Question

the focus, (0, -2), and directrix, x = 6, of a parabola are graphed below. which equation represents the parabola? x = 1/12(y + 2)^2 + 3 x = -1/12(y + 2)^2 + 3 x = -1/12(y - 2)^2 - 3 x = 1/12(y - 2)^2 - 3

Explanation:

Step1: Recall the formula for a parabola with a horizontal axis

The standard - form equation of a parabola with a horizontal axis is \((y - k)^2=4p(x - h)\), where \((h,k)\) is the vertex and \(p\) is the distance from the vertex to the focus (or from the vertex to the directrix). The vertex of a parabola is the mid - point between the focus \((x_f,y_f)\) and the point on the directrix closest to the focus. The focus is \((0, - 2)\) and the directrix is \(x = 6\). The \(y\) - coordinate of the vertex is the same as the \(y\) - coordinate of the focus, so \(k=-2\). The \(x\) - coordinate of the vertex \(h\) is given by \(h=\frac{0 + 6}{2}=3\). So the vertex is \((3,-2)\).

Step2: Calculate the value of \(p\)

The distance \(p\) from the vertex \((3,-2)\) to the focus \((0,-2)\) is \(p=0 - 3=-3\) (negative because the focus is to the left of the vertex).

Step3: Substitute \(h\), \(k\), and \(p\) into the standard form

Starting with \((y - k)^2 = 4p(x - h)\), substitute \(h = 3\), \(k=-2\), and \(p=-3\). First, rewrite the equation in terms of \(x\): \(x=h+\frac{1}{4p}(y - k)^2\). Substituting the values, we get \(x = 3+\frac{1}{4\times(-3)}(y+2)^2=3-\frac{1}{12}(y + 2)^2\).

Answer:

\(x=-\frac{1}{12}(y + 2)^2+3\)