Find the Standard Form of the Parabola f(x)=4x^2+8x+5

$f (x) = 4 x^{2} + 8 x + 5$

Step 1

Write $f (x) = 4 x^{2} + 8 x + 5$ as an equation.

$y = 4 x^{2} + 8 x + 5$

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.

$4 x^{2} + 8 x + 5 = y$

Step 3

Subtract $5$ from both sides of the equation.

$4 x^{2} + 8 x = y - 5$

Step 4

Complete the square.

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

$a = 4$

$b = 8$

$c = 0 = y - 5$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 5$

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

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

$d = \frac{8}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot 4}{2 (4)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{4}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$d = \frac{4}{4}$

Rewrite the expression.

$d = 1$

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

$e = 0 - \frac{8^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

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

$4 {(x + 1)}^{2} - 4 = y - 5$

Step 5

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

Add $4$ to both sides of the equation.

$4 {(x + 1)}^{2} = y - 5 + 4$

Add $- 5$ and $4$ .

$4 {(x + 1)}^{2} = y - 1$

Step 6

Divide each term in $4 {(x + 1)}^{2} = y - 1$ by $4$ and simplify.

Divide each term in $4 {(x + 1)}^{2} = y - 1$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 7

Factor.

Combine the numerators over the common denominator.

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

Reorder terms.

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

Find the Standard Form of the Parabola f(x)=4x^2+8x+5

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