1. Rational Functions Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as 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 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). 3.3: Rational Functions and Their Graphs

2. EXAMPLE: Finding the Domain of a Rational Function Find the domain of each rational function. a. Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. a. The denominator of is 0 if x = 3. Thus, x cannot equal 3. The domain of f consists of all real numbers except 3, written {x | x  3}. 3.3: Rational Functions and Their Graphs more

3. EXAMPLE: Finding the Domain of a Rational Function Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. Find the domain of each rational function. a. b. The denominator of is 0 if x =  3 or x  3. Thus, the domain of g consists of all real numbers except  3 and 3, written {x | x   {x | x   3, x  3}. 3.3: Rational Functions and Their Graphs more

4. EXAMPLE: Finding the Domain of a Rational Function Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. Find the domain of each rational function. a. c. No real numbers cause the denominator of to equal zero. The domain of h consists of all real numbers. 3.3: Rational Functions and Their Graphs

5. Rational Functions Unlike the graph of a polynomial function, the graph of the reciprocal function has a break in it and is composed of two distinct branches. We use a special arrow notation to 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. 3.3: Rational Functions and Their Graphs

6. 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)   or f(x)    as x approaches a from either the left or the right. f a y x x = a f a y x 3.3: Rational Functions and Their Graphs more

7. 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)   or 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  3.3: Rational Functions and Their Graphs

8. Vertical Asymptotes of Rational Functions If the graph of a rational function has vertical asymptotes, they can be located by using the following theorem. 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), the denominator, then x  a is a vertical asymptote of the graph of f. 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), the denominator, then x  a is a vertical asymptote of the graph of f. 3.3: Rational Functions and Their Graphs

9. Finding Vertical Asymptotes Example 1 If First simplify the function. Factor both numerator and denominator and cancel any common factors.

10. The asymptote(s) occur where the simplified denominator equals 0. The vertical line x=3 is the only vertical asymptote for this function. As the input value x to this function gets closer and closer to 3 the function itself looks more and more like the vertical line x=3.

11. Graph of Example 1 The vertical dotted line at x = 3 is the vertical asymptote

12. Finding Vertical Asymptotes Example 2 If Factor both the numerator and denominator and cancel any common factors. In this case there are no common factors to cancel.

13. The denominator equals zero whenever either or This function has two vertical asymptotes, one at x = -2 and the other at x = 3

14. Graph of Example 2 The two vertical dotted lines at x = -2 and x = 3 are the vertical asymptotes

15. 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   3.3: Rational Functions and Their Graphs

16. Horizontal Asymptotes of Rational Functions If the graph of a rational function has a horizontal asymptote, it can be located by using the following theorem. 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 + 1, the quotient is in the form ax + b, and the line y = ax + b is called an oblique (or slant) asymptote. 4.If n  m + 1, the graph of f has neither a horizontal asymptote or an oblique 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 + 1, the quotient is in the form ax + b, and the line y = ax + b is called an oblique (or slant) asymptote. 4.If n  m + 1, the graph of f has neither a horizontal asymptote or an oblique asymptote. 3.3: Rational Functions and Their Graphs

17. Finding Horizontal Asymptotes Example 3 If then there is a horizontal asymptote at the line y=0 because the degree of the numerator (2) is less than the degree of the denominator (3). This means that as x gets larger and larger in both the positive and negative directions (x → ∞ and x → -∞) the function itself looks more and more like the horizontal line y = 0

18. Graph of Example 3 The horizontal line y = 0 is the horizontal asymptote.

19. Finding Horizontal Asymptotes Example 4 If then because the degree of the numerator (2) is equal to the degree of the denominator (2) there is a horizontal asymptote at the line y= 6 / 5. Note, 6 is the leading coefficient of the numerator and 5 is the leading coefficient of the denominator. As x→∞ and as x→-∞ g(x) looks more and more like the line y= 6 / 5

20. Graph of Example 4 The horizontal dotted line at y = 6 / 5 is the horizontal asymptote.

21. 3.3: Rational Functions and Their Graphs Strategy for Graphing a Rational Function Suppose that where p(x) and q(x) are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f (  x)  f (x): y-axis symmetry f (  x)  f (x): origin symmetry 2. Find the y-intercept (if there is one) by evaluating f (0). 3. Find the x-intercepts (if there are any) by solving the equation p(x)  0. 4. Find any vertical asymptote(s) by solving the equation q (x)  0. 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes. Strategy for Graphing a Rational Function Suppose that where p(x) and q(x) are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f (  x)  f (x): y-axis symmetry f (  x)  f (x): origin symmetry 2. Find the y-intercept (if there is one) by evaluating f (0). 3. Find the x-intercepts (if there are any) by solving the equation p(x)  0. 4. Find any vertical asymptote(s) by solving the equation q (x)  0. 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

22. EXAMPLE:Graphing a Rational Function Step 4Find the vertical asymptotes: Set q(x)  0. x 2  4  0 Set the denominator equal to zero. x 2  4 x   2 Vertical asymptotes: x  2 and x  2. Solution more Step 3Find the x-intercept: 3x 2  0, so x  0: x-intercept is 0. Step 1Determine symmetry: f (  x)    f (x): Symmetric with respect to the y-axis. Step 2Find the y-intercept: f (0)  0: y-intercept is 0. 3.3: Rational Functions and Their Graphs

23. EXAMPLE:Graphing a Rational Function Solution The figure shows these points, the y-intercept, the x-intercept, and the asymptotes. x 33 11 134 f(x) 11 11 4 more Step 6Plot points between and beyond the x-intercept and the vertical asymptotes. With an x-intercept at 0 and vertical asymptotes at x  2 and x  2, we evaluate the function at  3,  1, 1, 3, and 4. Step 5Find the horizontal asymptote: y  3 / 1  3. -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 Vertical asymptote: x = 2 Vertical asymptote: x = -2 Horizontal asymptote: y = 3 x-intercept and y-intercept 3.4: Rational Functions and Their Graphs

24. EXAMPLE:Graphing a Rational Function Solution Step 7Graph the function. The graph of f (x) is shown in the figure. The y- axis symmetry is now obvious. -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 x = -2 y = 3 x = 2 -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 Vertical asymptote: x = 2 Vertical asymptote: x = -2 Horizontal asymptote: y = 3 x-intercept and y-intercept 3.4: Rational Functions and Their Graphs

25. 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 3.3: Rational Functions and Their Graphs more

26. 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 3.3: Properties of Rational Function and Their Graphs