Questions: Match each polynomial function to its graph. g(x) = x^2 - 9x^2 + 27x - 20 h(x) = -x^3 + 14x - 32 k(x) = -x^2 - 34x^2 - 64x - 9y

Transcript text: Match each polynomial function to its graph. \[ g(x)=x^{2}-9 x^{2}+27 x-20 \] \[ h(x)=-x^{3}+14 x-32 \] \[ k(x)=-x^{2}-34 x^{2}-64 x-9 y \]

Solution

Solution Steps

Step 1: Analyze f(x)

The function \(f(x) = x^2 + 12x + 31\) is a quadratic function with a positive leading coefficient. Therefore, its graph will be a parabola opening upwards. Completing the square gives \(f(x) = (x+6)^2 - 5\), so the vertex is at \((-6, -5)\).

Step 2: Analyze g(x)

The function \(g(x) = x^3 - 9x^2 + 27x - 28\) is a cubic function. The graph shown in the top right is the only cubic graph. Thus, \(g(x)\) corresponds to the top right graph.

Step 3: Analyze h(x)

The function \(h(x) = -x^2 + 14x - 52\) is a quadratic function with a negative leading coefficient. Its graph will be a parabola opening downwards. Completing the square gives \(h(x) = -(x-7)^2 - 3\), so its vertex is at \((7, -3)\).