Question

which equation best matches the graph shown below?
answer
$y = 0.5(x + 5)^2 + 1$ $y = -0.5(x - 5)^2 + 1$
$y = 0.5(x - 5)^2 + 1$ $y = -0.5(x + 5)^2 + 1$

Explanation:

Step1: Analyze the parabola's direction

The parabola opens upwards, so the coefficient of the squared term should be positive. This eliminates the options with \(-0.5\) (i.e., \(y = -0.5(x - 5)^2 + 1\) and \(y = -0.5(x + 5)^2 + 1\)).

Step2: Find the vertex form 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 \((-5, 1)\) (since it's 5 units to the left of the origin on the x - axis and 1 unit up on the y - axis). In the vertex form, \(h=-5\) and \(k = 1\), and \(a = 0.5\) (positive for upward opening). So substituting \(h=-5\), \(k = 1\) and \(a = 0.5\) into the vertex form, we get \(y=0.5(x + 5)^2+1\) (because \(x-(-5)=x + 5\)).

Answer:

\(y = 0.5(x + 5)^2 + 1\) (the first option in the list of answers)