Q. 48 - Precalculus Enhanced with Graphing Utilities

Question

Find a rational function that might have the given graph. (More than one answer might be possible).

Step-by-Step Solution

Answer

The rational function of the given graph is : 

$\frac{3 {(x-1)}^2 (x+2)}{(x+3) {(x-4)}^2}$

Step 1. Given information

given,

A graph is given with detailed information, on the basis of the given graph we have to find the rational function.

Step 2. First, find the x - intercept of a given graph

The numerator of a rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ in the lowest terms determines the $x-$intercepts of its graph. The graph shown in Figure 48 has $x-$intercepts 1 (even multiplicity; graph touches the $x-$axis) and -2 (odd multiplicity; graph crosses the $x-$axis). So one possibility for the numerator is  

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

Step 3. Find the Vertical asymptotes of the given graph

The denominator of a rational function in the lowest terms determines the vertical asymptotes of its graph. The vertical asymptotes of the graph are $x=-3 and x=4$. Since $R\left(x\right)$ approaches $\infty$ from the left of $x=-3$ and $R\left(x\right)$ approaches $-\infty$ from the right of $x=-3$, we know that $(x+3)$ is a factor of odd multiplicity in $q\left(x\right)$.

Also,  $R\left(x\right)$ approaches $\infty$ from both sides of $x=4$, so $(x-4)$) is a factor of even multiplicity in $q\left(x\right)$.

A possibility for the denominator is $q\left(x\right)=(x+3) {(x-4)}^2$

Step 4. Hence we find the rational function

So far we have $R\left(x\right)=\frac{{(x-1)}^2 (x+2)}{(x+3) {(x-4)}^2}$ . The horizontal asymptote of the graph given in Figure 48 is $y=3$, so we know that the degree of the numerator must equal the degree of the denominator, and the quotient of leading coefficients must be $\frac{3}{1}$ .

This leads to $R\left(x\right)=\frac{3 {(x-1)}^2 (x+2)}{(x+3) {(x-4)}^2}$ .