Graphing Rational Functions

Transcription

1 Unit 1 R a t i o n a l F u n c t i o n s Graphing Rational Functions Objectives: 1. Graph a rational function given an equation 2. State the domain, asymptotes, and any intercepts Why? The function describes the concentration of a drug in the blood stream over time. If we graph the function we will see what the concentration would be at any time, t. 1

2 What are rational functions? A function that is the ratio of two polynomials. The denominator cannot equal 0. image from mathisfun.com 2

3 The domain is found by finding the excluded values. These are values that would make the denominator zero. To find, set the denominator equal to zero and solve to find the excluded values of the function. These are also the vertical asymptotes. Examples 1. 2x 2. x x x x 2 = -4 x = -2 no excluded no excluded excluded value value value -2 3

4 Find the domain and vertical asymptote(s): a) b) c) D:{x x=3} VA: x=3 D:{x x=-2,1} VA: x=-2, 1 d:{x x=3} VA: x=3 as as 4

5 Find the domain and vertical asymptotes. a. b. c. D: {x x=-7} D: {x x=-4,4} D:{x x=2/5} VA: x=-7 VA: x=-4,4 VA: x=2/5 5

6 Horizontal Asymptotes: If the degree of the denominator is larger than the numerator, the asymptote is the x axis or y=0. If the degree of the numerator and the denominator are the same, the asymptote is the ratio of the leading coeff. of numerator coefficients. (y= coeff. of denominator ) If the degree of the numerator is larger than the denominator, there is an oblique or slant asymptote. (divide the denominator into the numerator to find) There can only be one horizontal or one slant asymptote and never both. Find the horizontal or slant asymptotes: a) b) c) 6

7 Crossing an Asymptote: The graph of a rational function never crosses a vertical asymptote. The graph of a rational function might cross a horizontal asymptote but does not necessarily do so. 7

8 A tale of discontinuity... Example Picture 1. Holes - x value that does not exist. Occurs when the numerator and denominator have a common factor Find the common factors in the numerator and denominator Set the factor equal to zero and solve The domain cannot contain holes or vertical asymptotes 8

9 Intercepts Example X - Intercepts when f(x) = 0 How to find: 1. Set the numerator equal to zero when x = 0 How to find: Y-Intercepts 1. Let x = 0 and plug it in for all x values. Simplify. 9

10 Steps for graphing: How to FACTOR the numerator and denominator 2. Find points of discontinuity 3. Find x and y intercepts 4. Find all asymptotes 5. Graph curves 10

11 1. Factor the numerator and denominator. 2. Find points of discontinuity (common factors) the hole(s) 3. Determine the domain. (no excluded values or holes) Set the denominator equal to zero and solve. 4. Find vertical asymptotes. X = excluded values above. 5. Find the horizontal or oblique asymptote, if there is one, and sketch it. Look at degree of top and bottom. 6. Find the x intercepts or zeros. Set the numerator = 0 & solve. 7. Find the y intercept. Plug in zero for x's. 8. Find other function values to determine the general shape. Then draw the graph. 11

12 Graph 1. Factor: 2. No common factors so no holes. 3. Domain: find excluded values: D: x 1/2 or 3 4. VA: x= x=3 5. HA:y=0 because bottom degree is larger than top. 6. X intercepts: Set top = 0 x int. = ( 3,0) 7. Y intercepts: Plug in 0 for x's. y int. = (0, 1) 8. Make an x y chart to find other values to graph. 12

13 Example: Back to Steps 13

14 1 Holes: y 6 5 V.A: H.A: X-Intercept: x Y-Intercept: Domain: Range: 14

15 2 Holes: y 6 5 V.A: H.A: X-Intercept: x Y-Intercept: Domain: 15

16 3 Holes: y 6 5 V.A: H.A: X-Intercept: x Y-Intercept: Domain: 16

17 4 Holes: y 6 5 V.A: H.A: X-Intercept: x Y-Intercept: Domain: 17

18 5 Holes: y 6 5 V.A: H.A: X-Intercept: x Y-Intercept: Domain: 18

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