Precalculus Examples

$f (x) = - \frac{1}{2} x^{2} - 2 x + 6$

Step 1

Write $f (x) = - \frac{1}{2} x^{2} - 2 x + 6$ as an equation.

$y = - \frac{1}{2} x^{2} - 2 x + 6$

Step 2

Combine $x^{2}$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2} - 2 x + 6$

Step 3

Complete the square for $- \frac{x^{2}}{2} - 2 x + 6$ .

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Step 3.1

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

$a = - \frac{1}{2}$

$b = - 2$

$c = 6$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

Cancel the common factor.

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

Step 3.3.2.1.2.2

Rewrite the expression.

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

Step 3.3.2.2

Dividing two negative values results in a positive value.

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

Step 3.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Step 3.3.2.4

Multiply $2$ by $1$ .

$d = 2$

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

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

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

Step 3.4.2.1.2

Simplify the denominator.

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Step 3.4.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 3.4.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

$e = 6 - \frac{4}{\frac{- 4}{2}}$

Step 3.4.2.1.3

Divide $- 4$ by $2$ .

$e = 6 - \frac{4}{- 2}$

Step 3.4.2.1.4

Divide $4$ by $- 2$ .

$e = 6 - - 2$

Step 3.4.2.1.5

Multiply $- 1$ by $- 2$ .

$e = 6 + 2$

Step 3.4.2.2

Add $6$ and $2$ .

$e = 8$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 2)}^{2} + 8$ .

$- \frac{1}{2} {(x + 2)}^{2} + 8$

Step 4

Set $y$ equal to the new right side.

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