Answer
From the far left to the far right, the graph
- falls from +$\infty$ on the far left,
- crosses the x-axis at $(-2,0)$, falling
- turns and rises through $(-1,0)$
- crosses the y-axis at $(0,12)$ still rising,
- turns and falls through $(2,0)$ below the x-axis,
- turns back rising through $(3,0),$
- continues rising to the upper far right.
Work Step by Step
End behavior:
When $ x\rightarrow-\infty$ , all four factors are negative. $P(x)$ is positive to the far left.
When $ x\rightarrow+\infty$ , all four factors are positive. $P(x)$ is positive to the far right.
Intercepts:
$x+2=0,\quad x+1=0\qquad x-2=0\quad x-3=0$
x-intercepts: at $x=-2, x=-1,\ x=2$and $x=3$, all single$.\\\\$
y-intercept: $P(0)=2(1)(-2)(-3)=+12$
Behavior around the y-intercept:
$P(-0.5)\approx 6.5, P(0)=12, P(0.5)\approx 14,$
so the graph rises through the y-intercept
From the far left to the far right, the graph
- falls from +$\infty$ on the far left,
- crosses the x-axis at $(-2,0)$, falling
- turns and rises through $(-1,0)$
- crosses the y-axis at $(0,12)$ still rising,
- turns and falls through $(2,0)$ below the x-axis,
- turns back rising through $(3,0),$
- continues rising to the upper far right.
