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5. Rational Functions

Graphing Rational Functions

5. Rational Functions

Graphing Rational Functions

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Graphing Rational Functions Using Transformations

Graphing Rational Functions Using Transformations

How to Graph Rational Functions

How to Graph Rational Functions

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Multiple Choice

Graph the rational function using transformations.

$f\left(x\right)=-\frac{1}{x}+3$

Multiple Choice

Graph the rational function using transformations.

$f\left(x\right)=\frac{1}{\left(x+3\right)^2}-2$

Multiple Choice

Graph the rational function.

$f\left(x\right)=\frac{x+3}{x^2+5x+6}$

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=[(x-5)(x-2)]/(x^2+9)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(3x^2+3x-6)/(x^2-x-12)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=x/(4-x^2)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-3)

Textbook Question

Solve each problem. This rational function has two holes and one vertical asymptote. ƒ(x)=(x^3+7x^2-25x-175)/(x^3+3x^2-25x-75)

What are the x-values of the holes?

Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (2x - 4)/(x + 3)

Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)

Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of $y=f\left(x\right)$ . State the domain of $f$ .

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Textbook Question

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

Textbook Question

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x^3/(3x^2+1)

Textbook Question

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x^2/(3x^2+1)

Textbook Question

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x^2+4)

Textbook Question

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=x/x(x+4)

Textbook Question

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x+3)/x(x+4)

Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (4x^2 - 16x + 16)/(2x - 3)

Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. r(x) = (x^2 + 4x + 3)/(x + 2)^2

Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

Textbook Question

Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=(x^2+3x+4)/(x-5)

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=3/(x-5)

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(4-3x)/(2x+1)

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(2x+6)/(x-4)

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(4x^2+25)/(x^2+9)

Textbook Question

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). ƒ(x)=1/(x-2)^2

Textbook Question

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). ƒ(x)=1/(x-2)

Textbook Question

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). ƒ(x)=-1/(x-2)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(6-3x)/(4-x)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=5x/(x^2-1)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=[(x+6)(x-2)]/[(x+3)(x-4)]

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=[(x+3)(x-5)]/[(x+1)(x-4)]

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(x^2+8x+16)/(x^2+4x-5)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(x^2+2x+1)/(x^2-x-6)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(20+6x-2x^2)/(8+6x-2x^2)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(18+6x-4x^2)/(4+6x+2x^2)

Textbook Question

Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x^2? What is its range?

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(9x^2-1)/(x^2-4)

Textbook Question

Graph each rational function. See Examples 5–9. ƒ(x)=(16x^2-9)/(x^2-9)

Textbook Question

Provide a short answer to each question. What is the equation of the vertical asymptote of the graph of y=1/(x-3)+2? Of the horizontal asymptote?

Textbook Question

Work each problem. Which function has a graph that does not have a vertical asymptote? A. ƒ(x)=1/(x^2+2) B. ƒ(x)=1/(x^2-2) C. ƒ(x)=3/x^2 D. ƒ(x)=(2x+1)/(x-8)

Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

Textbook Question

Find a rational function ƒ having a graph with the given features. x-intercepts: (-1, 0) and (3, 0) y-intercept: (0, -3) vertical asymptote: x=1 horizontal asymptote: y=1

Textbook Question

Find a rational function ƒ having a graph with the given features. x-intercepts: (1, 0) and (3, 0) y-intercept: none vertical asymptotes: x=0 and x=2 horizontal asymptote: y=1

Textbook Question

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Textbook Question

Provide a short answer to each question. Is ƒ(x)=1/x an even or an odd function? What symmetry does its graph exhibit?

Textbook Question

Graph each rational function. ƒ(x)=4/(x-1)

Textbook Question

Graph each rational function. ƒ(x)=(4x-2)/(3x+1)

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.

As x -> -3^-, f(x) -> __

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x -> ∞, f(x) -> __

Textbook Question

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x−1)

Textbook Question

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/(x−3)^2+1

Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=4x/(x−2)

Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=2/(x^2+x−2)

Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=2x^2/(x^2+4)

Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2−1)/x

Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+1)/x

Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+x−6)/(x−3)

Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^3+1)/(x^2+2x)

Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x^2+x−12)/(x^2−4)

Textbook Question

In Exercises 95–98, use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x)=(3x−7)/(x−2)

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