Precalculus Examples

$f (x) = 6 - 2 x + 4 x^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify $6 - 2 x + 4 x^{2}$ .

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Step 1.1.1

Move $6$ .

$y = - 2 x + 4 x^{2} + 6$

Step 1.1.2

Reorder $- 2 x$ and $4 x^{2}$ .

$y = 4 x^{2} - 2 x + 6$

Step 1.2

Complete the square for $4 x^{2} - 2 x + 6$ .

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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 $c$ .

$a = 4$

$b = - 2$

$c = 6$

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

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

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 $- 2$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

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

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

Move the negative in front of the fraction.

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

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

$e = 6 - \frac{{(- 2)}^{2}}{4 \cdot 4}$

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 $2$ .

$e = 6 - \frac{4}{4 \cdot 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

$e = 6 - \frac{4}{16}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $16$ .

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = 6 - \frac{4 (1)}{16}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $4$ out of $16$ .

$e = 6 - \frac{4 \cdot 1}{4 \cdot 4}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = 6 - \frac{4 \cdot 1}{4 \cdot 4}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = 6 - \frac{1}{4}$

Step 1.2.4.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 6 \cdot \frac{4}{4} - \frac{1}{4}$

Step 1.2.4.2.3

Combine $6$ and $\frac{4}{4}$ .

$e = \frac{6 \cdot 4}{4} - \frac{1}{4}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{6 \cdot 4 - 1}{4}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $6$ by $4$ .

$e = \frac{24 - 1}{4}$

Step 1.2.4.2.5.2

Subtract $1$ from $24$ .

$e = \frac{23}{4}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - \frac{1}{4})}^{2} + \frac{23}{4}$ .

$4 {(x - \frac{1}{4})}^{2} + \frac{23}{4}$

Step 1.3

Set $y$ equal to the new right side.

$y = 4 {(x - \frac{1}{4})}^{2} + \frac{23}{4}$

Step 2

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

$a = 4$

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

$k = \frac{23}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{4}, \frac{23}{4})$

Step 4