Question

1. write a rational function that has the end behavior like $y = -\frac{1}{3}x^{2}$.

Explanation:

Step1: Recall end - behavior of rational functions

For a rational function $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, the end - behavior is determined by the leading terms of $P(x)$ and $Q(x)$. If the degree of $P(x)$ is $m$ and the degree of $Q(x)$ is $n$, when $m = n$, the end - behavior is a non - zero constant; when $m>n$, the end - behavior is a non - constant polynomial; when $m < n$, $y = 0$ is the horizontal asymptote. We want an end - behavior like $y=-\frac{1}{3}x^{2}$, so the degree of the numerator should be 2 more than the degree of the denominator.

Step2: Construct the rational function

Let the denominator $Q(x)=1$ (a polynomial of degree 0) and the numerator $P(x)=-\frac{1}{3}x^{2}$. So a rational function with the desired end - behavior is $f(x)=-\frac{1}{3}x^{2}=\frac{-\frac{1}{3}x^{2}}{1}$. In general, we can also have $f(x)=\frac{-\frac{1}{3}x^{2}+x}{1}$ or $f(x)=\frac{-\frac{1}{3}x^{2}+5}{1}$ etc. As long as the leading term of the numerator is $-\frac{1}{3}x^{2}$ and the denominator is a non - zero constant (degree 0 polynomial), the end - behavior will be like $y = -\frac{1}{3}x^{2}$. A simple example is $f(x)=-\frac{1}{3}x^{2}=\frac{-\frac{1}{3}x^{2}}{1}$.

Answer:

$f(x)=\frac{-\frac{1}{3}x^{2}}{1}$ (or any rational function where the leading term of the numerator is $-\frac{1}{3}x^{2}$ and the denominator is a non - zero constant polynomial)