A parabola opening up or down has vertex (0, 6) and passes through (−6, 3). Write its equation in vertex form. Simplify any fractions.

Steps to Solve

1. Write down the vertex form of a quadratic function

The vertex form of a quadratic function is given by:

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

where $(h, k)$ is the vertex of the parabola.

2. Substitute the vertex values into the equation

Given the vertex is $(0, 6)$, we can substitute $h = 0$ and $k = 6$ into the equation:

$$ y = a(x - 0)^2 + 6 $$

This simplifies to:

$$ y = ax^2 + 6 $$

3. Substitute the point into the equation to find 'a'

We know the parabola passes through the point $(-6, 3)$. By substituting $x = -6$ and $y = 3$ into the equation, we can find $a$:

$$ 3 = a(-6)^2 + 6 $$

This simplifies to:

$$ 3 = 36a + 6 $$

4. Solve for 'a'

Next, we solve the equation:

$$ 3 - 6 = 36a $$

Which gives us:

$$ -3 = 36a $$

Dividing both sides by 36:

$$ a = -\frac{1}{12} $$

5. Write the final equation in vertex form

Now that we have the value of $a$, we can write the complete equation:

$$ y = -\frac{1}{12}x^2 + 6 $$