Question

Write an equation for a rational function with: Vertical asymptotes at x = -4 and x = 2 x-intercepts at x = -2 and x = -3 Horizontal asymptote at y = 3

Answer 1

Answer:

Correct answer:  y = ( 3x² + 15x + 18) / (x² + 2x - 8)

Step-by-step explanation:

The function has vertical asymptotes at points where it is not defined.

In our case it is at  x₁ = -4 and x₂ = 2

This means that the function in the denominator has a quadratic function whose roots are (x + 4) · (x - 2) = x² + 2x - 8

The function intercepts x axis at x₀₁ = -2 and x₀₂ = -3

This means that the function in the numerator has a quadratic function whose roots are (x + 2) · (x + 3) = x² + 5x + 6

The function currently looks like this:

y = (x² + 5x + 6) / (x² + 2x - 8)

Since the function has a horizontal asymptote y = 3, this means when x strive to + - infinite or x -> + - ∞  then it is

lim x -> + - ∞ (x² + 5x + 6) / (x² + 2x - 8) = 3

This means that the function in the numerator must has term 3x² which we will get when we multiply the currently function y = (x² + 5x + 6) / (x² + 2x - 8)  by the number 3 and get :

y = 3 · (x² + 5x + 6) / (x² + 2x - 8) = (3x² + 15x + 18) / (x² + 2x - 8)

y = ( 3x² + 15x + 18) / (x² + 2x - 8)

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