Chapter 4.5, Problem 62AYU

To determine

To create: A rational function with the given characteristics of the function.

Answer to Problem 62AYU

  $R\left(x\right)=\frac{2x^3-x^2-9x-6}{x^2-x-6}$

Explanation of Solution

Given: Three real zeros, one of multiplicity 2; y -intercept 1; vertical asymptotes, $x=-2\text{ and }x=3$ ; oblique asymptote, $y=2x+1$ .

Zeros of the rational function where cross or touch x -axis. If multiplicity is even then graph touches x -axis.

y -intercept: $\left(0,1\right)$

VA: $x=-2\text{ and }x=3$

Oblique Asymptote: $y=2x+1$

Rational function, $R\left(x\right)=\text{Oblique Asymptote + }\frac{p\left(x\right)}{VA}$

  $R\left(x\right)=2x+1+\frac{p\left(x\right)}{\left(x+2\right)\left(x-3\right)}$

There are three real zeros. Rational function at numerator maximum three degree.

y -intercept: $\left(0,1\right)$

  $\begin{array}{l}R\left(0\right)=2\left(0\right)+1+\frac{p\left(0\right)}{\left(0+2\right)\left(0-3\right)}\\ 1=1+\frac{p\left(0\right)}{-6}\\ p\left(0\right)=0\end{array}$

Therefore, $p\left(x\right)=ax$

The value of a depends on zeros. Using slider to set the value of a if the rational function touches x -axis.

  

Therefore, the ration function, $R\left(x\right)=\frac{2x^3-x^2-9x-6}{x^2-x-6}$