Question

which equation best matches the graph shown below?

answer
( y = 0.4(x + 5)^2 - 6 ) ( y = 0.4(x - 5)^2 - 6 )
( y = -0.4(x + 5)^2 - 6 ) ( y = -0.4(x - 5)^2 - 6 )

Explanation:

Step1: Determine the direction of the parabola

The graph of the parabola opens downward, so the coefficient of the squared term should be negative. This eliminates the options \( y = 0.4(x + 5)^2 - 6 \) and \( y = 0.4(x - 5)^2 - 6 \) since their coefficients are positive.

Step2: Find the vertex of the 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 \((-5, -6)\). In the vertex form, \( h = -5 \) and \( k = -6 \). Substituting these values into the vertex form, we get \( y = a(x - (-5))^2 + (-6) = a(x + 5)^2 - 6 \). Since we already determined that \( a \) is negative (from the direction of the parabola), and among the remaining options (\( y = -0.4(x + 5)^2 - 6 \) and \( y = -0.4(x - 5)^2 - 6 \)), the one with \( h = -5 \) is \( y = -0.4(x + 5)^2 - 6 \).

Answer:

\( y = -0.4(x + 5)^2 - 6 \)