9.3 Intro to Rational Functions
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This document discusses rational functions and their graphs. It defines a rational function as a function made by dividing polynomials. It examines where discontinuities occur in rational functions, which is when the denominator equals 0, creating vertical asymptotes. Holes are created when both the numerator and denominator equal 0. The document provides examples of finding vertical asymptotes and holes. It also discusses how the highest powers in the numerator and denominator determine if there is a horizontal asymptote and, if so, its value. Examples are given of finding horizontal asymptotes of various rational functions. The document concludes with an example of sketching the graph of a rational function using all identified features.
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9.3 Intro to Rational Functions
- 2. 9.3 Rational Functions and Their Graphs
- 3. A rational function is. . .
- 4. Examples of Rational Functions Where does the denominator = 0?
- 5. Examples of Rational Functions Where does the denominator = 0?
- 6. Examples of Rational Functions Where does the denominator = 0?
- 7. A point of discontinuity occurs when. . . It causes or in the graph.
- 8. Example 1. Find the points of discontinuity. a. 1x2x 1 y 2 ++ = b. 1x 1x y 2 + +− =
- 9. If a number is a solution to only the denominator of a rational function, then it creates an __________________ in the function. ex. )2x)(1x( 1x y +− + =
- 10. If a number is a solution to both the numerator and denominator of a rational function, then it creates a _____________ in the function. ex. )1x)(2x( 2x y −− − =
- 11. Example 2. Describe the vertical asymptotes and holes for the graph of each rational function. a. b.)3x)(2x( 1x y −− + = 2x )1x)(2x( y − −− =
- 13. Horizontal Asymptotes A horizontal asymptote depends on the biggest power of the numerator and denominator. 1. If n > d, 2. If n < d, 3. If n = d,
- 14. Example 3. Find the horizontal asymptotes of each function. a. b. c. 1x 6x2 y − +− = 2 x 1 y = x5 x2x y 3 + =
- 15. Example 4. Sketch the graph of Points of discontinuity: Holes: Vertical Asymptotes: Horizontal Asymptotes: )4x)(3x( 2x y −+ + =
- 16. Assignment: