Question

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: Analyze parabola direction

The parabola opens downward, so the coefficient of the squared term should be negative. Eliminate options with positive coefficients (0.4), so remove \( y = 0.4(x + 6)^2 + 1 \) and \( y = 0.4(x - 6)^2 + 1 \).

Step2: Find vertex form and vertex position

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. The graph's vertex is at \((-6, 1)\) (since it's left of the y - axis, \( h=-6 \)). For \( y=-0.4(x + 6)^2+1 \), \( h=-6 \) (because \( x+6=x - (-6) \)) and \( k = 1 \), matching the vertex. For \( y=-0.4(x - 6)^2+1 \), \( h = 6 \), which is right of the y - axis, not matching the graph.

Answer:

\( y = -0.4(x + 6)^2 + 1 \)