Given the vertex at $(0, 1)$ and the focus at $(0, 2.5)$, we can write the equation of the parabola as follows:

• The parabola opens upwards because the focus is above the vertex.
• The distance between the vertex and the focus is $2.5 - 1 = 1.5$ units. This distance represents $\frac{1}{4a}$, so $a = \frac{1}{4 \times 1.5} = \frac{1}{6}$.

Using the vertex form of the equation $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex:

$y = \frac{1}{6}(x - 0)^2 + 1$

Simplifying this:

$y = \frac{1}{6}x^2 + 1$

Therefore, the sentence should read:

"An equation of the parabola is $y = \frac{1}{6}x^2 + 1$."

Brief Explanation:

The equation was derived using the vertex form $y = a(x-h)^2 + k$, with the vertex at $(0, 1)$ and the value of $a$ calculated based on the distance between the vertex and the focus, which is $1.5$ units.

Would you like any more details or have any questions?

Here are some related questions:

1. How does the distance between the vertex and the focus affect the shape of the parabola?
2. How can you determine whether a parabola opens upwards or downwards?
3. What is the significance of the vertex in determining the equation of a parabola?
4. How would the equation change if the parabola opened downwards?
5. How do you find the directrix of a parabola given the vertex and focus?

Tip: Remember that the distance between the vertex and the focus helps determine the steepness of the parabola. The larger this distance, the narrower the parabola.