A parabola opening up or down has vertex (5, 4) and passes through (8, 23/8). Write its equation in vertex form.

To write the equation of the parabola in vertex form, we start with the general form of a parabola that opens up or down:

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

Here, \((h, k)\) is the vertex of the parabola. Given that the vertex is \((5, 4)\), we substitute \(h = 5\) and \(k = 4\) into the equation:

\[
y = a(x - 5)^2 + 4
\]

Next, we use the point \((8, \frac{23}{8})\) that lies on the parabola to find the value of \(a\). Substituting \(x = 8\) and \(y = \frac{23}{8}\) into the equation:

\[
\frac{23}{8} = a(8 - 5)^2 + 4
\]

Simplify inside the parentheses and square the result:

\[
\frac{23}{8} = a(3)^2 + 4
\]

\[
\frac{23}{8} = 9a + 4
\]

Next, subtract 4 from both sides:

\[
\frac{23}{8} - 4 = 9a
\]

To combine the fractions on the left-hand side, express 4 as \(\frac{32}{8}\):

\[
\frac{23}{8} - \frac{32}{8} = 9a
\]

\[
\frac{-9}{8} = 9a
\]

Now, solve for \(a\) by dividing both sides by 9:

\[
a = \frac{-9}{8 \times 9} = \frac{-1}{8}
\]

Now that we have \(a = -\frac{1}{8}\), the equation of the parabola is:

\[
y = -\frac{1}{8}(x - 5)^2 + 4
\]

This is the equation of the parabola in vertex form.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How do you determine if the parabola opens upwards or downwards?
2. Can you convert this equation into standard form \(y = ax^2 + bx + c\)?
3. What is the axis of symmetry for this parabola?
4. How does the value of \(a\) affect the width of the parabola?
5. If the parabola passes through another point, how can you verify the equation?

**Tip:** When the coefficient \(a\) is negative, the parabola opens downwards; when positive, it opens upwards.

A parabola opening up or down has vertex (5, 4) and passes through (8, 23/8). Write its equation in vertex form.

Learn how to find the equation of a parabola with vertex (5, 4) and passing through the point (8, 23/8) in vertex form. This problem explores quadratic functions, vertex form, and involves solving for the constant 'a'. Suitable for students in Grades 9-11.

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