Question

which equation best matches the graph shown below?
answer
\\( y = -0.2(x - 6)^2 + 2 \\) \\( y = -0.2(x + 6)^2 + 2 \\)
\\( y = 0.2(x + 6)^2 + 2 \\) \\( y = 0.2(x - 6)^2 + 2 \\)

Explanation:

Step1: Determine the direction of the parabola

The graph opens downward, so the coefficient of the squared term should be negative. This eliminates the options \( y = 0.2(x + 6)^2 + 2 \) and \( y = 0.2(x - 6)^2 + 2 \).

Step2: Find the vertex of the parabola

The vertex of the parabola is at \((-6, 2)\) (since it's shifted 6 units to the left and 2 units up). The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For a vertex at \((-6, 2)\), \( h = -6 \) and \( k = 2 \), so the equation should be \( y = a(x - (-6))^2 + 2 = a(x + 6)^2 + 2 \). Since the parabola opens downward, \( a \) is negative, so the equation is \( y = -0.2(x + 6)^2 + 2 \).

Answer:

\( y = -0.2(x + 6)^2 + 2 \)