Question

1. give the end behavior of the following functions: a. $r(t)=3t^{4}+12t^{2}+10$ b. $f(x)=-3(x - 1)^{12}(x - 3)^{13}$

Explanation:

Step1: Recall end - behavior rules for polynomials

The end - behavior of a polynomial $y = a_nx^n+\cdots+a_0$ is determined by the leading term $a_nx^n$, where $n$ is the degree and $a_n$ is the leading coefficient. If $n$ is even and $a_n>0$, as $x\to\pm\infty$, $y\to+\infty$. If $n$ is even and $a_n < 0$, as $x\to\pm\infty$, $y\to-\infty$. If $n$ is odd and $a_n>0$, as $x\to+\infty$, $y\to+\infty$ and as $x\to-\infty$, $y\to-\infty$. If $n$ is odd and $a_n < 0$, as $x\to+\infty$, $y\to-\infty$ and as $x\to-\infty$, $y\to+\infty$.

Step2: Analyze $R(t)$

For $R(t)=3t^4 + 12t^2+10$, the leading term is $3t^4$. The degree $n = 4$ (even) and the leading coefficient $a_n=3>0$. So, as $t\to+\infty$, $R(t)\to+\infty$ and as $t\to-\infty$, $R(t)\to+\infty$.

Step3: Analyze $F(x)$

First, expand the leading - term of $F(x)=-3(x - 1)^{12}(x - 3)^{13}$. The degree of the polynomial is $n=12 + 13=25$ (odd). The leading coefficient of $(x - 1)^{12}$ is $1$, the leading coefficient of $(x - 3)^{13}$ is $1$, and the overall leading coefficient of $F(x)$ is $-3\times1\times1=-3<0$. So, as $x\to+\infty$, $F(x)\to-\infty$ and as $x\to-\infty$, $F(x)\to+\infty$.

Answer:

a. As $t\to+\infty$, $R(t)\to+\infty$; as $t\to-\infty$, $R(t)\to+\infty$.
b. As $x\to+\infty$, $F(x)\to-\infty$; as $x\to-\infty$, $F(x)\to+\infty$.