Graphs of Other Trigonometric Functions 4 6 The

Graphs of Other Trigonometric Functions 4. 6

The Tangent Curve: The Graph of y=tanx and Its Characteristics y Period: Domain: All real numbers except /2 + k , k an integer 1 – 2 – 5 2 – – 3 2 – 1 2 Range: All real numbers 2 0 3 2 5 2 x Symmetric with respect to the origin Vertical asymptotes at odd multiples of /2

Graphing y = A tan(Bx – C) y = A tan (Bx – C) Bx – C = - /2 x-intercept between asymptotes 1. Find two consecutive asymptotes by setting the variable expression in the tangent equal to - /2 and solving Bx – C = /2 Bx – C = - /2 and Bx – C = /2 2. Identify an x-intercept, midway between consecutive asymptotes. x 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.

Text Example Graph y = 2 tan x/2 for – < x < 3 Solution Step 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 cont. Solution Step 3 Find points on the graph 1/4 and 1/4 of the way between an xintercept 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. 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 = 2 tan x/2 y 4 2 -˝ ˝ -2 -4 3˝ x

The Cotangent Curve: The Graph of y = cotx and Its Characteristics The Graph of y = cot x and Its Characteristics y 4 2 - - /2 -2 -4 /2 ˝ 3 /2 2 x Period: Domain: All real numbers except integral multiples of Range: All real numbers Vertical asymptotes: at integral multiples of n x-intercept occurs midway between each pair of consecutive asymptotes. Odd function with origin symmetry Points on the graph 1/4 and 3/4 of the way between consecutive asymptotes have ycoordinates of – 1 and 1.

Graphing y=Acot(Bx-C) 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. x 3. Find the points on the graph 1/4 and 3/4 of the y-coordway between an x-intercept and the asymptotes. inate is -A. 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. y = A cot (Bx – C) y-coordinate is A. x-intercept between asymptotes Bx – C =0

Example Graph y = 2 cot 3 x Solution: 3 x=0 and 3 x= x=0 and x = /3 are vertical asymptotes An x-intercepts occurs between 0 and /3 so an xintercepts 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 3 x

Graph of the Secant Function The graph y = sec x, use the identity . At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y Properties of y = sec x 1. domain : all real x 2. range: (– , – 1] [1, + ) 3. period: 4. vertical asymptotes: x

Graph of the Cosecant Function To graph y = csc x, use the identity . At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. y Properties of y = csc x 1. domain : all real x 2. range: (– , – 1] [1, + ) 3. period: 4. vertical asymptotes: where sine is zero. x

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