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 - 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 \((h,k)=(- 6,1)\).

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, \(p<0\). Let's use another point on the parabola. Suppose we take a point \((-5, - 1)\) (approximate from the graph). Substitute \(x=-5,y = - 1,h=-6,k = 1\) into \((y - k)^2=4p(x - h)\).

$$(-1 - 1)^2=4p(-5+6)$$

$$4 = 4p\times1$$

$$p = 1$$

(but since it opens to the left \(p=-1\))

Step4: Write the equation of the parabola

Substitute \(h=-6,k = 1,p=-1\) into \((y - k)^2=4p(x - h)\)
\((y - 1)^2=-4(x + 6)\)

Answer:

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