1. Rational Functions MATH 109 - Precalculus S. Rook

2. Overview Section 2.6 in the textbook: – Vertical asymptotes & holes – Horizontal asymptotes – Slant asymptotes – Graphing rational functions 2

3. Vertical Asymptotes & Holes

4. 4 Definition of a Rational Function Recall f(x) = N(x) / D(x) is a rational function for polynomials N(x) and D(x) – Domain is where D(x) ≠ 0

5. 5 Undefined and No Common Factors – Vertical Asymptote Vertical Asymptote: a vertical line x = k where the value of f(x) either dives down to -oo or soars to +oo f(x) gets “extremely close” but can never touch the line x = k Factor D(x) if possible and look for values such that D(x) = 0 If the rational function f(x) = N(x) / D(x) does NOT contain x – k as a common factor [in both N(x) and D(x)]: f(x) has a vertical asymptote at x = k

6. Undefined, but with a Common Factor – If the rational function f(x) = N(x) / D(x) DOES contain x – k as a common factor [in both N(x) and D(x)]: f(x) will contain a hole at x = k e.g. 6

7. Vertical Asymptotes & Holes (Example) Ex 1: i) Find the domain ii) Identify any vertical asymptotes: a)b) c) 7

8. Horizontal Asymptotes

9. 9 Horizontal Asymptote: a horizontal line y = k where the value of f(x) is EVENTUALLY bounded by k as x approaches -oo or +oo Unlike a vertical asymptote, f(x) IS ALLOWED TO CROSS a horizontal asymptote – Just so long as x becomes infinitely large or as x becomes infinitely small, f(x) is bounded by y = k

10. 10 Horizontal Asymptotes (Continued) Given the rational function f(x) = N(x) / D(x): – Let a n x n and b m x m be the leading terms of N(x) and D(x) respectively N(x) and D(x) MUST be in descending degree! – Then the horizontal asymptote of f(x) is: y = 0 (the x-axis) if n < m – i.e. Degree of the numerator is less than the degree of the denominator y = a n / b m if n = m – i.e. Degree of the numerator equals the degree of the denominator Nonexistent if m > n – i.e. Degree of the numerator is greater than the degree of the denominator

11. Horizontal Asymptotes (Example) Ex 2: Identify the horizontal asymptote if it exists: a)b) c) 11

12. Slant Asymptotes

13. 13 Slant Asymptotes Some rational functions have neither vertical nor horizontal asymptotes, but asymptotes of the form y = mx + b Given f(x) = N(x) / D(x), let a n x n and b m x m be the leading terms of N(x) and D(x) (in degree order) respectively f(x) has a slant asymptote if m = n + 1 i.e. the degree of the numerator is ONE greater than the denominator

14. 14 Slant Asymptotes (Continued) To find the slant asymptote of the rational function f(x) = N(x) / D(x): – Ensure that f(x) meets the criteria for having a slant asymptote – Perform polynomial long division of D(x) into N(x) The quotient is the equation of the slant asymptote in y = mx + b format

15. Slant Asymptotes (Example) Ex 3: i) State whether or not the function has a slant asymptote and ii) if it does, find it a)b) 15

16. Graphing Rational Functions

17. 17 Graphing Rational Functions To graph a rational function f(x) = N(x) / D(x): – Simplify f(x) by factoring and dividing out common factors if they exist – Sketch the vertical asymptotes for x – k that are not common factors and holes for x – k that are common factors – Sketch the horizontal asymptote or slant asymptote if it exists – Plot the y-intercept if it exists – Find the x-intercepts Those values of x such that N(x) = 0 and D(x) ≠ 0

18. Graphing Rational Functions (Continued) – Use the zeros and asymptotes to divide (-oo, +oo) into subintervals – Pick additional points in each subinterval, especially near any vertical asymptotes Recall that the value of the function has the same sign for EVERY value in a particular interval 18

19. Graphing Rational Functions (Example) Ex 4: i) State the domain ii) Identify all intercepts iii) Find any vertical, horizontal, or slant asymptotes iv) Plot additional points in each subinterval to sketch the function a)b) c) 19

20. Summary After studying these slides, you should be able to: – Identify the domain of a rational function – State where the vertical asymptotes and/or holes lie on a rational function – Find the horizontal asymptote if it exists – Find the slant asymptote if it exists – Graph a rational function Additional Practice – See the list of suggested problems for 2.6 Next lesson – Nonlinear Inequalities (Section 2.7) 20