Graphs of Other Trigonometric Functions Section 4.6

Graph of the Tangent Function 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. y x Properties of y = tan x 1. domain : all real x 2. range: (–, +) 3. period:  4. vertical asymptotes: period:

Steps in Graphing y = a tan bx. 1. Determine the period . 2. Locate two adjacent vertical asymptotes by solving for x: 3. Sketch the two vertical asymptotes found in Step 2. 4. Divide the interval into four equal parts. 5. Evaluate the function for the first – quarter point, midpoint, and third - quarter point, using the x – values in Step 4. 6. Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

Example: Find the period and asymptotes and sketch the graph of 1. Period of y = tan x is  . 2. Find consecutive vertical asymptotes by solving for x: Vertical asymptotes: 3. Plot several points in 4. Sketch one branch and repeat.

Example: Find the period and asymptotes and sketch the graph of 1. Period of y = tan x is . of Period ® is 2. Find consecutive vertical asymptotes by solving for x: Vertical asymptotes: 3. Divide - to  into four equal parts. 4. Sketch one branch and repeat.

Graph 1. Period is or 4. 2. Vertical asymptotes are x = - 2 x = 2 y x Graph 1. Period is or 4. 2. Vertical asymptotes are 3. Divide the interval - 2 to 2 into four equal parts.