Lesson 4.6 Graphs of Other Trigonometric Functions Essential Question: How do you sketch the graphs of other trigonometric functions?

Before we start… On your graphing calculator, graph: 𝑦= tan 𝜃

How do you sketch the graphs of other trigonometric functions? Identify: Intercepts Asymptotes Shapes of the graphs

Graph of the Tangent Function The tangent function is odd. Consequently, the graph of 𝑦= tan 𝑥 is symmetric with respect to the origin. You also know from the identity tan 𝑥 = sin 𝑥 cos 𝑥 that the tangent function is undefined when cos 𝑥=0 . Two such values are 𝑥=± 𝜋 2 ≈±1.5708.

Graph of the Tangent Function As indicated in the table, tan 𝑥 increases without bound as x approaches 𝜋 2 from the left, and it decreases without bound as x approaches − 𝜋 2 from the right. So, the graph of 𝑦= tan 𝑥 has vertical asymptotes at 𝑥= 𝜋 2 and 𝑥=− 𝜋 2 . Because the period of the tangent function is 𝜋, vertical asymptotes also occur at 𝑥= 𝜋 2 +𝑛𝜋, where n is an integer. x − 𝜋 2 – 1.57 – 1.5 − 𝜋 4 𝜋 4 1.5 1.57 𝜋 2 𝐭𝐚𝐧 𝒙 Undef. – 1,255.8 – 14.1 – 1 1 14.1 1,255.8

Library of Parent Functions: Tangent Function Graph of 𝑓 𝑥 = tan 𝑥 Domain: All real numbers 𝑥, 𝑥≠ 𝜋 2 +𝑛𝜋 Range: −∞,∞ Period: 𝜋 x-intercepts: 𝑛𝜋,0 y-intercept: (0,0) Vertical asymptote: 𝑥= 𝜋 2 +𝑛𝜋 Odd function Origin symmetry

Sketch the graph of 𝑦=−3 tan 2𝑥 .

Sketch the graph of 𝑦= tan 𝑥 4 .

Sketch the graph of 𝑦= tan 2𝑥 .

Graph of the Cotangent Function The graph of the parent cotangent function is similar to the graph of the parent tangent function. It also has a period of 𝜋. From the identity 𝑓 𝑥 = cot 𝑥 = cos 𝑥 sin 𝑥 you can see that the cotangent function has vertical asymptotes when sin 𝑥 equal zero, which occurs at 𝑥=𝑛𝜋, where n is an integer.

Library of Parent Functions: Cotangent Function Graph of 𝑓 𝑥 = cot 𝑥 Domain: All real numbers 𝑥, 𝑥≠𝑛𝜋 Range: −∞,∞ Period: 𝜋 x-intercepts: 𝜋 2 +𝑛𝜋,0 Vertical asymptotes: 𝑦=𝑛𝜋 Odd function Origin symmetry

Library of Parent Functions: Cosecant Function Graph of 𝑓 𝑥 = csc 𝑥 Domain: All real numbers 𝑥, 𝑥≠𝑛𝜋 Range: −∞, −1 ∪ 1 ,∞ Period: 2𝜋 No intercepts Vertical asymptotes: 𝑥=𝑛𝜋 Odd function Origin symmetry

Library of Parent Functions: Secant Function Graph of 𝑓 𝑥 = sec 𝑥 Domain: All real numbers 𝑥, 𝑥≠ 𝜋 2 +𝑛𝜋 Range: −∞, −1 ∪ 1 ,∞ Period: 2𝜋 y-intercept: (0,1) Vertical asymptotes: 𝑦= 𝜋 2 +𝑛𝜋 Even function y-axis symmetry

Sketch the graph of 𝑦= 2cot 𝑥 3 .

Sketch the graph of 𝑦= cot 𝑥 4 .

Sketch the graph of 𝑦= cot 𝜋𝑥 .

Sketch the graph of 𝑦=− csc 𝜋𝑥 2 .

Sketch the graph of 𝑦=2 csc 𝑥+ 𝜋 4 .

Sketch the graph of 𝑦= sec 2𝑥 .

Damping Trigonometric Graphs The product of two functions can be graphed using properties of the individual function. In the function 𝑓 𝑥 =𝑥 sin 𝑥 , the factor x is called the damping factor.

Analyze the graph of 𝑓 𝑥 = 𝑒 −𝑥 sin 3𝑥 .

Analyze the graph of 𝑓 𝑥 = 𝑒 𝑥 sin 4𝑥 .

Analyze the graph of 𝑓 𝑥 = 𝑒 −𝑥 cos 𝑥 .

Analyze the graph of 𝑓 𝑥 = 𝑥 2 cos 𝑥 .

How do you sketch the graphs of other trigonometric functions?

Ticket Out the Door Sketch 𝑦=4 tan 2𝑥