Find the Vertex f(x)=3x^2-2x-2

$f (x) = 3 x^{2} - 2 x - 2$

Step 1

Rewrite the equation in vertex form.

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

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

$a = 3$

$b = - 2$

$c = - 2$

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

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

Simplify the right side.

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

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 \cdot 3}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 1}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{- 1}{3}$

Move the negative in front of the fraction.

$d = - \frac{1}{3}$

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $3$ .

$e = - 2 - \frac{4}{12}$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$e = - 2 - \frac{4 (1)}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

$e = - 2 - \frac{1}{3}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e = - 2 \cdot \frac{3}{3} - \frac{1}{3}$

Combine $- 2$ and $\frac{3}{3}$ .

$e = \frac{- 2 \cdot 3}{3} - \frac{1}{3}$

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 3 - 1}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$e = \frac{- 6 - 1}{3}$

Subtract $1$ from $- 6$ .

$e = \frac{- 7}{3}$

Move the negative in front of the fraction.

$e = - \frac{7}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{1}{3})}^{2} - \frac{7}{3}$ .

$3 {(x - \frac{1}{3})}^{2} - \frac{7}{3}$

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{1}{3})}^{2} - \frac{7}{3}$

Step 2

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

$a = 3$

$h = \frac{1}{3}$

$k = - \frac{7}{3}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{3}, - \frac{7}{3})$

Step 4

Find the Vertex f(x)=3x^2-2x-2

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