Question

what are the left and right end behaviors of the rational function (y values)?

1. $m(x)=\frac{3(x - 1)^2(x + 5)}{(2x + 3)^2}$

left:
right:
left
right

Explanation:

Step1: Expand the numerator and denominator

The numerator $3(x - 1)^2(x + 5)=3(x^{2}-2x + 1)(x + 5)=3(x^{3}+5x^{2}-2x^{2}-10x+x + 5)=3(x^{3}+3x^{2}-9x + 5)=3x^{3}+9x^{2}-27x + 15$. The denominator $(2x + 3)^2=4x^{2}+12x + 9$.

Step2: Determine the degrees of numerator and denominator

The degree of the numerator $n = 3$ and the degree of the denominator $m=2$. Since $n>m$, we consider the leading - coefficient of the numerator and denominator. The leading - term of the numerator is $3x^{3}$ and the leading - term of the denominator is $4x^{2}$.

Step3: Find the end - behavior as $x\to-\infty$

As $x\to-\infty$, we consider the ratio of the leading terms $\frac{3x^{3}}{4x^{2}}=\frac{3}{4}x$. Since $x\to-\infty$, $y\to-\infty$.

Step4: Find the end - behavior as $x\to+\infty$

As $x\to+\infty$, the ratio of the leading terms is $\frac{3x^{3}}{4x^{2}}=\frac{3}{4}x$. Since $x\to+\infty$, $y\to+\infty$.

Answer:

Left: $y\to-\infty$
Right: $y\to+\infty$