Graphs of Other Trigonometric Functions 4.6

The Tangent Curve: The Graph of y=tanx and Its Characteristics 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 1 –2  –   2  x 5  3    3  5  – – – 2 2 2 2 2 2 –1

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 = /2 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 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. y = A tan (Bx – C) Bx – C =  /2 Bx – C = - /2 x x-intercept between asymptotes

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 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. 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

The Cotangent Curve: The Graph of y = cotx and Its Characteristics 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 y- coordinates of –1 and 1. y -4 -2 2 4 x -  -  /2  /2 ˝ 3  /2 2 

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. 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. y = A cot (Bx – C) Bx – C =  y-coord-inate is A. x x-intercept between asymptotes y-coord-inate is -A. Bx – C = 0

Example Graph y = 2 cot 3x Solution: 3x=0 and 3x= x=0 and x = /3 are vertical asymptotes An 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

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 x Properties of y = sec x 1. domain : all real x 2. range: (–,–1]  [1, +) 3. period:  4. vertical asymptotes: Secant Function

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. x y Properties of y = csc x 1. domain : all real x 2. range: (–,–1]  [1, +) 3. period:  4. vertical asymptotes: where sine is zero. Cosecant Function

Text Example Use the graph of y = 2 sin 2x to obtain the graph of y = 2 csc 2x. y -2 2 x ˝ -˝ y -2 2 x ˝ 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.