Question

Use the given information to create the equation for the rational function. The function is written in factored form to help you see how the given information shapes our equation. Enter values for factors from smallest to largest.Vertical asymptotes at x=-4 and x=-1, x-intercepts at (1,0) and (5,0), y-intercept at (0,7).The numerator is: Answer (x-Answer )(x-Answer )The denominator is: (x+Answer )(x+Answer )

Answer 1

The numerator is 28(x - 1)(x - 5)

The denominator is (x + 4)(x + 1)

Explanations:

We are to use the following information to create the equation of a rational function

• vertical asymptotes, at x = -4 and x = -1

,

• x-intercept, at (1, 0) and (5, 0)

,

• y-intercept, at (0, 7)

Since the vertical asymptotes of a rational function occur when the denominator is zero, hence the factors at the denominator will be (x + 4) and (x + 1) (As seen from the vertical asymptote values)

The x-intercept given at (1, 0) and (5, 0) shows that f(1) = 0 and f(5) = 0, hence the factors that will be at the numerator with the y-intecept will be (x - 1) (x - 5)

Given that the y-intercept is at (0, 7), this shows that f(0) = 7. Note that the rational function required will be given as:

`f(x)=(a(x-1)(x-5))/((x+4)(x+1))`

Determine the value of the constant "a". Since f(0) = 7, then;

`\begin{gathered} f(0)=(a(0-1)(0-5))/((0+4)(0+5)) \\ 7=(a(-1)(-5))/(4(5)) \\ 7=(5a)/(20) \\ 5a=140 \\ a=28 \end{gathered}`

Hence the required rational function will be expressed as:

`f(x)=(28(x-1)(x-5))/((x+4)(x+1))`