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. Identify the Vertex and a Point The vertex of the parabola is given as $(h, k) = (6, -4)$. Another point on the parabola is $(x, y) = \left(10, -\frac{8}{3}\right)$.

2. Write the Vertex Form of the Equation The vertex form of a parabola is given by the equation: $$ y = a(x - h)^2 + k $$ Substituting the vertex into the equation: $$ y = a(x - 6)^2 - 4 $$

3. Substitute the Other Point Now we will use the point $(10, -\frac{8}{3})$ to find the value of $a$. Substitute $x = 10$ and $y = -\frac{8}{3}$: $$ -\frac{8}{3} = a(10 - 6)^2 - 4 $$

4. Simplify the Equation Simplifying the equation: $$ -\frac{8}{3} = a(4) - 4 $$

5. Isolate the Variable a To isolate $a$, first add $4$ to both sides: $$ -\frac{8}{3} + 4 = 4a $$ Convert $4$ to a fraction: $$ -\frac{8}{3} + \frac{12}{3} = 4a $$ Therefore, $$ \frac{4}{3} = 4a $$

6. Solve for a Divide both sides by $4$: $$ a = \frac{4/3}{4} = \frac{1}{3} $$

7. Write the Final Equation Substitute $a$ back into the vertex form: $$ y = \frac{1}{3}(x - 6)^2 - 4 $$