5.3 Graphs of rational functions

Let [math]c[/math] be a positive real number. If rational function [math]f(x)=\displaystyle{\frac{1}{x^2+cx+5}}[/math] has only one asymptote when [math]c>\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

Let [math]f(x)=\dfrac{(x-1)^2(x+2)(2x+3)}{(x-1)(x+2)^2(x-3)}[/math]. Which of the following statements is true?

Let [math]f(x)=\dfrac{x^4+x^3-x^2-x-5}{x^2+x+2}[/math] and [math]g(x)[/math] be a polynomial function. If [math]h(x)=f(x)-g(x)[/math] has a horizontal asymptote [math]y=0[/math], find [math]g(x)[/math].

If [math]y=a[/math] is a horizontal asymptote of [math]\displaystyle{\frac{-5x^2+4x-3}{3x^2-4x+5}}[/math] where [math]a[/math] is a constant, what is the value of [math]a[/math]?

If [math]y=ax+b[/math] is a slant asymptote of [math]\displaystyle{\frac{2x^3-9x^2-2x-15}{x^2-4x-5}}[/math] where [math]a[/math] and [math]b[/math] are constants, what is the value of [math]a+b[/math]?

How many asymptotes does [math]\displaystyle{\frac{x-1}{x^3+1}}[/math] have?

Write the title of the appropriate graph below the corresponding formula. Please write Roman numerals

(a) [math]y=\dfrac{x^2-x-2}{x^2-3x}[/math]

(b) [math]y=\dfrac{2x^2+6x+4}{x^2+3x}[/math]

(c) [math]y=\dfrac{x-1}{x^2-2x-3}[/math]

(d) [math]y=\dfrac{x-2}{x^2-3x+2}[/math]

(e) [math]y=\dfrac{1-x}{x^2+x-6}[/math]

(f) [math]y=\dfrac{x^2+x+1}{x+1}[/math]

(g) [math]y=\dfrac{2x^2-7x+3}{x-3}[/math]

(h) [math]y=\dfrac{6x}{x^2+1}[/math]

(i) [math]y=\dfrac{-x^2+4x-3}{x-2}[/math].

What is the [math]y[/math]-coordinate of the hole in the graph of [math]f(x)=\dfrac{x-5}{x^2-25}[/math]?

The function [math]f(x)=\dfrac{ax^2+x-7}{9x^2+bx+4}[/math] has a horizontal asymptote at [math]y=c[/math] and exactly one vertical asymptote. If [math]b[/math] is positive, what is [math]\dfrac{a}{bc}[/math]?

Find the minimum value of [math]c[/math] such that the graph of [math]f(x)=\dfrac{x}{7x^3-5x^2+cx}[/math] has only one vertical asymptotes

Generate a plausible rational function for the graph above.

Identify any [math]x[/math] values for which the expression is undefined: [math]\dfrac{x-2}{x+1}\div\dfrac{2x-8}{x^2-4}[/math]

What is the equation of the graph as a transformation of the parent graph [math]f(x)=\dfrac{1}{x}[/math]? Assuming there is no scaling.

Which [math]g(x)[/math] represents the function [math]f(x)=x^2[/math] with holes at [math]-2[/math] and 2?

How many integers [math]c[/math] such that the graph [math]f(x)=\dfrac{x^2-1}{x^2-cx+4}[/math] has no vertical asymptotes?

Which of the following graphs have a horizontal asymptote?
I. [math]f(x)=\displaystyle{\frac{2x^2}{(x+3)(x-5)}}[/math]
II. [math]g(x)=\displaystyle{\frac{-8(x+5)(x-2)(3x+1)}{x^2-16}}[/math]
III. [math]h(x)=\displaystyle{\frac{-1}{x^2+7x+6}}[/math]
IV. [math]k(x)=\displaystyle{\frac{4x^2-25}{2x^2-3x-5}}[/math]

Let [math]f(x)=\displaystyle{\frac{x^2}{x+5}}[/math]. Which of the following statements is true about the graph?

Given [math]f(x)=\displaystyle{\frac{(2x+3)(x-5)}{(x+1)(x-5)}}[/math], find the exact location of the hole.

Match the function to the graph. Not all graphs will be used.

(a) [math]f(x)=\displaystyle{\frac{x+2}{x^2-4}}[/math]

(b) [math]f(x)=\displaystyle{\frac{4-x^2}{x+2}}[/math]

(c) [math]f(x)=\displaystyle{\frac{x^2}{(x-2)(x+3)}}[/math]

(d) [math]f(x)=\displaystyle{\frac{x}{(x-2)(x+3)}}[/math]

(e) [math]f(x)=\displaystyle{\frac{1}{x(x-2)(x+3)}}[/math]

(f) [math]f(x)=\displaystyle{\frac{x^2}{x-2}}[/math]