College Algebra, Chapter 4, 4.2, Section 4.2, Problem 22

Sketch the graph of polynomial function $\displaystyle P(x) = \frac{1}{4} (x + 1)^3(x-3)$ make sure the graph shows all intercepts and exhibits the proper end behaviour.
The function has an even degree 4 and a positive leading coefficient. Thus, its end behaviour is $y \rightarrow \infty \text{ as } x \rightarrow -\infty \text{ and } y \rightarrow \infty \text{ as } x \rightarrow \infty$.
To solve for the $x$-intercept, we set $y = 0$.



$
\begin{equation}
\begin{aligned}
0 &= \frac{1}{4}(x + 1)^3 (x - 3)\\
\\
0 &= (x+1)^3 (x - 3)
\end{aligned}
\end{equation}
$


We have,
$x = -1$ and $x = 3$

To solve for the $y$-intercept, we set $x = 0$

$
\begin{equation}
\begin{aligned}
y &= \frac{1}{4} (0 + 1)^3(0-3)\\
\\
y &= \frac{1}{4} (1)^3 (-3) = -\frac{3}{4}
\end{aligned}
\end{equation}
$