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1 Section 5.3 Graphs of other Trigonometric Functions Tangent and Cotangent Functions sin( x) Tangent function: f( x) tan( x) ; cos( x) 3 5 Vertical asymptotes: when cos( x ) 0, that is x,,, Domain: 3 5 x,,, (All real numbers except where there are vertical asymptotes) Range:, x-intercepts: when sin( x ) 0, that is x 0,, 2, Period: x tan x Point tan 1 4, tan 0 0 (0,0) 0 4 tan 1 4,

2 Big picture: f ( x) tan( x) 2

3 Often you will need to graph the function over just one period. In this case, you ll use the interval,. Here s the graph of f ( x) tan( x) over this interval, with 2 2 pertinent points marked. 3

4 Transformations: To graph f ( x) A tan( Bx C) D ; The period is: B Find two consecutive asymptotes by solving: Bx C and 2 Bx C. 2 Find an x-intercept by taking the average of the consecutive asymptotes. Find the x - coordinates of the points halfway between the asymptotes and the x- intercept. Evaluate the function at these values to find two more points on the graph of the function (key points). Note: If B > 1, it s a horizontal shrink. If 0 < B < 1, it s a horizontal stretch. Example: f( x) 5tan2x 1 4 Period: 2 Transformations: Vertical stretch; Horizontal shrinking; Phase shift: 4 to the right; 2 8 Vertical Shift: 1 up. 4

5 . Example 1: Find the following for f ( x) 4tan2x Period: Two Vertical Asymptotes over one period: Evaluate: f 8 5

6 Example 2: Which of the following is a vertical asymptote for f( x) 4tan2x 1? 4 A) B) C) D) x 2 x 2 x 8 3 x 8 6

7 x Example 3: Sketch f( x) 2tan 4 over one period. Period: Vertical asymptotes over one period: 7

8 POPPER for Section 5.3 Question#1: Which of the following is a point ON the graph of f ( x) 2tan(4 x) 1? A) 0,2 B),3 4 C),1 4 D),3 2 E),2 16 F) None of these 8

9 cos( x) Graph of Cotangent Function: f( x) cot( x) ; sin( x) Vertical asymptotes: when sin( x ) 0, that is x 0,, 2, Domain: x 0,, 2, (All real numbers except for the values that are vertical asymptotes.) Range:, x-intercepts: when cos( x ) 0, that is x,,, Period:

10 Often you will need to graph the function over just one period. In this case, you ll use the interval 0,. Here s the graph of f ( x) cot( x) over this interval. You can take the graph of either of these basic functions and draw the graph of a more complicated function by making adjustments to the key elements of the basic function. The key elements will be the location(s) of the asymptote(s), x intercepts, and the - 3 translations of the points at, 1 and either, 1 or,

11 To graph g( x) Acot( Bx C) D ; The period is: B Find two consecutive asymptotes by solving: Bx C 0 and BxC. Find an x-intercept by taking the average of the consecutive asymptotes. Find the x -coordinates of the points halfway between the asymptotes and the x- intercept. Evaluate the function at these values to find two more points on the graph of the function. Note: If B > 1, it s a horizontal shrink. If 0 < B < 1, it s a horizontal stretch Example: f( x) 2cot x 1 6 Period: 1 Transformations: Vertical stretch; Horizontal shrinking; 1 Phase shift: 6 to the left; 6 Vertical Shift: 1 down. 11

12 Example 4: f( x) 4cotx 6 2 Period: Two vertical asymptotes: Find 3 f? 4 12

13 Example 5: Sketch f ( x) 3cot(2 x) 13

14 Example 6: Give an equation of the form f ( x) Atan( BxC) D or f ( x) Acot( BxC) D that could represent the following graph. 14

15 Example 7: Give an equation of the form f ( x) Atan( BxC) Dor f ( x) Acot( BxC) Dthat could represent the following graph. A) f( x) 3cotx 8 B) f( x) 3tanx 8 C) f( x) 3cotx 8 D) f( x) 2cotx 1 8 E) None of these 15

16 POPPER for Section 5.3 Question#2: What is the period of the function: f ( x) 2cot(4 x ) 1? A) B) 2 C) 4 D) 2 E) 8 F) None of these 16

17 Section 5.3 Continued - Graphs of Secant and Cosecant Functions Graph of Cosecant Function: gx ( ) csc( x) 1 sin( x) Using the identity csc( x), you can conclude that the graph of g will have a vertical asymptote whenever sin( x ) 0. This means that the graph of g will have vertical asymptotes at x 0,, 2,. The easiest way to draw a graph of g( x) csc( x) is to draw the graph of f ( x) sin( x), sketch asymptotes at each of the zeros of f ( x) sin( x), then sketch in the cosecant graph. 1 gx ( ) csc( x) ; if sin( x) 0, then g(x) has a vertical asymptote. sin( x) Here s the graph of f ( x) sin( x) on the interval 5 5,

18 Next, we ll include the asymptotes for the cosecant graph at each point where sin( x ) 0. Now we ll include the graph of the cosecant function. 18

19 COSECANT FUNCTION: gx ( ) csc( x) Period: 2 Vertical Asymptote: x k, k is an integer x-intercepts: None y-intercept: None Domain: x k, k is an integer Range: (,1] [1,) Typically, you ll just graph over one period. 19

20 Transformations: To graph y Acsc( BxC) D, first graph, THE HELPER GRAPH: y Asin( BxC) D. Example: Which function can be used as the helper graph to sketch the graph of f ( x) 2csc(4 x ) 2? Answer: y2sin(4 x ) 2 20

21 Example: Which of the following is a vertical asymptote for the function f ( x) 5csc(2 x )? A) B) C) x 4 3 x 4 x 2 21

22 POPPER for Section 5.3 Question#3: What is the phase shift for the function: f ( x) 2csc(4 x ) 1? A) to the left B) 2 to the right C) to the left D) 4 to the right E) 4 to the left F) None of these 22

23 Graph of Secant Function: gx ( ) sec( x) You ll also be able to take advantage of what you know about the graph of 1 f ( x) cos( x) to help you graph g( x) sec( x). Using the identity sec( x), you cos( x) can conclude that the graph of g will have a vertical asymptote whenever cos( x ) 0. 3 This means that the graph of g will have vertical asymptotes at x,,. 2 2 The easiest way to draw a graph of g( x) sec( x) is to draw the graph of f ( x) cos( x), sketch asymptotes at each of the zeros of f ( x) cos( x), then sketch in the secant graph. 1 gx ( ) sec( x) ; if cos( x) 0, then g(x) has a vertical asymptote. cos( x) 5 5 Here s the graph of f ( x) cos( x) on the interval,

24 Next, we ll include the asymptotes for the secant graph. Now we ll include the graph of the secant function. 24

25 SECANT FUNCTION: gx ( ) sec( x) Period: 2 Vertical Asymptote:2k x k /2is an odd integer x-intercepts: None y-intercept: (0, 1) Domain: x k / 2, k is an odd integer Range: (,1] [1,) 25

26 Transformations: To graph y Asec( BxC) D, first graph, THE HELPER GRAPH: y Acos( BxC) D. Example: Which function can be used as the helper graph to sketch the graph of f ( x) 2sec(5 x) 1? Answer: y2cos(5 x) 1 27

27 Example: Which of the following is a vertical asymptote for the function f ( x) 5sec( x) 1? 1 A) x 4 1 B) x 4 C) x 1 1 D) x 2 28

28 Example: Sketch f ( x) csc2x over one period. 29

29 Example: Sketch x f ( x) 2sec over the interval 3,

30 Example: Give an equation of the form y Acsc( BxC) D or y Asec( BxC) Dthat could describe the following graph. 31

31 Example: Give an equation of the form y Acsc( BxC) D or y Asec( BxC) Dthat could describe the following graph. A) f( x) 2secx 4 B) f( x) 3secx 1 4 C) f( x) 4secx 4 D) f( x) secx

32 POPPER for Section 5.3 Question#4: Which of the following is a vertical asymptote for the function: f ( x) 4sec(2 x )? A) x B) x 2 C) x 2 3 D) x 4 E) x 0 F) None of these 33

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