Graph the rational function.
-\9\begin{array} { c }
y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \

\end{array}\)

Question

Graph the rational function.
-\9\begin{array} { c } 
y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \

\end{array}\)

Quizplus · Accepted Answer

To graph the rational function $y = \frac{x^2 + x - 42}{x^2 - x - 56}$, we need to analyze its behavior near vertical and horizontal asymptotes, $x$- and $y$-intercepts, and critical points.

First, we factor the numerator and denominator: $y = \frac{(x + 7)(x - 6)}{(x - 8)(x + 7)}$. Notice that the factor $(x + 7)$ cancels out, leaving us with $y = \frac{x - 6}{x - 8}$, except at the point $x = -7$, where the function is undefined due to the vertical asymptote $x = -7$.

To find the horizontal asymptote(s), we look at the degree of the numerator and denominator. Since they are the same, the function has a horizontal asymptote at $y = \frac{	ext{leading coefficient of numerator}}{	ext{leading coefficient of denominator}} = \frac{1}{1} = 1$.

To find the $x$-intercept(s), we set $y=0$ and solve for $x$. This gives us $0 = \frac{(x + 7)(x - 6)}{(x - 8)(x + 7)}$, which means either $x + 7 = 0$ (yielding $x = -7$) or $x - 6 = 0$ (yielding $x = 6$). Note that $x = -7$ is a vertical asymptote, so it cannot be an $x$-intercept. Thus, the only $x$-intercept is $(6,0)$.

To find the $y$-intercept, we set $x=0$ and solve for $y$. This gives us $y = \frac{-42}{-56} = \frac{3}{4}$. Thus, the $y$-intercept is $(0,\frac{3}{4})$.

We also note that the function has critical points at $x = -1$ and $x = 7$, where the denominator is zero. We can use a sign analysis or a sign chart to determine the behavior of the function in each interval.

Sign chart:

| x  | -\infty |   -7   |  -1  | 6 |   7   |   8   | \infty | 
|----|-------:|-------:|-----:|---|------:|------:|-------:|
| y' |       - |  -     |  +   | + |    -  |  +    |     + |
| y  |   -     | undefined |  -1| 0 |  -infinity | 1 |   + |

From the sign chart, we can see that the function is decreasing from $-\infty$ to the vertical asymptote $x = -7$, then increasing from $-\infty$ to the critical point $x = -1$, where it has a local minimum of $-1$. It then decreases from $-1$ to the $x$-intercept $x = 6$, where it has a local maximum of $\frac{5}{2}$. After that, it decreases again from $\frac{5}{2}$ to the vertical asymptote $x = 8$, then increases from $+\infty$.

Based on this analysis, we can graph the function as follows:

Answer: D

Graph the Rational Function