Question

the function f(x) is shown on the coordinate plane below.
y=f(x)
which of the following is the equation of the function f(x) graphed above?
$f(x) = -x(x + 3)(x - 1)(x - 2)^2$
$f(x) = -x(x + 3)(x + 1)(x + 2)^2$
$f(x) = x(x - 3)(x + 1)(x + 2)^2$
$f(x) = x(x + 3)(x - 1)(x + 2)^2$
clear my selection

Explanation:

Step1: Identify x-intercepts from graph

The graph crosses/touches x-axis at $x=-3$, $x=-1$, $x=0$, $x=2$.

Step2: Match intercepts to factors

For intercept $x=a$, factor is $(x-a)$. So factors are $(x+3)$, $(x+1)$, $x$, $(x-2)$.

Step3: Check multiplicity behavior

At $x=2$, graph touches axis (even multiplicity); others cross (odd multiplicity). So $(x-2)^2$.

Step4: Determine leading coefficient

As $x\to+\infty$, $f(x)\to+\infty$. For degree 4 (even), leading coefficient positive.

Step5: Match to options

Only $f(x)=x(x+3)(x-1)(x-2)^2$ fits all criteria.

Answer:

$f(x) = x(x + 3)(x - 1)(x - 2)^2$