To write the equation of the parabola shown in the image, let's analyze the details:

• Vertex: The vertex of the parabola is at the point $(0, 2)$.
• Focus: The focus of the parabola is at the point $(0, 3)$.

Since the parabola opens upwards and its vertex form equation is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex, the equation can be determined by the distance between the vertex and the focus.

The distance between the vertex and the focus is 1 unit (from $y = 2$ to $y = 3$). This distance is equal to $\frac{1}{4a}$, so $a = \frac{1}{4 \times 1} = \frac{1}{4}$.

Now, substituting the values into the vertex form equation:

$y = \frac{1}{4}(x - 0)^2 + 2$

Simplifying this:

$y = \frac{1}{4}x^2 + 2$

Thus, the sentence should read:

"An equation of the parabola is $y = \frac{1}{4}x^2 + 2$."

Brief Explanation:

The equation was derived using the vertex form $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex $(0, 2)$. The value of $a$ was determined by the distance between the vertex and the focus, which is $\frac{1}{4}$, given the focus is 1 unit above the vertex.

Would you like any more details or have any questions?

Here are some related questions:

1. How do you find the value of $a$ in a parabola?
2. What is the significance of the focus in a parabola?
3. How does the direction a parabola opens affect its equation?
4. How can you find the equation of a parabola if given the directrix instead of the focus?
5. What role does the vertex play in the equation of a parabola?

Tip: When writing the equation of a parabola, always ensure you correctly identify the vertex and the direction in which the parabola opens.