Warm Up Find the zeros of the following function F(x) = x2 -1 Factor the following function X2 + x – 2 Simplify the expression: x2 – x – 6

Math IV Lesson12 Rational functions and asymptotes 2.6 Essential Question: How do you find the domain and asymptotes of rational functions? Standard: MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

Rational Functions and Their Graphs 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).

EXAMPLE: Finding the Domain of a Rational Function Rational Functions and Their Graphs 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}. more

EXAMPLE: Finding the Domain of a Rational Function Rational Functions and Their Graphs 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. 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}. more

EXAMPLE: Finding the Domain of a Rational Function 3.6: Rational Functions and Their Graphs 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. c. No real numbers cause the denominator of to equal zero. The domain of h consists of all real numbers.

Rational Functions Arrow Notation 3.6: Rational Functions and Their Graphs 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 Symbol Meaning 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.

Vertical Asymptotes of Rational Functions 3.6: Rational Functions and Their Graphs Vertical Asymptotes of Rational Functions 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 - f a y x x = a f a y x x = a Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right. more

Vertical Asymptotes of Rational Functions 3.6: Rational Functions and Their Graphs Vertical Asymptotes of Rational Functions 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 - f a y x x = a x = a f a y x Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.

Vertical Asymptotes of Rational Functions 3.6: Rational Functions and Their Graphs 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.

Horizontal Asymptotes of Rational Functions 3.6: Rational Functions and Their Graphs Horizontal Asymptotes of Rational Functions A rational function may have several vertical asymptotes, but it can have at most one horizontal asymptote. 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. f y x y = b x y f y = b f y x y = b f (x)  b as x   f (x)  b as x   f (x)  b as x  

Horizontal Asymptotes of Rational Functions 3.6: Rational Functions and Their Graphs 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. If n < m, the x-axis is the horizontal asymptote of the graph of f. If n = m, the line y = is the horizontal asymptote of the graph of f. If n > m, the graph of f has no horizontal asymptote.

f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry 3.6: 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.

EXAMPLE: Graphing a Rational Function 3.6: Rational Functions and Their Graphs EXAMPLE: Graphing a Rational Function Solution Step 1 Determine symmetry: f (-x) = = = f (x): Symmetric with respect to the y-axis. Step 2 Find the y-intercept: f (0) = = 0: y-intercept is 0. Step 3 Find the x-intercept: 3x2 = 0, so x = 0: x-intercept is 0. Step 4 Find the vertical asymptotes: Set q(x) = 0. x2 - 4 = 0 Set the denominator equal to zero. x2 = 4 x = 2 Vertical asymptotes: x = -2 and x = 2. more

EXAMPLE: Graphing a Rational Function 3.6: Rational Functions and Their Graphs EXAMPLE: Graphing a Rational Function Solution Step 5 Find the horizontal asymptote: y = 3/1 = 3. Step 6 Plot 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. -5 -4 -3 -2 -1 1 2 3 4 5 7 6 Vertical asymptote: x = 2 Vertical asymptote: x = -2 Horizontal asymptote: y = 3 x-intercept and y-intercept x -3 -1 1 3 4 f(x) = The figure shows these points, the y-intercept, the x-intercept, and the asymptotes. more

EXAMPLE: Graphing a Rational Function 3.6: Rational Functions and Their Graphs EXAMPLE: Graphing a Rational Function Solution Step 7 Graph the function. The graph of f (x) = 2x is shown in the figure. The y-axis symmetry is now obvious. -5 -4 -3 -2 -1 1 2 3 4 5 7 6 Vertical asymptote: x = 2 Vertical asymptote: x = -2 Horizontal asymptote: y = 3 x-intercept and y-intercept -5 -4 -3 -2 -1 1 2 3 4 5 7 6 x = -2 y = 3 x = 2

EXAMPLE: Finding the Slant Asymptote of a Rational Function 3.6: Rational Functions and Their Graphs 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 x2 - 4x - 5: 1 -4 -5 3 -3 1 -1 -8 3 Remainder more

EXAMPLE: Finding the Slant Asymptote of a Rational Function 3.6: Rational Functions and Their Graphs 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. -2 -1 4 5 6 7 8 3 2 1 -3 Vertical asymptote: x = 3 Slant asymptote: y = x - 1