Question

question 4 of 10
which polynomial function is graphed below?
(graph of a polynomial function with a grid, x-intercepts at -3 (with a touch point, indicating a double root) and 2, and y-intercept around -12. options:
a. $f(x) = (x - 2)^2(x + 3)$
b. $f(x) = (x - 3)(x + 2)^2$
c. $f(x) = (x - 2)(x + 3)^2$
d. $f(x) = (x - 3)^2(x + 2)$)

Explanation:

Step1: Find x-intercepts

From the graph, x-intercepts are at $x=-3$ (touching, even multiplicity) and $x=2$ (crossing, odd multiplicity).

Step2: Match with options

Check options:
A: Roots at 2 (multiplicity 2, touch) and -3 (multiplicity 1, cross) → matches intercepts and multiplicities.
B: Roots at 3 (cross) and -2 (touch) → intercepts wrong.
C: Roots at 2 (cross) and -3 (touch) → but leading coefficient sign: as $x→∞$, $f(x)→∞$ (correct), but let's confirm multiplicity: $x=-3$ is squared (touch), $x=2$ is linear (cross) → wait, check end behavior and y-intercept. Wait, original graph: when $x=0$, y-value is negative. For option C: $f(0)=(0-2)(0+3)^2=(-2)(9)=-18$ (negative). Option A: $f(0)=(0-2)^2(0+3)=4*3=12$ (positive, contradicts graph). Oh correction: Step1 end behavior: leading term of polynomial: for option C: $(x)(x^2)=x^3$, as $x→∞$, $x^3→∞$; $x→-∞$, $x^3→-∞$ → matches graph. Option A: $(x^2)(x)=x^3$, same end behavior, but y-intercept of graph is negative. So y-intercept check: graph at x=0 is negative. Option C: $f(0)=-18$ (negative), option A: 12 (positive). So correct is C.

Answer:

C. $f(x)=(x - 2)(x + 3)^2$