5.3 Graphs of rational functions (in class)

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=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]?

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].

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]?

Generate a plausible rational function for the graph above.

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.