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

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

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

4. 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 ≈±

5. 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

6. 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

7. Sketch the graph of 𝑦=−3 tan 2𝑥 .

8. Sketch the graph of 𝑦= tan 𝑥 4 .

9. Sketch the graph of 𝑦= tan 2𝑥 .

10. 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.

11. 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

12. 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

13. 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

14. Sketch the graph of 𝑦= 2cot 𝑥 3 .

15. Sketch the graph of 𝑦= cot 𝑥 4 .

16. Sketch the graph of 𝑦= cot 𝜋𝑥 .

17. Sketch the graph of 𝑦=− csc 𝜋𝑥 2 .

18. Sketch the graph of 𝑦=2 csc 𝑥+ 𝜋 4 .

19. Sketch the graph of 𝑦= sec 2𝑥 .

20. 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.

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

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

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

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

26. How do you sketch the graphs of other trigonometric functions?

27. Ticket Out the Door
Sketch 𝑦=4 tan 2𝑥