Question

sketch the graph of the function below, including correct signs, ( x )-intercepts and ( y )-intercepts.

( f(x) = -x^2(2x + 12)(x + 4) )

plot the ( y )-intercept and the roots. click on the graph to plot a point. click
a point again to delete it.

graph with axes and grid, and a done plotting button

Explanation:

Step1: Find x-intercepts (roots)

Set $f(x)=0$:
$$-x^2(2x+12)(x+4)=0$$
Solve each factor:
$x^2=0 \implies x=0$ (double root)
$2x+12=0 \implies x=-6$
$x+4=0 \implies x=-4$
So x-intercepts are $(-6,0)$, $(-4,0)$, $(0,0)$.

Step2: Find y-intercept

Set $x=0$:
$$f(0)=-(0)^2(2(0)+12)(0+4)=0$$
So y-intercept is $(0,0)$.

Step3: Confirm end behavior

Expand leading terms: $-x^2(2x)(x)=-2x^4$. As $x\to\pm\infty$, $f(x)\to-\infty$. For the double root at $x=0$, the graph touches the x-axis and turns around.

Answer:

x-intercepts: $(-6, 0)$, $(-4, 0)$, $(0, 0)$
y-intercept: $(0, 0)$