Question

which equation best matches the graph shown below? answer \\( y = 4(x - 5)^2 + 2 \\) \\( y = -4(x + 5)^2 + 2 \\) \\( y = 4(x + 5)^2 + 2 \\) \\( y = -4(x - 5)^2 + 2 \\)

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.

Step2: Determine the vertex

From the graph, the vertex is at \((-5, 2)\) (since it's the highest point on the downward - opening parabola). So \( h=-5\) and \( k = 2\). Substituting into the vertex form, we get \( y=a(x + 5)^2+2\) (because \(x-(-5)=x + 5\)).

Step3: Determine the direction and the value of \(a\)

The parabola opens downward, so \(a<0\). Among the options with \(h=-5\) and \(k = 2\), the equation \(y=-4(x + 5)^2+2\) has \(a=-4<0\) and the correct vertex \((-5,2)\). The other option \(y = 4(x + 5)^2+2\) has \(a = 4>0\) which would open upward, so it's incorrect. The options with \(h = 5\) have the vertex at \((5,2)\) which does not match the graph.

Answer:

\(y=-4(x + 5)^2+2\) (the option \(y = - 4(x + 5)^2+2\))