A parabola opening up or down has vertex (6, -4) and passes through (10, -8/3). Write its equation in vertex form. Simplify any fractions.

Steps to Solve

1. Start with the vertex form of a quadratic equation The vertex form of a quadratic equation is given by:

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

where ( (h, k) ) is the vertex of the parabola. In this case, ( (h, k) = (6, -4) ).

2. Substitute the vertex into the equation Substituting the vertex coordinates into the vertex form, we get:

$$ y = a(x - 6)^2 - 4 $$

3. Use the given point to find ( a ) Now we will use the point ( (10, -\frac{8}{3}) ) to find the value of ( a ). Substitute ( x = 10 ) and ( y = -\frac{8}{3} ) into the equation:

$$ -\frac{8}{3} = a(10 - 6)^2 - 4 $$

This simplifies to:

$$ -\frac{8}{3} = a(4) - 4 $$

4. Solve for ( a ) Rearranging the equation gives us:

$$ a(4) = -\frac{8}{3} + 4 $$

Convert 4 to a fraction with a common denominator:

$$ 4 = \frac{12}{3} $$

So we have:

$$ a(4) = -\frac{8}{3} + \frac{12}{3} $$

This simplifies to:

$$ a(4) = \frac{4}{3} $$

To find ( a ):

$$ a = \frac{4}{3} \div 4 = \frac{4}{3} \cdot \frac{1}{4} = \frac{1}{3} $$

5. Write the final equation Now substitute ( a ) back into the equation:

$$ y = \frac{1}{3}(x - 6)^2 - 4 $$