Chapter 4.5, Problem 60AYU

In Problems 51-54, find a rational function that might have the given graph. (More than one answer might be possible.)

To determine

To find: A rational function.

Answer to Problem 60AYU

$f\left(x\right)=\frac{3{\left(x-1\right)}^2\left(x+2\right)}{{\left(x-4\right)}^2\left(x+3\right)}$

Explanation of Solution

Given:

Since the graph is vertical asymptotes at $x=4$ and $x=-3$, therefore, the rational function must have $x=-4$ and $x+3$ factors. In that case, its denominator must be at least 2.

Now, let’s assume its denominator only has $x-4$ and $x+3$ as factors, then the degree is 2 with leading coefficient 1, in order have horizontal asymptote $y=3$, its numerator must be degree 2 with leading coefficient 3 also. By observing that it has 1 and $-2$ as the $x\text{-intercepts}$, we have to use $3\left(x-1\right)\left(x+2\right)$as the numerator.

We can see that the $x\text{-intercepts}$ 1 is a turning point, so the factor $x-1$ must have even power, so we can change the numerator $3\left(x-1\right)\left(x+2\right)$to $3{\left(x-1\right)}^2\left(x+2\right)$.

However, we have to keep the denominator the same degree as the numerator, so we have to change the denominator $\left(x-4\right)\left(x+3\right)$ to ${\left(x-4\right)}^2\left(x+3\right)$.

Therefore

$f\left(x\right)=\frac{3{\left(x-1\right)}^2\left(x+2\right)}{{\left(x-4\right)}^2\left(x+3\right)}$