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?

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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?

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Answer 1 · 2018-01-31T04:41:23

Rational function is $(2x^2-22x+56)/(5x^2-20x-105)$

Explanation:

A zero at $x=4$ means we have $(x-4)$ as a factor in numerator;

a hole at $x=7$ means, we have $x-7$ a factor both in numerator as well as denominator;

a vertical asymptote at $x=-3$ means $x+3$ a factor in denominator only

a horizontal asymptote at $y=2/5$ means highesr degrees in both numerator and denominator are equal and their coefficients are in ratio of $2:5$

Hence desired rational function is $(2(x-4)(x-7))/(5(x-7)(x+3))$

i.e. $(2x^2-22x+56)/(5x^2-20x-105)$

See its graph down below. Observe vertical asymptote $x=-3$ and horizontal asymptote $y=2/5$ . We have a zero at $x=4$ as functtion passes through $(4,0)$ . Hole is not seen as $x-7$ cancels out, but we know, the function is not definedat this point.

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

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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?

1 Answer

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