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Hitman42 [59]
1 year ago
13

Use a given information to create equation for the rational function. The function is written in factored form to help see how t

hey given information shapes the equation. If the leading coefficient is not an integer enter the value as a fraction.

Mathematics
1 answer:
Mars2501 [29]1 year ago
4 0

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)

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