Answer
See graph and explanations.
![](https://gradesaver.s3.amazonaws.com/uploads/solution/08e642e5-fa04-42bb-9c52-3f92c6d1043f/result_image/1586174649.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T014517Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=a008b10286b9eac85de3699e22b1ee201679a714215788ebd16a598e194b103e)
Work Step by Step
Step 1. Given $f(x)=-2(x-1)^2(x+3)$, we test for symmetry:
$f(-x)=-2(-x-1)^2(-x+3)=2(x+1)^2(x-3)$; no symmetry.
Step 2. End behavior: the leading term $-2x^3$, rises to the left and falls to the right.
Step 3. The x-intercepts are: $x=-3, 1$. The graph crosses the x-axis at $x=-3$, and touches/returns at $x=1$.
Step 4. The y-intercept: $f(0)=-2(1)(3)=-6$
Step 5. The maximum number of turns: $n-1=2$
Step 6. We use the test points as necessary to graph the function as shown in the figure.