Q. 47 - 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 :

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

Step 1. Given information

given,

A graph is given in the question with the complete graph details, we have to find a rational function of the given graph.

Step 2. First, find the x - intercepts of the 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 47 has $x-$intercepts 1 (odd multiplicity; graph crosses the $x-$axis) and 3 (odd multiplicity; graph crosses the $x-$axis). So one possibility for the numerator is $p\left(x\right)=$$(x-1) (x-3)$

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 $x=-1 and x=2$, Since width="33" height="20" style="max-width: none; vertical-align: -5px;" $R\left(x\right)$  approaches $\infty$  from both sides of $x=-1$ 

So, $(x+1)$ is a factor of even multiplicity in $q\left(x\right)$.

Also, $R\left(x\right)$ approaches $-\infty$ from both sides of $x=2$.  

So, $(x-2)$ is a factor of even multiplicity in $q\left(x\right)$. A possibility for the denominator is $q\left(x\right)=$ ${(x+1)}^2 {(x-2)}^2$ 

Step 4. Hence the rational function is :

So far we have,

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