How do you create a rational function that includes the following: crosses x-axis at 4, Touches the x-axis at -3, One vertical asymptote at x=1 and another at x=6, one horizontal asymptote at y= -3?

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Donald Erickson · Accepted Answer

The denominator of the rational function is primarily responsible for vertical asymptotes: If the denominator is zero and the numerator non-zero then there will be a vertical asymptote. So given that we want vertical asymptotes at x=1 and x=6, let&#039;s consider a denominator (x−1)(x−6)=x2−7x+6 Given that we want our function to cross the x axis at x=4, we need a factor (x-4) in the numerator. Given that we want it to touch the x axis at x=-3 we need factors (x+3)(x+3) So the numerator needs to be a multiple of: (x−4)(x+3)(x+3)=(x−4)(x2+6x+9)=x3+2x2−15x−36 So far, our function looks like: x3+2x2−15x−36x2−7x+6 This has all of the attributes that we want except the horizontal asymptote. To get a horizontal asymptote at a non-zero value of y, we need the numerator and denominator to have the same degree. To avoid adding any extra vertical asymptotes, we can duplicate one of the existing linear factors in the denominator, say (x-1) to get: x3+2x2−15x−36(x−1)(x2−7x+6)=x3+2x2−15x−36x3−8x2+13x−6 This has horizontal asymptote y=1. To change it to y=-3 just multiply the whole function by -3 to get: f(x0)=−3(x3+2x2−15x−36)x3−8x2+13x−6=−3x3−6x2+45x+108x3−8x2+13x−6

How do you create a rational function that includes the

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