1. Rational Functions - Rational functions are quotients of polynomial functions: where P(x) and Q(x) are polynomial functions and Q(x)  0. -The domain of a rational function is the set of all real numbers except for the x- values that make the denominator zero. For example, the domain of the rational function is the set of all real numbers except 0, 2, and -5. This is P(x). This is Q(x). 3.6: Rational Functions and Their Graphs

2. Rational Functions Graphs of rational functions have breaks in them and can have distinct branches. We use a special arrow notation to help describe this situation symbolically: Arrow Notation SymbolMeaning x  a  x approaches a from the right. x  a  x approaches a from the left. x   x approaches infinity; that is, x increases without bound. x    x approaches negative infinity; that is, x decreases without bound. Arrow Notation SymbolMeaning x  a  x approaches a from the right. x  a  x approaches a from the left. x   x approaches infinity; that is, x increases without bound. x    x approaches negative infinity; that is, x decreases without bound. 3.6: Rational Functions and Their Graphs

3. 3.6 Rational Functions and Graphs Consider the graph of : 1. What happens as ? 2. What happens as 3. What happens as 4. What happens as

4. Definition of a Vertical Asymptote The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  Definition of a Vertical Asymptote The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  Vertical Asymptotes of Rational Functions Thus, f (x)   as x approaches a from either the left or the right. f a y x x = a f a y x 3.6: Rational Functions and Their Graphs more

5. Definition of a Vertical Asymptote The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. Definition of a Vertical Asymptote The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. Vertical Asymptotes of Rational Functions Thus, f(x)    as x approaches a from either the left or the right. x = a f a y x f a y x f (x)    as x  a  f (x)    as x  a  3.6: Rational Functions and Their Graphs

6. Vertical Asymptotes of Rational Functions If the graph of a rational function has vertical asymptotes, they can be located in the following way: Locating Vertical Asymptotes If is a rational function in which p(x) and q(x) have no common factors and a is a zero of Q(x), then x  a is a vertical asymptote of the graph of f(x). Locating Vertical Asymptotes If is a rational function in which p(x) and q(x) have no common factors and a is a zero of Q(x), then x  a is a vertical asymptote of the graph of f(x). 3.6: Rational Functions and Their Graphs

7. Definition of a Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. Definition of a Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. Horizontal Asymptotes of Rational Functions A rational function may have several vertical asymptotes, but it can have at most one horizontal asymptote. f y x y = b x y f f y x f (x)  b as x   f (x)  b as x   f (x)  b as x   3.6: Rational Functions and Their Graphs

8. Horizontal Asymptotes of Rational Functions If the graph of a rational function has a horizontal asymptote, it can be located in the following way: Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n  m, the x-axis is the horizontal asymptote of the graph of f. 2.If n  m, the line y  is the horizontal asymptote of the graph of f. 3.If n  m, the graph of f has no horizontal asymptote. Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n  m, the x-axis is the horizontal asymptote of the graph of f. 2.If n  m, the line y  is the horizontal asymptote of the graph of f. 3.If n  m, the graph of f has no horizontal asymptote. 3.6: Rational Functions and Their Graphs

9. Strategy for Graphing a Rational Function Suppose that where P(x) and Q(x) are polynomial functions with no common factors. 1. Factor: Factor numerator and denominator. 2. Intercepts: Find the y-intercept by evaluating f (0). Find the x-intercepts by finding the zeros of the numerator. 3. Asymptotes: Find the horizontal asymptote by using the rule for determining the horizontal asymptote of a rational function. Find the vertical asymptote(s) by finding the zeros of the denominator. 4. Behavior: check the behavior on each side of vertical asymptote(s). 5. Sketch graph. (Plot at least one point between and beyond each x-intercept and vertical asymptote.) Strategy for Graphing a Rational Function Suppose that where P(x) and Q(x) are polynomial functions with no common factors. 1. Factor: Factor numerator and denominator. 2. Intercepts: Find the y-intercept by evaluating f (0). Find the x-intercepts by finding the zeros of the numerator. 3. Asymptotes: Find the horizontal asymptote by using the rule for determining the horizontal asymptote of a rational function. Find the vertical asymptote(s) by finding the zeros of the denominator. 4. Behavior: check the behavior on each side of vertical asymptote(s). 5. Sketch graph. (Plot at least one point between and beyond each x-intercept and vertical asymptote.)

10. EXAMPLE:Finding the Slant Asymptote of a Rational Function Find the slant asymptotes of f (x)  Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x  3 into x 2  4x  5: 2 1  4  5 1 3  3 1  1  8 3 Remainder 3.6: Rational Functions and Their Graphs more

11. EXAMPLE:Finding the Slant Asymptote of a Rational Function Find the slant asymptotes of f (x)  Solution The equation of the slant asymptote is y  x  1. Using our strategy for graphing rational functions, the graph of f (x)  is shown. -245678321 7 6 5 4 3 1 2 -3 -2 Vertical asymptote: x = 3 Vertical asymptote: x = 3 Slant asymptote: y = x - 1 Slant asymptote: y = x - 1 3.6: Rational Functions and Their Graphs

12. Locating Horizontal Asymptotes: Divide the numerator and denominator by the highest power of x that appears in denominator and then let Horizontal Asymptotes of Rational Functions If the graph of a rational function has a horizontal asymptote, it can be located in the following way: 3.6: Rational Functions and Their Graphs