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?

Nov 16, 2017

$f \left(x\right) = \frac{- 3 {x}^{3} - 6 {x}^{2} + 45 x + 108}{{x}^{3} - 8 {x}^{2} + 13 x - 6}$

Explanation:

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's consider a denominator $\left(x - 1\right) \left(x - 6\right) = {x}^{2} - 7 x + 6$

Given that we want our function to cross the $x$ axis at $x = 4$, we need a factor $\left(x - 4\right)$ in the numerator.

Given that we want it to touch the $x$ axis at $x = - 3$ we need factors $\left(x + 3\right) \left(x + 3\right)$

So the numerator needs to be a multiple of:

$\left(x - 4\right) \left(x + 3\right) \left(x + 3\right) = \left(x - 4\right) \left({x}^{2} + 6 x + 9\right) = {x}^{3} + 2 {x}^{2} - 15 x - 36$

So far, our function looks like:

$\frac{{x}^{3} + 2 {x}^{2} - 15 x - 36}{{x}^{2} - 7 x + 6}$

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 $\left(x - 1\right)$ to get:

$\frac{{x}^{3} + 2 {x}^{2} - 15 x - 36}{\left(x - 1\right) \left({x}^{2} - 7 x + 6\right)} = \frac{{x}^{3} + 2 {x}^{2} - 15 x - 36}{{x}^{3} - 8 {x}^{2} + 13 x - 6}$

This has horizontal asymptote $y = 1$. To change it to $y = - 3$ just multiply the whole function by $- 3$ to get:

$f \left(x\right) = \frac{- 3 \left({x}^{3} + 2 {x}^{2} - 15 x - 36\right)}{{x}^{3} - 8 {x}^{2} + 13 x - 6}$

$\textcolor{w h i t e}{f \left(x\right)} = \frac{- 3 {x}^{3} - 6 {x}^{2} + 45 x + 108}{{x}^{3} - 8 {x}^{2} + 13 x - 6}$

graph{(-3x^3-6x^2+45x+108)/(x^3-8x^2+13x-6) [-25.05, 25.07, -12, 12]}