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

$f(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex, and $a$ is a constant that determines the width and direction of the parabola.

Given:

• Vertex: $(0, 3)$, so $h = 0$ and $k = 3$.
• Point on the parabola: $(1, -4)$.

Step 1: Substitute the vertex into the vertex form

Since $h = 0$ and $k = 3$, the equation becomes: $f(x) = a(x - 0)^2 + 3 \quad \Rightarrow \quad f(x) = a x^2 + 3$

Step 2: Use the point $(1, -4)$ to find $a$

Substitute $x = 1$ and $f(x) = -4$ into the equation: $-4 = a(1)^2 + 3$

Solve for $a$: $-4 = a + 3$ $a = -4 - 3 = -7$

Step 3: Write the final equation

Now that we know $a = -7$, the equation of the parabola is: $f(x) = -7x^2 + 3$

This is the equation of the parabola in vertex form.

Would you like more details or have any questions?

Here are 5 questions that could further expand your understanding:

1. What happens to the parabola if the value of $a$ changes?
2. How would the equation change if the vertex were different, say $(2, 5)$?
3. Can you graph the equation $f(x) = -7x^2 + 3$ and describe its shape?
4. What is the significance of the sign of $a$ in the equation?
5. How do we convert a standard form equation of a parabola to vertex form?

Tip: The vertex form of a parabola is useful because it immediately gives you the vertex and helps you analyze the direction and width of the parabola easily.