Question

Use a given information to create equation for the rational function. The function is written in factored form to help see how they given information shapes the equation. If the leading coefficient is not an integer enter the value as a fraction.

Answer 1

Recall that a rational function:

$\frac{P(x)}{Q(x)},$

has a vertical asymptote at x₀ if and only if:

$Q(x_0)=0.$

Also, the roots of the above rational function are the same as P(x).

Since the rational function has a vertical asymptote at x=-1, we get that its denominator must be:

$Q(x)=x+1\text{.}$

Since the rational function has a double zero at x=2 we get that its numerator must be of the form:

$P(x)=k(x-2)(x-2)\text{.}$

Finally, since the rational function has y-intercept at (0,2) we get that:

$2=\frac{P(0)}{Q(0)}=\frac{k(0-2)(0-2)}{0+1}\text{.}$

Simplifying the above equation we get:

$\begin{gathered} \frac{k(-2)(-2)}{1}=2, \\ 4k=2. \end{gathered}$

Dividing the above equation by 4 we get:

$\begin{gathered} \frac{4k}{4}=\frac{2}{4}, \\ k=\frac{1}{2}\text{.} \end{gathered}$

Therefore, the rational function that satisfies the given conditions is:

$f(x)=\frac{\frac{1}{2}(x-2)(x-2)}{x+1}\text{.}$

Answer:

The numerator is:

$\frac{1}{2}(x-2)(x-2)$

The denominator is:

$(x+1)$