Questions: Which of the following functions could be the function graphed to the right? g(x) = (x-3)(x^2+4) g(x) = x^4+4x^3-3x^2+12 g(x) = (x+3)(x+2)(x-2) g(x) = -x^3-3x^2-4x+12 g(x) = (x-3)(x^2-4)

Transcript text: Which of the following functions could be the function graphed to the right? $g(x)=(x-3)\left(x^{2}+4\right)$ $g(x)=x^{4}+4 x^{3}-3 x^{2}+12$ $g(x)=(x+3)(x+2)(x-2)$ $g(x)=-x^{3}-3 x^{2}-4 x+12$ $g(x)=(x-3)\left(x^{2}-4\right)$

Solution

Solution Steps

Step 1: Analyze the graph's roots

The graph of the function has roots at approximately x = -3, x = 2, and x = 2. Thus, the factored form of any polynomial representing this graph must be in the form of _a_(x+3)(x-2)..., where _a_ is a constant, since x = -3 and x = 2 are roots.

Step 2: Eliminate choices based on roots

The first option, g(x) = (x-3)(x²+4), has only one real root at x = 3, because the term (x²+4) yields no real roots.
The second option is a standard polynomial form which gives little information without factoring.
The third option, g(x) = (x+3)(x+2)(x-2), has roots of x = -3, x = -2, and x = 2.
The fourth option gives little information.
The fifth option, g(x) = (x-3)(x²-4), which can be rewritten as g(x) = (x-3)(x-2)(x+2), has roots of x = -2, x=2, and x=3.

Step 3: Match the roots to the correct factored form.

Only the third option, g(x) = (x+3)(x+2)(x-2), provides the roots observed in the graph.