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The problem shows a parabola with its vertex at \( (1, 2) \) and another point on the parabola at \( (5, 3) \). To find the equation of the parabola, we can use the vertex form of a parabola, which is:

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

where \( (h, k) \) is the vertex of the parabola. In this case, the vertex is \( (1, 2) \), so the equation becomes:

\[
y = a(x - 1)^2 + 2
\]

Now, we need to determine the value of \( a \). To do this, we use the given point \( (5, 3) \). Substitute \( x = 5 \) and \( y = 3 \) into the equation:

\[
3 = a(5 - 1)^2 + 2
\]

Simplifying:

\[
3 = a(4)^2 + 2
\]
\[
3 = 16a + 2
\]
\[
1 = 16a
\]
\[
a = \frac{1}{16}
\]

Thus, the equation of the parabola is:

\[
y = \frac{1}{16}(x - 1)^2 + 2
\]

Let me know if you need further details or explanations!

Here are five related questions to expand on this concept:
1. How would you find the focus and directrix of this parabola?
2. What would happen to the parabola if the vertex were moved to \( (2, 3) \)?
3. How do you convert the vertex form of a parabola to the standard form \( y = ax^2 + bx + c \)?
4. What is the general formula for the axis of symmetry in the vertex form of a parabola?
5. How does the value of \( a \) affect the shape and direction of the parabola?

**Tip**: The value of \( a \) in the vertex form controls the "width" of the parabola. A smaller \( |a| \) makes the parabola wider, while a larger \( |a| \) makes it narrower.

A continuación se muestra una parábola. Su vértice y otro punto de la parábola están indicados. Escribir una ecuación de la parábola. El vértice es (1, 2) y otro punto es (5, 3).

Solve the problem of finding the equation of a parabola given its vertex at (1, 2) and another point at (5, 3). The solution uses the vertex form of a parabola and shows how to calculate the value of 'a'. Suitable for Grades 9-10.

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