Precalculus Examples

$f (x) = x^{2} - x - 2$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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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 $c$ .

$a = 1$

$b = - 1$

$c = - 2$

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 $d = \frac{b}{2 a}$ .

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 1.1.3.2.1.1

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

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

Step 1.1.3.2.1.2

Cancel the common factor.

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

Step 1.1.3.2.1.3

Rewrite the expression.

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

Step 1.1.3.2.2

Move the negative in front of the fraction.

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

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

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

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

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

Step 1.1.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.1.4.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 1.1.4.2.3

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

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

Step 1.1.4.2.4

Combine the numerators over the common denominator.

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

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $- 2$ by $4$ .

$e = \frac{- 8 - 1}{4}$

Step 1.1.4.2.5.2

Subtract $1$ from $- 8$ .

$e = \frac{- 9}{4}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{9}{4}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{1}{2})}^{2} - \frac{9}{4}$ .

${(x - \frac{1}{2})}^{2} - \frac{9}{4}$

Step 1.2

Set $y$ equal to the new right side.

$y = {(x - \frac{1}{2})}^{2} - \frac{9}{4}$

Step 2

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

$a = 1$

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

$k = - \frac{9}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{9}{4})$

Step 4