Find the Vertex 8/3x^2-8/3x

$\frac{8}{3} x^{2} - \frac{8}{3} x$

Step 1

Rewrite the equation in vertex form.

Simplify each term.

Combine $\frac{8}{3}$ and $x^{2}$ .

$y = \frac{8 x^{2}}{3} - \frac{8}{3} x$

Combine $x$ and $\frac{8}{3}$ .

$y = \frac{8 x^{2}}{3} - \frac{x \cdot 8}{3}$

Move $8$ to the left of $x$ .

$y = \frac{8 x^{2}}{3} - \frac{8 x}{3}$

Complete the square for $\frac{8 x^{2}}{3} - \frac{8 x}{3}$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = \frac{8}{3}$

$b = - \frac{8}{3}$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- \frac{8}{3}}{2 (\frac{8}{3})}$

Simplify the right side.

Cancel the common factor of $\frac{8}{3}$ .

Cancel the common factor.

$d = \frac{- \frac{8}{3}}{2 (\frac{8}{3})}$

Rewrite the expression.

$d = \frac{- 1}{2}$

Move the negative in front of the fraction.

$d = - \frac{1}{2}$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- \frac{8}{3})}^{2}}{4 (\frac{8}{3})}$

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- \frac{8}{3})}^{2}$ and $\frac{8}{3}$ .

Factor $- 1$ out of $- \frac{8}{3}$ .

$e = 0 - \frac{{(- (\frac{8}{3}))}^{2}}{4 (\frac{8}{3})}$

Rewrite $- (\frac{8}{3})$ as $- 1 (\frac{8}{3})$ .

$e = 0 - \frac{{(- 1 (\frac{8}{3}))}^{2}}{4 (\frac{8}{3})}$

Apply the product rule to $- 1 (\frac{8}{3})$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{8}{3})}^{2}}{4 (\frac{8}{3})}$

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 {(\frac{8}{3})}^{2}}{4 (\frac{8}{3})}$

Multiply ${(\frac{8}{3})}^{2}$ by $1$ .

$e = 0 - \frac{{(\frac{8}{3})}^{2}}{4 (\frac{8}{3})}$

Factor $\frac{8}{3}$ out of ${(\frac{8}{3})}^{2}$ .

$e = 0 - \frac{\frac{8}{3} \cdot \frac{8}{3}}{4 (\frac{8}{3})}$

Cancel the common factors.

Factor $\frac{8}{3}$ out of $4 (\frac{8}{3})$ .

$e = 0 - \frac{\frac{8}{3} \cdot \frac{8}{3}}{\frac{8}{3} \cdot 4}$

Cancel the common factor.

$e = 0 - \frac{\frac{8}{3} \cdot \frac{8}{3}}{\frac{8}{3} \cdot 4}$

Rewrite the expression.

$e = 0 - \frac{\frac{8}{3}}{4}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{8}{3} \cdot \frac{1}{4})$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$e = 0 - (\frac{4 (2)}{3} \cdot \frac{1}{4})$

Cancel the common factor.

$e = 0 - (\frac{4 \cdot 2}{3} \cdot \frac{1}{4})$

Rewrite the expression.

$e = 0 - \frac{2}{3}$

Subtract $\frac{2}{3}$ from $0$ .

$e = - \frac{2}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{8}{3} {(x - \frac{1}{2})}^{2} - \frac{2}{3}$ .

$\frac{8}{3} {(x - \frac{1}{2})}^{2} - \frac{2}{3}$

Set $y$ equal to the new right side.

$y = \frac{8}{3} \cdot {(x - \frac{1}{2})}^{2} - \frac{2}{3}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = \frac{8}{3}$

$h = \frac{1}{2}$

$k = - \frac{2}{3}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{2}{3})$

Step 4

Find the Vertex 8/3x^2-8/3x

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