Question

A parabola opening up or down has vertex (2,
–
4) and passes through 8,
1
2
. Write its equation in vertex form.
Simplify any fractions.

Answer 1

To write the equation of a parabola in vertex form, we use the formula:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola. Given that the vertex is \((2, -4)\), we can substitute \(h = 2\) and \(k = -4\):

\[
y = a(x - 2)^2 - 4
\]

Next, we need to determine the value of \(a\). We can do this by using the other point through which the parabola passes, which is \((8, \frac{1}{2})\). We substitute \(x = 8\) and \(y = \frac{1}{2}\) into the equation:

\[
\frac{1}{2} = a(8 - 2)^2 - 4
\]

First, simplify \( (8 - 2)^2 \):

\[
(8 - 2)^2 = 6^2 = 36
\]

Now substitute this back into the equation:

\[
\frac{1}{2} = a(36) - 4
\]

Next, solve for \(a\):

\[
\frac{1}{2} + 4 = 36a
\]

Convert \(4\) to a fraction:

\[
\frac{1}{2} + \frac{8}{2} = 36a
\]

This gives:

\[
\frac{9}{2} = 36a
\]

Now, solve for \(a\) by dividing both sides by 36:

\[
a = \frac{9}{2 \times 36} = \frac{9}{72} = \frac{1}{8}
\]

Now that we have \(a\), we can write the final equation:

\[
y = \frac{1}{8}(x - 2)^2 - 4
\]

This is the equation of the parabola in vertex form:

\[
\boxed{y = \frac{1}{8}(x - 2)^2 - 4}
\]