Precalculus Examples

$f (x) = - 2 x^{2} + 2 x - 3$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- 2 x^{2} + 2 x - 3$ .

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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 $c$ .

$a = - 2$

$b = 2$

$c = - 3$

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

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $2$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.1.2

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

$e = - 3 - \frac{2^{2}}{4 \cdot - 2}$

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

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

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Step 1.1.4.2.1.1.1

Rewrite $2$ as $- 1 (- 2)$ .

$e = - 3 - \frac{{(- 1 (- 2))}^{2}}{4 \cdot - 2}$

Step 1.1.4.2.1.1.2

Apply the product rule to $- 1 (- 2)$ .

$e = - 3 - \frac{{(- 1)}^{2} {(- 2)}^{2}}{4 \cdot - 2}$

Step 1.1.4.2.1.1.3

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

$e = - 3 - \frac{1 {(- 2)}^{2}}{4 \cdot - 2}$

Step 1.1.4.2.1.1.4

Multiply ${(- 2)}^{2}$ by $1$ .

$e = - 3 - \frac{{(- 2)}^{2}}{4 \cdot - 2}$

Step 1.1.4.2.1.1.5

Factor $- 2$ out of ${(- 2)}^{2}$ .

$e = - 3 - \frac{- 2 \cdot - 2}{4 \cdot - 2}$

Step 1.1.4.2.1.1.6

Cancel the common factors.

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Step 1.1.4.2.1.1.6.1

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

$e = - 3 - \frac{- 2 \cdot - 2}{- 2 (- 2 \cdot - 2)}$

Step 1.1.4.2.1.1.6.2

Cancel the common factor.

$e = - 3 - \frac{- 2 \cdot - 2}{- 2 (- 2 \cdot - 2)}$

Step 1.1.4.2.1.1.6.3

Rewrite the expression.

$e = - 3 - \frac{- 2}{- 2 \cdot - 2}$

Step 1.1.4.2.1.2

Cancel the common factor of $- 2$ .

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Step 1.1.4.2.1.2.1

Cancel the common factor.

$e = - 3 - \frac{- 2}{- 2 \cdot - 2}$

Step 1.1.4.2.1.2.2

Rewrite the expression.

$e = - 3 - \frac{1}{- 2}$

Step 1.1.4.2.1.3

Move the negative in front of the fraction.

$e = - 3 - - \frac{1}{2}$

Step 1.1.4.2.1.4

Multiply $- - \frac{1}{2}$ .

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Step 1.1.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = - 3 + 1 (\frac{1}{2})$

Step 1.1.4.2.1.4.2

Multiply $\frac{1}{2}$ by $1$ .

$e = - 3 + \frac{1}{2}$

Step 1.1.4.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 3 \cdot \frac{2}{2} + \frac{1}{2}$

Step 1.1.4.2.3

Combine $- 3$ and $\frac{2}{2}$ .

$e = \frac{- 3 \cdot 2}{2} + \frac{1}{2}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 3 \cdot 2 + 1}{2}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $- 3$ by $2$ .

$e = \frac{- 6 + 1}{2}$

Step 1.1.4.2.5.2

Add $- 6$ and $1$ .

$e = \frac{- 5}{2}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{5}{2}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - \frac{1}{2})}^{2} - \frac{5}{2}$ .

$- 2 {(x - \frac{1}{2})}^{2} - \frac{5}{2}$

Step 1.2

Set $y$ equal to the new right side.

$y = - 2 {(x - \frac{1}{2})}^{2} - \frac{5}{2}$

Step 2

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

$a = - 2$

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

$k = - \frac{5}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{5}{2})$

Step 4