Answer:
The equation of the parabola is:
y
=
1
32
x
2
Explanation:
Please notice that we know that the parabola opens upward or downward, because the directrix is a horizontal line:
y
=
−
8
This tells use that the equation of the parabola of the form:
y
=
f
(
x
)
Not
x
=
f
(
y
)
.
Furthermore, we can see that it must open upward, because the y coordinate of the focus is above the directrix.
The vertex has the same x coordinate as the focus:
x
=
0
And its y coordinate is halfway between the focus and the directrix:
y
=
8
−
8
2
=
0
Therefore the vertex is
(
0
,
0
)
Use the vertex form of the equation of a parabola that opens upward or downward:
y
=
a
(
x
−
h
)
2
+
k
where the vertex is the point
(
h
,
k
)
.
Substitute the vertex into the equation:
y
=
a
(
x
−
0
)
2
+
0
Simplify:
y
=
a
x
2
The distance, f, from the vertex to the focus is,
f
=
8
; this allows use to find the value of
a
, using the equation:
a
=
1
4
f
a
=
1
32
The equation of the parabola is:
y
=
1
32
x
2
Step-by-step explanation:
