Question

which equation best matches the graph shown below?
answer
\\( y = -5(x + 6)^2 + 1 \\) \\( y = 5(x + 6)^2 + 1 \\)
\\( y = -5(x - 6)^2 + 1 \\) \\( y = 5(x - 6)^2 + 1 \\)

Explanation:

Step1: Identify vertex form of parabola

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 \((-6, 1)\), so \( h = -6 \) and \( k = 1 \). Substituting into the vertex form, we get \( y = a(x - (-6))^2 + 1 = a(x + 6)^2 + 1 \). This eliminates the options with \((x - 6)\) (third and fourth options).

Step2: Determine the direction (a's sign)

The parabola opens downward, so the coefficient \( a \) must be negative. Among the remaining options (\( y = -5(x + 6)^2 + 1 \) and \( y = 5(x + 6)^2 + 1 \)), the one with \( a = -5 \) (negative) matches the downward opening.

Answer:

\( y = -5(x + 6)^2 + 1 \) (the first option: \( y = -5(x + 6)^2 + 1 \))