Question

which equation best matches the graph shown below?
answer
$y = (x + 3)^2 - 6$ $y = -(x + 3)^2 - 6$
$y = -(x - 3)^2 - 6$ $y = (x - 3)^2 - 6$

Explanation:

Step1: Analyze the parabola's direction

The parabola opens upwards, so the coefficient of the squared term should be positive. This eliminates options with a negative coefficient, i.e., \( y = -(x + 3)^2 - 6 \) and \( y = -(x - 3)^2 - 6 \).

Step2: Identify the vertex form

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the graph, the vertex is at \((-3, -6)\). So \( h = -3 \) and \( k = -6 \). Substituting into the vertex form, we get \( y = (x - (-3))^2 - 6 = (x + 3)^2 - 6 \).

Answer:

\( y = (x + 3)^2 - 6 \) (corresponding to the first option: \( y = (x + 3)^2 - 6 \))