Precalculus Examples

$f (x) = - 2 x^{2} - 8 x + 9$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- 2 x^{2} - 8 x + 9$ .

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Step 1.1.1

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

$a = - 2$

$b = - 8$

$c = 9$

Step 1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

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

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Step 1.1.3.2.1.1

Factor $2$ out of $- 8$ .

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

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

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

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

Step 1.1.3.2.1.2.2

Cancel the common factor.

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

Step 1.1.3.2.1.2.3

Rewrite the expression.

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

Step 1.1.3.2.2

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

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Step 1.1.3.2.2.1

Factor $- 2$ out of $- 4$ .

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

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor $- 2$ out of $- 2$ .

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

Step 1.1.3.2.2.2.2

Cancel the common factor.

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

Step 1.1.3.2.2.2.3

Rewrite the expression.

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

Step 1.1.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 1.1.4

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

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Step 1.1.4.1

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

$e = 9 - \frac{{(- 8)}^{2}}{4 \cdot - 2}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

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

$e = 9 - \frac{64}{4 \cdot - 2}$

Step 1.1.4.2.1.2

Multiply $4$ by $- 2$ .

$e = 9 - \frac{64}{- 8}$

Step 1.1.4.2.1.3

Divide $64$ by $- 8$ .

$e = 9 - - 8$

Step 1.1.4.2.1.4

Multiply $- 1$ by $- 8$ .

$e = 9 + 8$

Step 1.1.4.2.2

Add $9$ and $8$ .

$e = 17$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + 2)}^{2} + 17$ .

$- 2 {(x + 2)}^{2} + 17$

Step 1.2

Set $y$ equal to the new right side.

$y = - 2 {(x + 2)}^{2} + 17$

Step 2

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

$a = - 2$

$h = - 2$

$k = 17$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 17)$

Step 4