Question

which equation best matches the graph shown below?
answer
\\( y = -0.4(x + 6)^2 + 1 \\) \\( y = -0.4(x - 6)^2 + 1 \\)
\\( y = 0.4(x + 6)^2 + 1 \\) \\( y = 0.4(x - 6)^2 + 1 \\)

Explanation:

Step1: 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 \((-6, 1)\), so \( h = -6 \) and \( k = 1 \). Substituting into the vertex form, we get \( y = a(x + 6)^2 + 1 \) (since \( x - (-6) = x + 6 \)).

Step2: Determine the direction of opening

The parabola opens downward, so the coefficient \( a \) must be negative. Among the options with \( (x + 6)^2 \), only \( y = -0.4(x + 6)^2 + 1 \) has a negative \( a \).

Answer:

\( y = -0.4(x + 6)^2 + 1 \) (the first option: \( y = -0.4(x + 6)^2 + 1 \))