Question

Create a rational function that has the following characteristics: crosses the x-axis at $2$; touches the x-axis at $-1$; one vertical asymptote at $x=-5$and another at $x=6$; and one horizontal asymptote, $y=3$. Compare your function to a fellow classmate’s. How do they differ? What are their similarities?

Short answer

The rational function is$\frac{3(x-2){(x+1)}^2}{{(x-6)}^a{(x+5)}^b}$.

Step-by-step solution

Step 1. Finding numerator of the rational function. 

Here we have to find the rational through the given information First of all the x intercept of the function is:

It touches the x-axis at $-1$.

It crosses the x-axis at $2$.

And it has a horizontal asymptote as $y=3$

From here we can write the numerator as $3(x-2){(x+1)}^2$.

Step 2. Finding denominator of the rational function.

Now for denominator we are given that the vertical asymptote is at $x=\{-5,6\}$and also it has been given that the horizontal asymptote of the function is $y=2$.

For any function to have a horizontal asymptote, the degree of the numerator and the denominator must be equal.

From this we have the denominator${(x-6)}^a{(x+5)}^b$

Step 3. Finding rational function.

Using numerator and denominator, the rational function obtained is

$\frac{3(x-2){(x+1)}^2}{{(x-6)}^a{(x+5)}^b}$

where a, b are arbitrary constants such that $a+b=3$.