Answer
From the far left to the far right, the graph
- rises from $-\infty$, the lower far left,
- flattens out near $(-2,0)$ where it crosses the x-axis,
- continues to rise above y=8 before it turns and
- crosses the y-axis at $(0,8)$, falling to
- to the point $(-1,0)$, where it touches x and turns back, rising
- and continues to rise to the far right.
Work Step by Step
End behavior:
When $ x\rightarrow-\infty$ , the squared factor is positive, the cubed is negative
$P(x)$ is negative to the far left.
When $ x\rightarrow+\infty$ , both factors are positive,
$P(x)$ is positive to the far right.
Intercepts:
$ x-1=0,\quad x+2=0$
x-intercepts:
$x=-2,$ triple (graph flattens out at the crossing)$.\\\\$
$x=1$, double (graph touches x and turns)$.\\\\$
y-intercept: $P(0)=(-1)^{2}(2^{3})=8$
$P(0.5)\approx 3.9$, so the graph falls through $(0,8)$
From the far left to the far right, the graph
- rises from $-\infty$, the lower far left,
- flattens out near $(-2,0)$ where it crosses the x-axis,
- continues to rise above y=8 before it turns and
- crosses the y-axis at $(0,8)$, falling to
- to the point $(-1,0)$, where it touches x and turns back, rising
- and continues to rise to the far right.