Question

Find a rational function that might have the given graph.

Short answer

One possibility of the rational function is $R\left(x\right)=\frac{3\left(x+2\right){\left(x-1\right)}^2}{\left(x+3\right){\left(x-4\right)}^2}$.

Step-by-step solution

Step 1. Given information.

Consider the given graph,

The numerator of a rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$

in lowest terms determines the x-intercepts of its graph.

The graph has x-intercept, localid="1646072236844" $-2$, as this touches the x-axis and is an even multiplicity and has x-intercept, localid="1646072278870" $1$, as this crosses the x-axis and is an odd multiplicity.

So, one possibility for the numerator is given below,

localid="1646073763896" $p\left(x\right)=\left(x+2\right){\left(x-1\right)}^2$.

Step 2. Determine the vertical asymptotes.

Consider the given question,

The denominator of a rational function in lowest terms determines the vertical asymptotes of its graph.

The vertical asymptotes are $x=-3,4$.

As $R\left(x\right)$approaches $\infty$to the left of $x=-3$and $R\left(x\right)$approaches $-\infty$to the right of localid="1646072581381" $x=-3$, then $x+3$is a factor of odd multiplicity of $q\left(x\right)$.

Similarly, localid="1646073819637" $\left(x-4\right)$is also a factor of odd multiplicity in $q\left(x\right)$.

So, one possibility for the denominator is given below,

$q\left(x\right)={\left(x-4\right)}^2\left(x+3\right)$.

Step 3. Form the rational function.

Consider the given question,

The horizontal asymptote of the given graph is $y=3$. Thus, the degree of the numerator must be equal to the degree of the denominator and that the quotient of the leading coefficients must be $3$.

Hence, a possible rational function is given below,

localid="1646145213137" $R\left(x\right)=\frac{3\left(x+2\right){\left(x-1\right)}^2}{\left(x+3\right){\left(x-4\right)}^2}$