Precalculus Examples

$f (x) = - \frac{1}{2} x^{2} - 4 x + 10$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

$y = - \frac{x^{2}}{2} - 4 x + 10$

Step 1.2

Complete the square for $- \frac{x^{2}}{2} - 4 x + 10$ .

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Step 1.2.1

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

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

$b = - 4$

$c = 10$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 4$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 4$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 2 (- 1 \cdot 2)$

Step 1.2.3.2.3

Multiply $- 2 (- 1 \cdot 2)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $2$ .

$d = - 2 \cdot - 2$

Step 1.2.3.2.3.2

Multiply $- 2$ by $- 2$ .

$d = 4$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = 10 - \frac{{(- 4)}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 1.2.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 10 - \frac{{(- 1 (4))}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 10 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.3

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

$e = 10 - \frac{1 \cdot 4^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 10 - \frac{4^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 10 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.6

Cancel the common factors.

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Step 1.2.4.2.1.1.6.1

Cancel the common factor.

$e = 10 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.6.2

Rewrite the expression.

$e = 10 - \frac{4}{- \frac{1}{2}}$

Step 1.2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = 10 - (4 (- 1 \cdot 2))$

Step 1.2.4.2.1.3

Multiply $- (4 (- 1 \cdot 2))$ .

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Step 1.2.4.2.1.3.1

Multiply $- 1$ by $2$ .

$e = 10 - (4 \cdot - 2)$

Step 1.2.4.2.1.3.2

Multiply $4$ by $- 2$ .

$e = 10 - - 8$

Step 1.2.4.2.1.3.3

Multiply $- 1$ by $- 8$ .

$e = 10 + 8$

Step 1.2.4.2.2

Add $10$ and $8$ .

$e = 18$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 4)}^{2} + 18$ .

$- \frac{1}{2} {(x + 4)}^{2} + 18$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + 4)}^{2} + 18$

Step 2

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

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

$h = - 4$

$k = 18$

Step 3

Find the vertex $(h, k)$ .

$(- 4, 18)$

Step 4