Question

Create a rational function that has the following characteristics: crosses the x-axis at $3$; touches the x-axis at $-2$; one vertical asymptote, $x=1$; and one horizontal asymptote, $y=2$. Give your rational function to a fellow classmate and ask for a written critique of your rational function.

Short answer

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

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 x intercept of the function is:

It touches the x-axis at $-2$.

It crosses the x-axis at $3$.

And also it has a horizontal asymptote as $y=2$

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

Step 2. Finding rational function.

Now for denominator we are given that the vertical asymptote is at $x=1$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 ${(x-1)}^3$.

So the rational function is$\frac{2(x+2){(x-1)}^2}{{(x-1)}^3}$.