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.