Answer
From the far left to the far right, the graph
- rises from $-\infty$ on the far left,
- crosses the x-axis at $(-2,0),$ rising
- continues to rise above 12, turns, and
- falling, crosses the y-axis at y=12,
- falls through $(\displaystyle \frac{2}{3},0)$ 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 three factors are negative. $P(x)$ is negative to the far left.
When $ x\rightarrow+\infty$ , all three factors are positive. $P(x)$ is positive to the far right.
Intercepts:
$x-3=0,\quad x+2=0\qquad 3x-2=0$
x-intercepts: at $x=3, x=-2$, and $x=\displaystyle \frac{2}{3}$, all single$.\\\\$
y-intercept: $P(0)=-3(2)(-2)=+12$
Behavior around the y-intercept:
$P(-1)=20, P(0)=12, P(1)=6,$
so the graph falls through the y-intercept
From the far left to the far right, the graph
- rises from $-\infty$ on the far left,
- crosses the x-axis at $(-2,0),$ rising
- continues to rise above 12, turns, and
- falling, crosses the y-axis at y=12,
- falls through $(\displaystyle \frac{2}{3},0)$ below the x-axis,
- turns back rising through $(3,0),$
- continues rising to the upper far right.