Find the Vertex y=-3x^2+4x

$y = - 3 x^{2} + 4 x$

Rewrite the equation in vertex form.

Complete the square for $- 3 x^{2} + 4 x$ .

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

$a = - 3$

$b = 4$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

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

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 \cdot - 3}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{4^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot - 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{4}{3}$ .

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$e = 0 + \frac{4}{3}$

Add $0$ and $\frac{4}{3}$ .

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

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{2}{3})}^{2} + \frac{4}{3}$ .

$- 3 {(x - \frac{2}{3})}^{2} + \frac{4}{3}$

Set $y$ equal to the new right side.

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

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

$a = - 3$

$h = \frac{2}{3}$

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

Find the vertex $(h, k)$ .

$(\frac{2}{3}, \frac{4}{3})$

Find the Vertex y=-3x^2+4x

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