Question

homework assignment 5.3 graphs of polynomial functions due friday by 11:59pm points 9 submitting an external tool homework assignment 5.3 graphs of polynomial functions score: 0/9 answered: 0/9 question 1 given the function $c(t)=-(t - 3)(t + 4)(t - 6)$: the coordinates of its $c$-intercept are the coordinates of its $t$-intercepts are the degree of $c$ is as $x\to-infty,f(x)\to?$ as $x\toinfty,f(x)\to?$ question help: video written example message instructor submit question

Explanation:

Step1: Find the $C$-intercept

Set $t = 0$ in $C(t)$.
\[

$$\begin{align*} C(0)&=-(0 - 3)(0+4)(0 - 6)\\ &=-(-3)\times4\times(-6)\\ &=-72 \end{align*}$$

\]
So the $C$-intercept is $(0,-72)$.

Step2: Find the $t$-intercepts

Set $C(t)=0$.
\[

$$\begin{align*} -(t - 3)(t + 4)(t - 6)&=0\\ t-3&=0\quad\text{or}\quad t + 4=0\quad\text{or}\quad t-6=0\\ t&=3\quad\text{or}\quad t=-4\quad\text{or}\quad t = 6 \end{align*}$$

\]
So the $t$-intercepts are $(3,0),(-4,0),(6,0)$.

Step3: Determine the degree

The degree of a polynomial is the sum of the powers of the factors when expanded. Here, since we have three linear - factors multiplied together, the degree of $C(t)$ is $1 + 1+1=3$.

Step4: Determine end - behavior

The leading coefficient of $C(t)=-(t - 3)(t + 4)(t - 6)=-t^{3}+5t^{2}+18t - 72$ is $- 1$ (the coefficient of the highest - degree term $-t^{3}$).
When $x\to-\infty$, for a polynomial $y = ax^{n}+\cdots$ with $a=-1$ and $n = 3$ (odd), $y\to\infty$ because $(-1)\times(-\infty)^{3}=\infty$.
When $x\to\infty$, for a polynomial $y = ax^{n}+\cdots$ with $a=-1$ and $n = 3$ (odd), $y\to-\infty$ because $(-1)\times(\infty)^{3}=-\infty$.

Answer:

The coordinates of its $C$-intercept are $(0,-72)$.
The coordinates of its $t$-intercepts are $(3,0),(-4,0),(6,0)$.
The degree of $C$ is $3$.
As $x\to-\infty$, $f(x)\to\infty$.
As $x\to\infty$, $f(x)\to-\infty$.