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Part 1
Write the equation of the parabola in vertex form.
vertex ​(0​,3​), point left parenthesis 1 comma negative 4 right parenthesis
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Part 1
​f(x)equals

enter your response here ​(Simplify your​ answer.)

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.

Write the equation of the parabola in vertex form. Vertex (0, 3), point (1, -4).

Learn how to write the equation of a parabola in vertex form given a vertex (0, 3) and a point (1, -4). This step-by-step guide explains the vertex form equation and uses algebra to solve for the constant a. Suitable for students in Grades 9-11.

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