Question

201805na - s3 - 12
the graph of the function ( f ) is shown in the ( xy )-plane above, where ( y = f(x) ). which of the following functions could define ( f )?
a) ( f(x)=(x - 3)(x - 1)^2(x + 2)^2 )
b) ( f(x)=(x - 3)^2(x - 1)(x + 2) )
c) ( f(x)=(x + 3)(x + 1)^2(x - 2)^2 )
d) ( f(x)=(x + 3)^2(x + 1)(x - 2) )

Explanation:

Step1: Identify x-intercepts

From the graph, x-intercepts are $x=-3$, $x=-1$, $x=2$.

Step2: Analyze intercept multiplicities

• $x=-3$: graph crosses x-axis → odd multiplicity (1)
• $x=-1$, $x=2$: graph touches x-axis → even multiplicity (2)

Step3: Match to function form

A function with these intercepts and multiplicities is $f(x)=(x+3)(x+1)^2(x-2)^2$, which matches option C.

Step4: Verify end behavior

As $x\to+\infty$, $f(x)\to+\infty$, consistent with leading term $x^5$ (positive coefficient, odd degree) from option C.

Answer:

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