For the following exercises, write an equation for a...

For the following exercises, write an equation for a rational function with the given characteristics.
Vertical asymptotes at $x=-4$
and $x=-5, x$ -intercepts at
$(4,0)$ and $(-6,0),$ horizontal
asymptote at $y=7$

Key Concepts

X-Intercepts

X-intercepts of a rational function are the points where the function equals zero, which occur when the numerator is zero and the corresponding factor does not cancel with any factor in the denominator. These intercepts can be used to determine the zeros of the numerator polynomial.

Rational Function

A rational function is defined as the ratio of two polynomials, typically expressed in the form f(x) = N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not zero. This structure allows the function to exhibit a variety of behaviors such as asymptotes and intercepts, which are dictated by the factors in the numerator and denominator.

Vertical Asymptote

Vertical asymptotes occur at values of x that make the denominator zero while not canceling with factors in the numerator. They indicate points where the function tends to infinity and are found by setting each factor in the denominator equal to zero, provided those factors do not appear in the numerator.

Horizontal Asymptote

Horizontal asymptotes describe the end behavior of a function as x approaches infinity or negative infinity. For rational functions, if the degrees of the numerator and denominator polynomials are equal, the horizontal asymptote is determined by the ratio of their leading coefficients. This feature is essential for understanding how the function behaves far from the origin.

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