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**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

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**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.

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**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).

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**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

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**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

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**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.

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**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.

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**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

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**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.

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**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.

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**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

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**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.

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**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) = Find any vertical asymptote(s) by solving the equation q (x) = 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.

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**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 = Set the denominator equal to zero. x2 = 4 x = 2 Vertical asymptotes: x = -2 and x = 2. more

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**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

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**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

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**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: 3 Remainder more

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**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

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