1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Graphs of the Circular Functions

2. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2 4.1 Graphs of the Sine and Cosine Functions 4.2 Translations of the Graphs of the Sine and Cosine Functions 4.3 Graphs of the Tangent and Cotangent Functions 4.4 Graphs of the Secant and Cosecant Functions 4.5Harmonic Motion 4 Graphs of the Circular Functions

3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Graphs of the Tangent and Cotangent Functions 4.3 Graph of the Tangent Function ▪ Graph of the Cotangent Function ▪ Graphing Techniques ▪ Connecting Graphs with Equations

4. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Graph of the Tangent Function A vertical asymptote is a vertical line that the graph approaches but does not intersect. As the x-values get closer and closer to the line, the function values increase or decrease without bound.

5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Tangent Function f(x) = tan x

6. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Tangent Function f(x) = tan x  The graph is discontinuous at values of x of the form and has vertical asymptotes at these values.  Its x-intercepts are of the form x = n .  Its period is .  Its graph has no amplitude, since there are no minimum or maximum values.  The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, tan(–x) = –tan(x).

7. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Cotangent Function f(x) = cot x

8. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Cotangent Function f(x) = cot x  The graph is discontinuous at values of x of the form x = n  and has vertical asymptotes at these values.  Its x-intercepts are of the form.  Its period is .  Its graph has no amplitude, since there are no minimum or maximum values.  The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, cot(–x) = –cot(x).

9. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Tangent and Cotangent Functions To graph the cotangent function, we must use one of the identities The tangent function can be graphed directly with a graphing calculator using the tangent key. because graphing calculators generally do not have cotangent keys.

10. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Guidelines for Sketching Graphs of Tangent and Cotangent Functions Step 2Sketch the two vertical asymptotes found in Step 1. Step 1Determine the period, To locate two adjacent vertical asymptotes, solve the following equations for x: Step 3Divide the interval formed by the vertical asymptotes into four equal parts.

11. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Guidelines for Sketching Graphs of Tangent and Cotangent Functions Step 4Evaluate the function for the first- quarter point, midpoint, and third- quarter point, using the x-values found in Step 3. Step 5Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

12. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Example 1 GRAPHING y = tan bx Graph y = tan 2x. Step 1The period of this function is To locate two adjacent vertical asymptotes, solve The asymptotes have equations and

13. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Example 1 GRAPHING y = tan bx (continued) Step 2Sketch the two vertical asymptotes.

14. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Example 1 GRAPHING y = tan bx (continued) Step 3Divide the interval into four equal parts. first-quarter value: middle value: 0 third-quarter value: Step 4Evaluate the function for the x-values found in Step 3.

15. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 1 GRAPHING y = tan bx (continued) Step 5Join these points with a smooth curve, approaching the vertical asymptotes. Graph another period by adding one half period to the left and one half period to the right.

16. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Example 2 GRAPHING y = a tan bx Divide the interval (– ,  ) into four equal parts to obtain the key x-values of The period is Adjacent vertical asymptotes are at x = –  and x = – . Evaluate the function for the x-values found in Step 3 to obtain the key points

17. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Example 2 GRAPHING y = a tan bx (continued) Plot the asymptotes and the points found in step 4. Join them with a smooth curve. Because the coefficient –3 is negative, the graph is reflected across the x-axis compared to the graph of

18. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Note The function defined by has a graph that compares to the graph of y = tan x as follows: The period is larger because The graph is “stretched” vertically because a = –3, and |–3| > 1.

19. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Each branch of the graph falls from left to right (the function decreases) between each pair of adjacent asymptotes because a = –3, and –3 < 0. When a < 0, the graph is reflected across the x-axis compared to the graph of y = |a| tan bx.

20. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 Example 3 GRAPHING y = a cot bx Divide the interval into four equal parts to obtain the key x-values of Evaluate the function for the x-values found in Step 3 to obtain the key points The period is To locate two adjacent vertical asymptotes, solve 2x = 0 and 2x =  to obtain x = 0 and

21. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 Example 3 GRAPHING y = a cot bx (continued) Plot the asymptotes and the points found in step 4. Join them with a smooth curve.

22. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 Example 4 GRAPHING y = c + tan x Graph y = 2 + tan x. Every y value for this function will be 2 units more than the corresponding y value in y = tan x, causing the graph to be translated 2 units up compared to the graph of y = tan x.

23. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 Example 4 To see the vertical translation, observe the coordinates displayed at the bottoms of the screens. GRAPHING y = c + tan x (cont)

24. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 Example 5 GRAPH y = c + acot(x – d) The period is  because b = 1. The graph will be translated down 2 units because c = –2. The graph will be reflected across the x-axis because a = –1. The phase shift is units to the right.

25. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 Example 5 GRAPH y = c + acot(x – d) (cont’d) To locate adjacent asymptotes, solve Divide the interval into four equal parts and evaluate the function at the three key x-values within the interval give these points.

26. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 Example 5 Plot the asymptotes and key points, then join them with a smooth curve. An additional period to the left has been graphed. GRAPH y = c + acot(x – d) (cont’d)

27. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 Example 6a Determine an equation for each graph. This graph is that of y = tan x but reflected across the x-axis and stretched vertically by a factor of 2. Therefore, an equation for this graph is y = –2 tan x. DETERMINING AN EQUATION FOR A GRAPH

28. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 Example 6b Determine an equation for each graph. This is the graph of a cotangent function, but the period is rather than. Therefore, the coefficient of x is 2. This graph is vertically translated 1 unit down compared to the graph of y = cot 2x. An equation for this graph is y = –1 + cot 2x. DETERMINING AN EQUATION FOR A GRAPH

29. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 Note Because the circular functions are periodic, there are infinitely many equations that correspond to each graph in Example 6. Confirm that both are equations for the graph in Example 6(b).

30. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 Note When writing the equation from a graph, it is practical to write the simplest form. Therefore, we choose values of b where b > 0 and write the function without a phase shift when possible.