Question

which equation represents the set of points equidistant from point f and line d?
a. $y = 3(x + 2)^2+2$
b. $y = 3(x - 2)^2+2$

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of points equidistant from a fixed - point (focus) and a fixed - line (directrix). The standard form of a parabola with vertex \((h,k)\) is \(y=a(x - h)^2+k\), where the distance from the vertex to the focus and from the vertex to the directrix is \(\frac{1}{4|a|}\).

Step2: Identify the vertex of the parabola

The vertex of the parabola is the mid - point between the focus \(F\) and the directrix \(d\). By observing the graph (assuming the focus \(F\) has \(x\) - coordinate \(- 2\) and \(y\) - coordinate \(5\) and the directrix \(y=-1\)), the \(x\) - coordinate of the vertex \(h=-2\) and the \(y\) - coordinate of the vertex \(k = 2\) (since \(\frac{5+( - 1)}{2}=2\)).

Step3: Determine the value of \(a\)

The distance from the vertex \((-2,2)\) to the focus \(F\) (or to the directrix) is \(3\). We know that the distance from the vertex to the focus is \(\frac{1}{4|a|}\), so \(\frac{1}{4|a|}=3\), then \(|a|=\frac{1}{12}\). But we can also use the general form and check the options. Since the vertex is \((h=-2,k = 2)\), the equation of the parabola is \(y=a(x+2)^2 + 2\).

Answer:

A. \(y = 3(x + 2)^2+2\)