Find the Standard Form of the Parabola f(x)=3x^2+12x+4

$f (x) = 3 x^{2} + 12 x + 4$

Step 1

Write $f (x) = 3 x^{2} + 12 x + 4$ as an equation.

$y = 3 x^{2} + 12 x + 4$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 x^{2} + 12 x + 4 = y$

Step 3

Subtract $4$ from both sides of the equation.

$3 x^{2} + 12 x = y - 4$

Step 4

Complete the square.

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

$a = 3$

$b = 12$

$c = 0 = y - 4$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 4$

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

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

$d = \frac{12}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot 3}$

Cancel the common factors.

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

$d = \frac{2 \cdot 6}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{6}{3}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = 2$

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

$e = 0 - \frac{12^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 0 - \frac{144}{12}$

Divide $144$ by $12$ .

$e = 0 - 1 \cdot 12$

Multiply $- 1$ by $12$ .

$e = 0 - 12$

Subtract $12$ from $0$ .

$e = - 12$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 2)}^{2} - 12$ .

$3 {(x + 2)}^{2} - 12 = y - 4$

Step 5

Move all terms not containing $x$ to the right side of the equation.

Add $12$ to both sides of the equation.

$3 {(x + 2)}^{2} = y - 4 + 12$

Add $- 4$ and $12$ .

$3 {(x + 2)}^{2} = y + 8$

Step 6

Divide each term in $3 {(x + 2)}^{2} = y + 8$ by $3$ and simplify.

Divide each term in $3 {(x + 2)}^{2} = y + 8$ by $3$ .

$\frac{3 {(x + 2)}^{2}}{3} = \frac{y}{3} + \frac{8}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 {(x + 2)}^{2}}{3} = \frac{y}{3} + \frac{8}{3}$

Divide ${(x + 2)}^{2}$ by $1$ .

${(x + 2)}^{2} = \frac{y}{3} + \frac{8}{3}$

Step 7

Factor.

Combine the numerators over the common denominator.

${(x + 2)}^{2} = \frac{y + 8}{3}$

Reorder terms.

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

Find the Standard Form of the Parabola f(x)=3x^2+12x+4

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