Question

the graph of a parabola is given below. match the graph to its equation.
choose the correct equation below
a. $(x + 6)^2 = -4(y - 1)$
b. $(x + 6)^2 = 4(y - 1)$
c. $(x - 6)^2 = 4(y + 1)$
d. $(x - 6)^2 = -4(y + 1)$
e. $(y - 1)^2 = 4(x + 6)$
f. $(y - 1)^2 = -4(x + 6)$
g. $(y + 1)^2 = 4(x - 6)$
h. $(y + 1)^2 = -4(x - 6)$

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens left or right is \((y - k)^2=4p(x - h)\), where \((h,k)\) is the vertex. If \(p>0\), it opens to the right; if \(p < 0\), it opens to the left.

Step2: Identify the vertex of the parabola

From the graph, the vertex of the parabola is \((- 6,1)\). So \(h=-6\) and \(k = 1\).

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

The parabola opens to the left. For a parabola of the form \((y - k)^2=4p(x - h)\), when it opens to the left, \(p<0\). Also, we can use another point on the parabola. Let's assume we use the symmetry. The general form becomes \((y - 1)^2=4p(x + 6)\).
Take a point, say when \(y = 1\), we can check the behavior. Since it opens to the left, comparing with the standard form \((y - k)^2=4p(x - h)\), we know that \(4p=-4\) (by observing the shape and direction, if we consider the basic transformation, for a parabola opening left with vertex \((-6,1)\) the equation is \((y - 1)^2=-4(x + 6)\)

Answer:

F. \((y - 1)^2=-4(x + 6)\)