5. Rational Functions

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).

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/(x²+4)

Graph each rational function. ƒ(x)=(3x²+3x-6)/(x²-x-12)

Graph each rational function. ƒ(x)=(20+6x-2x²)/(8+6x-2x²)

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)=(x²−1)/x

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)

Graph each rational function. ƒ(x)=(x²+8x+16)/(x²+4x-5)

Graph each rational function. ƒ(x)=(18+6x-4x²)/(4+6x+2x²)

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)

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

Follow the seven steps to graph each rational function. f(x)=2x²/(x²+4)

Follow the seven steps to graph each rational function. f(x)=− 1/(x²−4)

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

Graph the rational function.

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

Graph the rational function using transformations.

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