Question

sketch the graph of the function below, including correct signs, x-intercepts and y-intercepts.
$f(x) = (2x + 8)(x + 2)^2(x - 1)(x - 3)$
plot the y-intercept and the roots. click on the graph to plot a point. click
a point again to delete it.

Explanation:

Step1: Find x-intercepts (roots)

Set $f(x)=0$, solve $2x+8=0$, $(x+2)^2=0$, $x-1=0$, $x-3=0$:
$2x+8=0 \implies x=-4$; $(x+2)^2=0 \implies x=-2$; $x-1=0 \implies x=1$; $x-3=0 \implies x=3$

Step2: Find y-intercept

Set $x=0$, calculate $f(0)$:
$$f(0)=(2(0)+8)(0+2)^2(0-1)(0-3)=(8)(4)(-1)(-3)=96$$

Step3: Note sign behavior

For $x<-4$, all factors negative/even power: $f(x)>0$;
Between $-4$ and $-2$, $f(x)<0$;
At $x=-2$, graph touches x-axis (even root);
Between $-2$ and $1$, $f(x)<0$;
Between $1$ and $3$, $f(x)>0$;
For $x>3$, $f(x)>0$

Answer:

x-intercepts: $(-4, 0)$, $(-2, 0)$, $(1, 0)$, $(3, 0)$
y-intercept: $(0, 96)$