Graph of y = cot x and Its Characteristics (2 of 2) • Vertical asymptotes at integral multiples of • An x-intercept occurs midway between each pair of consecutive asymptotes • Odd function with origin symmetry • Points on the graph of the way between consecutive asymptotes have y-coordinates of 1 and − 1, respectively. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 13

Graphing Variations of y = cot x (1 of 2) Graphing y = A cot (Bx − C), B > 0 1. Find two consecutive asymptotes by finding an interval containing one full period: A pair of consecutive asymptotes occurs at 2. Identify an x-intercept, midway between the consecutive asymptotes 3. Find the points on the graph of the way between the consecutive asymptotes. These points have y-coordinates of A and −A, respectively. 4. Use steps 1 -3 to graph one full period of the function. Add additional cycles to the left or right as needed Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 14

Graphing Variations of y = cot x (2 of 2) Graphing y = A cot (Bx − C), B > 0 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 15

Example 2: Graphing a Cotangent Function (1 of 4) Graph Solution: Step 1 Find two consecutive asymptotes. An interval containing one period is (0, 2). Thus, two consecutive asymptotes occur at x = 0 and x = 2. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 16

Example 2: Graphing a Cotangent Function (2 of 4) Graph Step 2 Identify an x-intercept midway between the consecutive asymptotes. x = 1 is midway between x = 0 and x = 2. The graph passes through (1, 0). Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 17

Example 2: Graphing a Cotangent Function (3 of 4) Graph Step 3 Find points on the graph of the way between consecutive asymptotes. These points have y-coordinates of A and −A. The graph passes through Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 18

Example 2: Graphing a Cotangent Function (4 of 4) Graph Step 4 Use steps 1 -3 to graph one full period of the function. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 19

The Graphs of y = csc x and y = sec x We obtain the graphs of the cosecant and the secant curves by using the reciprocal identities We obtain the graph of y = csc x by taking reciprocals of the y-values in the graph of y = sin x. Vertical asymptotes of y = csc x occur at the x-intercepts of y = sin x. We obtain the graph of y = sec x by taking reciprocals of the y-values in the graph of y = cos x. Vertical asymptotes of y = sec x occur at the x-intercepts of y = cos x. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 20

The Cosecant Curve: The Graph of y = csc x and Its Characteristics (1 of 2) Period: Domain: All real numbers except integral multiples of Range: All real numbers y such that Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 21

The Cosecant Curve: The Graph of y = csc x and Its Characteristics (2 of 2) • Vertical asymptotes at integral multiples of • Odd functions, csc(−x) = −csc x, with origin symmetry Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 22

The Secant Curve: The Graph of y = sec x and Its Characteristics (1 of 2) • Period: • Domain: All real numbers except odd multiples of • Range: All real numbers y such that Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 23

The Secant Curve: The Graph of y = sec x and Its Characteristics (2 of 2) • Vertical asymptotes at odd multiples of • Even functions, sec(−x) = sec x, with y-axis symmetry Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 24

Example 3: Using a Sine Curve to Obtain a Cosecant Curve (1 of 2) Use the graph of to obtain the graph of The x-intercepts of the sine graph correspond to the vertical asymptotes of the cosecant graph. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 25

Example 3: Using a Sine Curve to Obtain a Cosecant Curve (2 of 2) Use the graph of to obtain the graph of Using the asymptotes as guides, we sketch the graph of Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 26

Example 4: Graphing a Secant Function (1 of 4) Graph y = 2 sec 2 x for Solution: We begin by graphing the reciprocal function, y = 2 cos 2 x. This equation is of the form y = A cos Bx, with A = 2 and B = 2. amplitude: period: We will use quarter-periods to find x-values for the five key points. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 27

Example 4: Graphing a Secant Function (2 of 4) Graph y = 2 sec 2 x for The x-values for the five key points are: Evaluating the function y = 2 cos 2 x at each of these values of x, the key points are: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 28

Example 4: Graphing a Secant Function (3 of 4) Graph y = 2 sec 2 x for The key points for our graph of y = 2 cos 2 x are: We draw vertical asymptotes through the x-intercepts to use as guides for the graph of y = 2 sec 2 x. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 29

Example 4: Graphing a Secant Function (4 of 4) Graph y = 2 sec 2 x for Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 30

The Six Curves of Trigonometry (1 of 6) Domain: all real numbers: Range: Period: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 31

The Six Curves of Trigonometry (2 of 6) Domain: all real numbers: Range: Period: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 32

The Six Curves of Trigonometry (3 of 6) Domain: all real numbers except odd multiples of Range: all real numbers Period: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 33

The Six Curves of Trigonometry (4 of 6) Domain: all real numbers except integral multiples of Range: all real numbers Period: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 34

The Six Curves of Trigonometry (5 of 6) Domain: all real numbers except integral multiples of Range: Period: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 35

The Six Curves of Trigonometry (6 of 6) Domain: all real numbers except odd multiples of Range: Period: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved Slide - 36