Tangent Section 4.6

Objectives I can use the characteristics of tangent to accurately graph I can determine the location of asymptotes to help graph

y x Tangent Function Graph of the Tangent Function 2. range: (– , +  ) 3. period:  4. vertical asymptotes: 1. domain : all real x Properties of y = tan x period: To graph y = tan x, use the identity. At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.

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

BIG Picture Tangent Plot x-intercept point ½ between vertical asymptotes Plot points at the ¼ and ¾ locations using value of |A| Connect with smooth curve similar to the cubic function

Example: Tangent Function 1. Find consecutive vertical asymptotes by solving for x: 3. Sketch one branch and repeat. Example: Find the period and asymptotes and sketch the graph of Vertical asymptotes: y x 2. Plot x-intercept ½ between asymptotes, then use value of A to find ¼ and ¾ points.

Homework Worksheet 8-2