Question

In Problems $7-44$, follow Steps $1$through $7$on page $228$to analyze the graph of each function.

$Q\left(x\right)=\frac{x^4-1}{x^2-4}$

Short answer

The graph for the function $Q\left(x\right)=\frac{x^4-1}{x^2-4}$is,

Step-by-step solution

Step 1 Given function is,

$Q\left(x\right)=\frac{x^4-1}{x^2-4}$

which may also written as,

$Q\left(x\right)=\frac{x^4-1}{\left(x+2\right)\left(x-2\right)}$

The domain of $Q\left(x\right)$is $\left\{x|x\ne \pm 2\right\}$.

Now we will write $Q$in lowest terms. Here, we can see that there are no common factors between numerator and denominator. Therefore,$Q$is in its lowest terms.

Step 2 The x- intercept can be found by determining the real zeros of the numerator of Q written in the lowest terms.

Therefore, by solving we get, $x=\pm 1$. From here we find the only real finite zero of the numerator as $\pm 1$.Hence, the $x$-intercept of $Q$is $\pm 1$. The $y-$intercept can be found by determining the value of $Q$when $x=0$and here the value of $Q$is $\frac{1}{4}$for$x=0$. Here, the$y-$intercept is,$\frac{1}{4}$.

Step 3 We can find the vertical asymptotes by finding the zeros of the denomination of the rational function in lowest terms.

The obtained vertical asymptotes are at$x=\pm 2$.

In this graph there is no horizontal asymptote. So, let us calculate the oblique asymptotes.

This can be found by dividing the numerator by denominator and the part except the remainder is the asymptote.

In $Q$when we do this procedure, we find that $Q$can be again written as,

$y=\frac{x^4-1}{x^2-4}$

Therefore, the function has neither horizontal nor oblique as the asymptote.

Step 4 Now let us draw the graph by using the known values.