which equation best matches the graph shown below?

Explanation:

Step1: Identify the type of graph

The graph is a parabola opening downward, so it's a quadratic function with a negative leading coefficient. The general form of a quadratic function is ( y = ax^2 + bx + c ), or in factored form ( y = a(x - r_1)(x - r_2) ), where ( r_1 ) and ( r_2 ) are the roots (x-intercepts).

Step2: Find the x-intercepts

From the graph, the parabola crosses the x-axis at ( x = -8 ) and ( x = -4 ) (approximate, looking at the grid). So the roots are ( r_1 = -8 ) and ( r_2 = -4 ).

Step3: Write the factored form

Using the factored form ( y = a(x - (-8))(x - (-4)) = a(x + 8)(x + 4) ). Since the parabola opens downward, ( a < 0 ). Let's expand this: ( y = a(x^2 + 12x + 32) ).

Step4: Check the y-intercept (optional, but helps)

The graph seems to have a y-intercept, but maybe we can assume a = -1 for simplicity (common case). Then ( y = - (x^2 + 12x + 32) = -x^2 - 12x - 32 ). Let's check the vertex or other points. The vertex of a parabola ( y = ax^2 + bx + c ) is at ( x = -\frac{b}{2a} ). For ( y = -x^2 - 12x - 32 ), ( x = -\frac{-12}{2(-1)} = -6 ), which matches the vertex at x = -6 (from the graph, the peak is around x = -6). So this equation fits.

Answer:

A possible equation is ( y = -x^2 - 12x - 32 ) (or equivalent factored form ( y = - (x + 8)(x + 4) ))