Section 5.3 Graphs of the Cosecant and Secant Functions 1

Transcription

1 Section 5.3 Graphs of the Cosecant, Secant, Tangent, and Cotangent Functions The Cosecant Graph RECALL: 1 csc x so where sin x 0, csc x has an asymptote. sin x To graph y Acsc( Bx C) D, first graph THE HELPER GRAPH, y Asin( Bx C) D. f ( x) cscx Period: 2 Vertical Asymptote: x k, k is an integer Example 1: Let f ( x) 4csc 2x. 2 a. Give two asymptotes. Section 5.3 Graphs of the Cosecant and Secant Functions 1

2 b. Sketch its graph by first stating and sketching the helper graph. Helper function: Amplitude: A = Period: 2 = B Phase Shift: C B = One cycle begins at the phase shift and ends at: C 2 B B Any other transformations? Section 5.3 Graphs of the Cosecant and Secant Functions 2

3 Example 2: Give an equation of the form following graph. y Acsc( Bx C) D that could describe the Let s begin by recalling one cycle of the basic sine graph. Then choose one cycle on the graph above. Amplitude: A = M m 2 = Vertical Shift, D: It ll be half-way between the maximum and the minimum values. Use the period to find B: Recall the period formula 2 = B Compare your chosen cycle to the basic one cycle of sine. Any other transformations? Cosecant Function: Section 5.3 Graphs of the Cosecant and Secant Functions 3

4 The Secant Graph RECALL: 1 sec x so where cos x 0, sec x has an asymptote. cos x To graph y Asec( Bx C) D, first graph, THE HELPER GRAPH, y Acos( Bx C) D. f ( x) secx Period: 2 Vertical Asymptote: k x, k is an odd integer 2 Example 3: Let f( x) sec x 2. a. State the transformations. b. Give two asymptotes. Section 5.3 Graphs of the Cosecant and Secant Functions 4

5 c. Sketch its graph by first stating and sketching its helper graph. Helper function: Amplitude: A = Period: 2 = B Phase Shift: C B = One cycle begins at the phase shift and ends at: C 2 B B Any other transformations? Section 5.3 Graphs of the Cosecant and Secant Functions 5

6 Example 4: Recall the following graph from Example 2. Give an equation of the form y Asec( Bx C) D that could describe the following graph. Let s begin by recalling one cycle of the basic cosine graph. Then choose one of these cycles on the graph above. A, D, B are the same as in Example 5: A = 4 D = 3 B = 4 Now compare your chosen cycle to the original one cycle of cosine. Any phase shift? Any other transformations? Secant Function: Section 5.3 Graphs of the Cosecant and Secant Functions 6

7 The Graph of Tangent sin x Recall: tan x so where cos x 0, tan x has an asymptote and where sin x 0, cos x tan x has an x- intercept. f ( x) tanx Period: k Vertical Asymptote: x, k is an odd integer. 2 The period for the function f x) Atan Bx D ( is B. How to graph f x) Atan Bx D ( : 1. Find two consecutive asymptotes by setting Bx C equal to solve for x. and and then Divide the interval connecting the asymptotes found in step 1 into four equal parts. This will create five points. The first and last point is where the asymptotes run through. The middle point is where the x-intercept is located, assuming no vertical shift. The y- coordinates of the second and fourth points are A and A, assuming no vertical shift. Section 5.3 Graphs of the Cosecant and Secant Functions 7

8 x Example 5: Given f( x) tan 1, state its period and sketch its graph. 4 Section 5.3 Graphs of the Cosecant and Secant Functions 8

9 The Graph of Cotangent cos x Recall: cot x so where cos x 0, cot x has an x- intercept and where sin x 0, sin x cot x has an asymptote. f ( x) cotx Period: Vertical Asymptote: x k, k is an integer. The period of the function f x) Acot Bx D ( is B. How to graph f ( x) Acot Bx D : 1. Find two consecutive asymptotes by setting Bx C equal to 0 and and then solve for x. 2. Divide the interval connecting the asymptotes found in step 1 into four equal parts. This will create five points. The first and last point is where the asymptotes run through. The middle point is where the x-intercept is located, assuming no vertical shift. The y- coordinates of the second and fourth points are A and A, assuming no vertical shift. Section 5.3 Graphs of the Cosecant and Secant Functions 9

10 Example 6: Let f( x) cot x, sketch its graph. 3 Note: Given f ( x) Atan Bx C D and f x) Acot Bx D ( if A is negative, then a transformation is an x-axis reflection. Hence, the tangent graph will look similar to the cotangent graph and vice versa. Section 5.3 Graphs of the Cosecant and Secant Functions 10

11 Example 7: Give an equation of the form f x) Atan Bx D f ( x) Acot Bx D that could represent the following graph. ( and Tangent: Choose a cycle. Find A by using the y values of the second and fourth points and plug big small them into A = 2 Vertical Shift, D: It ll be half-way between the big y-value and the small y-value. Use the period to find B: Recall the period formula = B Compare your chosen cycle with the basic one cycle of tangent. Any other transformations? Tangent Function: Cotangent: Choose a cycle. For cotangent, A, D, B will be the same. However, do we have an x-axis reflection? To determine if there is a phase shift, compare the current asymptotes to the original asymptotes of one cycle of cotangent. The easiest ones to use here are x = 0 and x =. Did they move? If so, if which direction and by how much? Cotangent Function: Section 5.3 Graphs of the Cosecant and Secant Functions 11

12 Try this one: Give an equation of the form f (x) = A csc(b x - C) + D which could be used to represent the given graph. (Note: C or D may be zero.) a. 1 f( x) 2csc x 1 2 b. 1 f( x) 4csc x 1 2 c. 1 f( x) 2csc x 2 d. 1 f ( x) 2csc x 1 2 e. 1 f( x) 4csc x 1 2 Section 5.3 Graphs of the Cosecant and Secant Functions 12