Graphs of Other Trigonometric

Functions

4.6

x

y

2–2 – 01

–152 3

2 2 – 5

2 – 32 –

2

Period:

Domain: All real numbers except /2 + k , k an integer

Range: All real numbers

Symmetric with respect to the origin

Vertical asymptotes at odd multiples of /2

The Tangent Curve: The Graph of y=tanx and Its Characteristics

Graphing y = A tan(Bx – C)

1. Find two consecutive asymptotes by setting the variable expression in the tangent equal to -/2 and /2 and solving

Bx – C = -/2 and Bx – C = /22. Identify an x-intercept, midway between consecutive asymptotes.3. Find the points on the graph 1/4 and 3/4 of the way between and x-intercept and the asymptotes. These points have y-coordinates of –A and A.4. Use steps 1-3 to graph one full period of the function. Add additional cycles to the left or

right as needed.

x

x-intercept between

asymptotes

y = A tan (Bx – C)

Bx – C = /2Bx – C = - /2

Graph y = 2 tan x/2 for – < x < 3 SolutionStep 1 Find two consecutive asymptotes.

Thus, two consecutive asymptotes occur at x = - and x = .

Step 2 Identify any x-intercepts, midway between consecutive asymptotes. Midway between x = - and x = is x = 0. An x-intercept is 0 and the graph passes through (0, 0).

Text Example

Step 3 Find points on the graph 1/4 and 1/4 of the way between an x-intercept and the asymptotes. These points have y-coordinates of –A and A. Because A, the coefficient of the tangent, is 2, these points have y-coordinates of -2 and 2.

Solution

Step 4 Use steps 1-3 to graph one full period of the function. We use the two consecutive asymptotes, x = - and x = , an x-intercept of 0, and points midway between the x-intercept and asymptotes with y-coordinates of –2 and 2. We graph one full period of y = 2 tan x/2 from – to . In order to graph for – < x < 3 , we continue the pattern and extend the graph another full period on the right.

y

-4

-2

2

4

˝x

-˝ 3˝

y = 2 tan x/2

Text Example cont.

The Graph of y = cot x and Its CharacteristicsCharacteristicsPeriod: Domain: All real numbers except integral multiples of Range: All real numbersVertical asymptotes: at integral

multiples of n x-intercept occurs midway between each pair of consecutive asymptotes.Odd function with origin symmetryPoints on the graph 1/4 and 3/4 of the way between consecutive asymptotes have y-coordinates of –1 and 1.

y

-4

-2

2

4

/2x

- /2 3 /2˝ 2 -

The Cotangent Curve: The Graph of y = cotx and Its Characteristics

1. Find two consecutive asymptotes by setting the variable expression in the cotangent equal to 0 and ˝ and solving

Bx – C = 0 and Bx – C = 2. Identify an x-intercept, midway between consecutive asymptotes.3. Find the points on the graph 1/4 and 3/4 of the way between an x-intercept and the asymptotes. These points have y-coordinates of –A and A.4. Use steps 1-3 to graph one full period of the function. Add additional cycles to the left or

right as needed.

x

x-intercept between

asymptotes

y = A cot (Bx – C)

Bx – C =

Bx – C = 0

y-coord-inate is A.

y-coord-inate is -A.

Graphing y=Acot(Bx-C)

ExampleGraph y = 2 cot 3xSolution:3x=0 and 3x=x=0 and x = /3 are vertical asymptotesAn x-intercepts occurs between 0 and /3 so an x-

intercepts is at (/6,0)The point on the graph midway between the asymptotes

and intercept are /12 and 3/12. These points have y-coordinates of -A and A or -2 and 2

Graph one period and extend as needed

Example cont• Graph y = 2 cot 3x

-3 -2 -1 1 2 3

-10

-8

-6

-4

-2

2

4

6

8

10

23

y

x

2

2

2 32

5

4

4

xy cos

Graph of the Secant Function

2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes:

kkx 2

1. domain : all real x)(

2 kkx

cos

1secx

x The graph y = sec x, use the identity .

Properties of y = sec x

xy sec

At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

2

3

x

2

2

22

5

y4

4

Graph of the Cosecant Function

2. range: (–,–1] [1, +) 3. period:

where sine is zero.

4. vertical asymptotes: kkx

1. domain : all real x kkx

sin

1cscx

x To graph y = csc x, use the identity .

Properties of y = csc x xy csc

xy sin

At values of x for which sin x = 0, the cosecant functionis undefined and its graph has vertical asymptotes.

Use the graph of y = 2 sin 2x to obtain the graph of y = 2 csc 2x.

Solution The x-intercepts of y = 2 sin 2x correspond to the vertical asymptotes of y = 2 csc 2x. Thus, we draw vertical asymptotes through the x-intercepts. Using the asymptotes as guides, we sketch the graph of y = 2 csc 2x.

y

-2

2

x˝-˝

y

-2

2

x˝

Text Example