Answer
please see image:
Work Step by Step
Factoring 2 out of the first parentheses, rewrite:
$P(x)=-2(x-\displaystyle \frac{1}{2})(x+1)(x+3)$
End behavior:
When $ x\rightarrow-\infty$ , observing just the signs,
we have
$-(-)(-)(-)$, so $ P(x)\rightarrow+\infty$
When $ x\rightarrow+\infty$ , observing just the signs,
we have
$-(+)(+)(+)$, so $ P(x)\rightarrow-\infty$
Intercepts:
x-intercepts: $x=-3, x=-1, x=\displaystyle \frac{1}{2}.$
y-intercept: $P(0)=-(-1)(1)(3)=3$
$P(2.9)=$3.132
$P(3.1)=$2.728,
so the graph is falling as it crosses the y-axis.
Short description of the graph, left to right:
- falls from the upper left,
- crosses the x-axis at x=-3,
- continues falling to a local minimum, turns and
- while rising, crosses the x-axis at x=-1,
- continues rising to a local maximum, turns, and
- falling, crosses the y-axis at y=3
- still falling, crosses the x-axis at x=$\displaystyle \frac{1}{2}$,
- continues falling indefinitely
