Question

Design a rational function with the following characteristics: three real zeros, one of multiplicity 2; y-intercept 1; vertical asymptotes $x=-2andx=3$; oblique asymptote $y=2x+1$. Is this rational function unique? Compare yours with those of other students. What will be the same as everyone else’s? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the rational function?

Short answer

The required function is$F\left(x\right)=\frac{A}{B}=\frac{-{(x-2)}^2(x+3)}{{(x+2)}^2(x-3)}$

Step-by-step solution

Step 1. Given Information

The given data is that it has three real zeros, one of multiplicity 2; y-intercept 1; vertical asymptotes $x=-2andx=3$; oblique asymptote$y=2x+1$

Step 2. Explanation

To create a rational function with the characterstics,

1. Three real zeroes: the numerator total degree is 3.

2. One of multiplicity 2: one of the numerator is a square.

3. y-intercept is 1, when $x=0,y=1$

4. Vertical asymptotes $x=-2andx=3$, denominators are in lowest terms possible ${(x+2)}^n{(x-3)}^m$

5. Oblique asymptote $y=2x+1$

From the first 4 condition, we can construct a possible function,

$F\left(x\right)=\frac{A}{B}=\frac{-{(x-2)}^2(x+3)}{{(x+2)}^2(x-3)}$

In order to satisfy the oblique asymptote, it has to satisfy the condition,

$\frac{A}{B}=2x+1+C$where C is the remainder.

There are many ways to construct the function, since the remainder C can be anything.