Final answer:
To verify if a graph represents the quadratic inequality y ≤ x² + 12x + 15, check if the shaded area includes the parabola that opens upwards with the vertex found by completing the square or the vertex formula.
Step-by-step explanation:
To determine if the graph represents the quadratic inequality y ≤ x² + 12x + 15, we need to verify if the shaded region on the graph includes all the points where y is less than or equal to the quadratic expression. This particular quadratic inequality forms a parabola that opens upwards, as the coefficient of the x² term is positive.
The vertex of this parabola can be found by completing the square or by using the vertex formula, h = -b/2a. For this inequality x² + 12x + 15, we can complete the square to rewrite it in vertex form, or we can find the vertex by using the formula h = -b/2a, which would be -12/2(1) giving us h = -6.
This is the x-coordinate of the vertex; the y-coordinate can be found by substituting x with -6 in the original equation. If the shaded region includes this point and all points below the parabola up to and including the parabola itself, then it would be the correct graph for the given inequality.