Precalculus Examples

$f (x) = \frac{1}{4} x^{2} - 2 x - 12$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Combine $\frac{1}{4}$ and $x^{2}$ .

$y = \frac{x^{2}}{4} - 2 x - 12$

Step 1.2

Complete the square for $\frac{x^{2}}{4} - 2 x - 12$ .

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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 $c$ .

$a = \frac{1}{4}$

$b = - 2$

$c = - 12$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $- 2$ .

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

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.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{- 1}{\frac{1}{4}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

Step 1.2.3.2.3

Multiply $- 1$ by $4$ .

$d = - 4$

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

$e = - 12 - \frac{{(- 2)}^{2}}{4 (\frac{1}{4})}$

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

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

$e = - 12 - \frac{4}{4 (\frac{1}{4})}$

Step 1.2.4.2.1.2

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

$e = - 12 - \frac{4}{\frac{4}{4}}$

Step 1.2.4.2.1.3

Divide $4$ by $4$ .

$e = - 12 - \frac{4}{1}$

Step 1.2.4.2.1.4

Divide $4$ by $1$ .

$e = - 12 - 1 \cdot 4$

Step 1.2.4.2.1.5

Multiply $- 1$ by $4$ .

$e = - 12 - 4$

Step 1.2.4.2.2

Subtract $4$ from $- 12$ .

$e = - 16$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x - 4)}^{2} - 16$ .

$\frac{1}{4} {(x - 4)}^{2} - 16$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{4} \cdot {(x - 4)}^{2} - 16$

Step 2

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

$a = \frac{1}{4}$

$h = 4$

$k = - 16$

Step 3

Find the vertex $(h, k)$ .

$(4, - 16)$

Step 4