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

Understand the Problem

The question is asking us to write the equation of a parabola in vertex form given its vertex and a point through which it passes. We need to apply the standard form of a parabola and use the provided vertex and point to derive the equation.

Answer

The equation of the parabola is $$ y = -\frac{1}{12}x^2 + 6. $$

Steps to Solve

Write the vertex form of a parabola

The vertex form of a parabola is given by the equation:

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

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

Substitute the vertex into the equation

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

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

This simplifies to:

$$ y = ax^2 + 6 $$

Use the second point to find 'a'

The parabola passes through the point $(-6, 3)$. We will substitute $x = -6$ and $y = 3$ into the equation:

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

This simplifies to:

$$ 3 = 36a + 6 $$

Solve for 'a'

Now, we isolate the variable $a$:

$$ 3 - 6 = 36a $$

This results in:

$$ -3 = 36a $$

Divide both sides by 36:

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

Write the final equation

Now that we have the value of $a$, we substitute it back into the equation for the parabola:

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

The equation of the parabola in vertex form is

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

More Information

The vertex form of a parabola is particularly useful because it highlights the vertex. The value of $a$ indicates that the parabola opens downward since it is negative, and it is relatively wide due to the small absolute value of $a$.

Tips

Forgetting to substitute both coordinates of the point correctly into the equation.
Confusing the vertex with the point on the parabola.

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