Chapter 3.5 Rational Functions

Pindling College Algebra   by Example  Series

Key Concepts: Understand rational functions, their asymptotes and properties for graphing them

Skills to Learn

1. Know the basic features of a simple rational function: e.g. Average Hourly Cost

2. Know how to determine the domain of rational functions

3. Know how to find the vertical asymptotes of rational functions

4. Know how to find the horizontal asymptote of rational functions

5. Know how to graph rational functions

6. Know how to graph missing point on a graph

Definition of rational function

A rational function, r(x) is the ratio of two polynomials, P(x) and Q(x): Examples are:

Basic Properties of Rational Functions

Example 1. If the average cost is the total cost, C divided by the number of items, n and the total cost is C = P(n), where P(n) = 2.25n + 5 show the following: (a) Formula for average cost: (b) What is the average cost when 10 items are sold? So n = 10 and (b) Find n to give and average cost of 5. So

(e) Graph the rational function

Domains of Rational Functions

Example 2. Find the domain of the rational function Note: the domain of a rational function is determined by the restriction imposed by the denominator; here (division by zero) And denominator is not equal to 0 when x = 2 and -2 Since So for the domain or Values of x within	Example 3. Find the domain of the rational function Note: the domain of a rational function is determined by the restriction imposed by the denominator; here (division by zero) And denominator is not equal to 0 when x = -3 Since So domain is all x except or or Values of x within
Example 4. Find the domain for Note: the domain of a rational function is determined by the restriction imposed by the denominator; here (division by zero) And denominator is not equal to 0 when x = -1, 0 and 1 Since So domain is or Values of x within	Example 5. Find the domain for Note: the domain of a rational function is determined by the restriction imposed by the denominator; here (division by zero) And denominator is not equal to 0 when x = - ½ and 3 Since So domain is or Values of x within

Vertical and Horizontal Asymptotes of Rational Functions

Vertical Asymptotes are found at values of x for which the r(x) gives the denominator = 0. So for , Vertical Asymptotes (VA) occurs at values of x when Horizontal Asymptotes: Take the ratio of the highest degrees of P(x) and Q(x) and if (1) The degree of for , then the Horizontal Asymptote, HA = 0 (2) The degree of for , there is no Horizontal Asymptote or (there is a slant asymptote, i.e. The asymptote is a linear function) (3) The degree of for , and the coefficients of P(x) is a and the coefficient of Q(x) is b, then the Horizontal Asymptote, HA =

Example 6. Find the asymptotes for VA at , since HA at y = 0: , since degree of

Example 7. Find the asymptotes for VA at , since No HA since , or degree of

Example 8. Find the asymptotes for VA at , since HA at y = 3: , since degree of

Example 9. Find the asymptotes for VA at , since HA at y = 1, since , or degree of

Example 10. Find the asymptotes for VA at , since HA at y = 0, =>0, or degree of

Example 11. Find the asymptotes for VA at , since HA at y = -½ , since , or degree of

Note: Slant Asymptotes to be shown in class.

Graphing Rational Functions:

Steps in Graphing Rational Functions: 1. Check for symmetry: (a) Symmetry about y-axis if or have only even powers of x. (b) Symmetry about the origin if 2. Find the vertical Asymptotes: VA are x values when Q(x) = 0 3. Find the Horizontal Asymptotes: HA given the ratio of highest degrees of P(x) and Q(x): (1) When degree of Q(x) > P(x): HA = 0 (2) When degree of P(x) > Q(x): There is no HA (however slant asymptote) (3) When degree of P(x) = Q(x): HA = a, where a is the ratio of the coefficients of the highest degree of P(x) / Q(x) 4. Find the y-intercept: value of y when x = 0 5. Find the x-intercept(s): values of x when P(x) = 0 6. Plots a few point: within the domains given by the VA.

Example 12. Graph (1) Symmetry about y-axis (even powers) (2) VA at x = 0 (since (3) HA at y = 3 ( (4) y-intercept: when x = 0 , y = Undefined (so no y-intercept) (5) x-intercepts: x = -2 and 2 when y = 0,

So graph is:

Example 12. Graph (1) No Symmetry about y-axis or origin (2) VA at x = 0,-2, 2, since (3) HA at y = 0 ( (4) y-intercept: when x = 0 , y = Undefined (so no y-intercept) (5) x-intercepts: x = -3 and 1 when y = 0,

So graph is:

Example 13. Graph (1) Symmetry about y-axis (even powers) (2) VA at x = -½ , ½, since (3) HA at y = ¼ ( (4) y-intercept: when x = 0 , y = 1 (5) x-intercepts: x = -1 and 1 when y = 0,

So graph is:

Graphs with Holes or Missing point - No x or y-values

Holes are missing points in rational functions where both P(x) and Q(x) share the same factor The hole is the point for the root of that common factor: Example 15: for the rational function , (x+1) is a common factor with root or x = -1 so the point for the hole is (-1, f(-1) ), note f(-1) has no solution.

Graph of function with hole:

Example 16. Grpah with hole at x = 1 , common factor (x -1)

Example 16. Grpah with hole at x = 0 Common factors of (x + 3) and (x - 5)