Question

which equation best matches the graph shown below?
answer
$y = 2(x - 4)^2 + 1$ $y = 2(x + 4)^2 + 1$
$y = -2(x + 4)^2 + 1$ $y = -2(x - 4)^2 + 1$

Explanation:

Step1: Analyze the parabola's direction

The graph opens downward, so the coefficient of the squared term should be negative. This eliminates the first two options (\(y = 2(x - 4)^2 + 1\) and \(y = 2(x + 4)^2 + 1\)) since their coefficients are positive.

Step2: Find the vertex of the parabola

The vertex of the parabola is at \((4, 1)\)? Wait, no, looking at the graph, the vertex is at \(x = 4\)? Wait, no, the graph's vertex is at \(x = 4\)? Wait, no, the x - coordinate of the vertex: the standard form of a parabola is \(y=a(x - h)^2 + k\), where \((h,k)\) is the vertex. From the graph, the vertex is at \((4,1)\)? Wait, no, looking at the graph, the vertex is at \(x = 4\), \(y = 1\)? Wait, no, the graph is symmetric around \(x = 4\)? Wait, no, let's check the options. The fourth option is \(y=-2(x - 4)^2+1\), which has vertex \((4,1)\) and opens downward (since \(a=-2<0\)). The third option is \(y=-2(x + 4)^2+1\) with vertex \((-4,1)\). From the graph, the vertex is at \(x = 4\) (since the parabola is on the right side, around \(x = 4\)). So the correct equation should be \(y=-2(x - 4)^2+1\).

Answer:

\(y=-2(x - 4)^2+1\) (the fourth option: \(y=-2(x - 4)^2+1\))