5.5 Key features of rational functions

Introduction

When a rational function has a

Sign charts and finding formulas for rational functions

Consider the rational function $r(x) = \frac{x^2 - 1}{x^2 - 3x - 4}$, and let $p(x) = x^2 - 1$ (the numerator of $r(x)$) and $q(x) = x^2 - 3x - 4$ (the denominator of $r(x)$).

Reasoning algebraically, for what values of $x$ is $p(x) = 0$?

Consider the rational function $r(x) = \frac{x^2 - 1}{x^2 - 3x - 4}$ from Preview Activity 5.5, whose numerator is $p(x) = x^2 - 1$ and whose denominator is $q(x) = x^2 - 3x - 4$. Explain why the graph of $r$ generated by Desmos or another computational device is incorrect, and also identify the locations of any zeros and vertical asymptotes of $r$.

$\displaystyle f(x) = \frac{x^3 - 6x^2 + 5x}{x^2-1}$

$\displaystyle g(x) = \frac{11(x^2 + 1)(x-7)}{23(x-1)(x^2+4)}$

$\displaystyle h(x) = \frac{x^2 - 8x + 12}{x^2 - 3x - 18}$

$\displaystyle q(x) = \frac{(x-2)(x^2-9)}{(x-3)(x^2 + 4)}$

$\displaystyle r(x) = \frac{19(x-2) (x-3)^2 (x+1)}{17(x+1)(x-4)^2(x-5)}$

$\displaystyle s(x) = \frac{1}{x^2 + 1}$

Construct a sign chart for the function $q(x) = \frac{(x-2)(x^2-9)}{(x-3)(x-1)^2}$. Then, graph the function $q$ and compare the graph and sign chart.

A rational function $r$ such that $r$ has a vertical asymptote at $x = -2$, a zero at $x = 1$, a hole at $x = 5$, and a horizontal asymptote of $y = -3$.

A rational function $u$ whose numerator has degree $3$, denominator has degree $3$, and that has exactly one vertical asymptote at $x = -4$ and a horizontal asymptote of $y = \frac{3}{7}$.

A rational function $w$ whose formula generates a graph with all of the characteristics shown in the following figure. Assume that $w(5) = 0$ but $w(x) \gt 0$ for all other $x$ such that $x \gt 3$.

A rational function $z$ whose formula satisfies the sign chart shown in the following figure, and for which $z$ has no horizontal asymptote and its only vertical asymptotes occur at the middle two values of $x$ noted on the sign chart.

A rational function $f$ that has exactly two holes, two vertical asymptotes, two zeros, and a horizontal asymptote.