Find the Vertex f(x)=3x^2-3x-6

$f (x) = 3 x^{2} - 3 x - 6$

Step 1

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} - 3 x - 6$ .

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

$a = 3$

$b = - 3$

$c = - 6$

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

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

Simplify the right side.

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

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

$e = - 6 - \frac{{(- 3)}^{2}}{4 \cdot 3}$

Simplify the right side.

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

Rewrite $- 3$ as $- 1 (3)$ .

$e = - 6 - \frac{{(- 1 (3))}^{2}}{4 \cdot 3}$

Apply the product rule to $- 1 (3)$ .

$e = - 6 - \frac{{(- 1)}^{2} \cdot 3^{2}}{4 \cdot 3}$

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

$e = - 6 - \frac{1 \cdot 3^{2}}{4 \cdot 3}$

Multiply $3^{2}$ by $1$ .

$e = - 6 - \frac{3^{2}}{4 \cdot 3}$

Factor $3$ out of $3^{2}$ .

$e = - 6 - \frac{3 \cdot 3}{4 \cdot 3}$

Cancel the common factors.

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

$e = - 6 - \frac{3 \cdot 3}{3 \cdot 4}$

Cancel the common factor.

$e = - 6 - \frac{3 \cdot 3}{3 \cdot 4}$

Rewrite the expression.

$e = - 6 - \frac{3}{4}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 6 \cdot \frac{4}{4} - \frac{3}{4}$

Combine $- 6$ and $\frac{4}{4}$ .

$e = \frac{- 6 \cdot 4}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 4 - 3}{4}$

Simplify the numerator.

Multiply $- 6$ by $4$ .

$e = \frac{- 24 - 3}{4}$

Subtract $3$ from $- 24$ .

$e = \frac{- 27}{4}$

Move the negative in front of the fraction.

$e = - \frac{27}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{1}{2})}^{2} - \frac{27}{4}$ .

$3 {(x - \frac{1}{2})}^{2} - \frac{27}{4}$

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{1}{2})}^{2} - \frac{27}{4}$

Step 2

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

$a = 3$

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

$k = - \frac{27}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{27}{4})$

Step 4

Find the Vertex f(x)=3x^2-3x-6

Related Questions

Support LCMGCF.com by disabling your adblocker.