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Part 1

Write the equation of the parabola in vertex form.

vertex 

​(00​,66​), 

point left parenthesis 1 comma negative 1 right parenthesis(1,−1)

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Part 1

​f(x)equals=enter your response here 

​(Simplify your​ answer.)

f(x)(Simplify your answer.)

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Let's solve the problem step by step to find the equation of the parabola in vertex form.

### Vertex Form of a Parabola:
The vertex form of a parabola is:

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

where \((h, k)\) is the vertex, and \(a\) is a constant.

### Given:
- Vertex \((h, k) = (0, 6)\)
- Point \((x, y) = (1, -1)\)

### Step 1: Plug the vertex into the vertex form equation
Using the vertex \((h, k) = (0, 6)\), the equation becomes:

\[
f(x) = a(x - 0)^2 + 6
\]

Simplifying:

\[
f(x) = ax^2 + 6
\]

### Step 2: Substitute the point \((x, y) = (1, -1)\) to find \(a\)
Substitute \(x = 1\) and \(y = -1\) into the equation:

\[
-1 = a(1)^2 + 6
\]

\[
-1 = a + 6
\]

### Step 3: Solve for \(a\)
\[
a = -1 - 6 = -7
\]

### Step 4: Write the final equation
Substitute \(a = -7\) back into the equation:

\[
f(x) = -7x^2 + 6
\]

So, the equation of the parabola in vertex form is:

\[
f(x) = -7x^2 + 6
\]

Would you like more details or have any other questions?

### Related Questions:
1. How does changing the vertex affect the equation of the parabola?
2. What does the value of \(a\) represent in the vertex form equation?
3. How can you find the focus and directrix of a parabola from its vertex form?
4. How do you determine if the parabola opens upwards or downwards?
5. How can you convert this vertex form equation to standard form?

### Tip:
Always double-check your substitution steps when finding the value of \(a\) using a given point. Small mistakes can lead to incorrect values.

Learn how to write the equation of a parabola in vertex form using the given vertex and a point on the parabola. This problem involves applying the vertex form formula f(x) = a(x - h)^2 + k and solving for the constant 'a' using substitution. Suitable for high school students.

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