# How do you write a rational function that has the following properties: a zero at x= 4, a hole at x= 7, a vertical asymptote at x= -3, a horizontal asymptote at y= 2/5?

Rational function is

graph{(2x^2-22x+56)/(5x^2-20x-105) [-10.67, 9.33, -4.4, 5.6]} 1

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To write a rational function with the given properties, we can start by considering the factors that contribute to each property.

To have a zero at x = 4, we include the factor (x - 4) in the numerator.

To have a hole at x = 7, we include the factor (x - 7) in both the numerator and denominator, canceling it out.

To have a vertical asymptote at x = -3, we include the factor (x + 3) in the denominator.

To have a horizontal asymptote at y = 2/5, we ensure that the degree of the numerator is less than or equal to the degree of the denominator.

Combining these factors, the rational function can be written as:

f(x) = (x - 4) / ((x - 7)(x + 3))

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