Precalculus Examples

$f (x) = 2 x^{2} - x + 1$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $2 x^{2} - x + 1$ .

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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 $c$ .

$a = 2$

$b = - 1$

$c = 1$

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

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Multiply $2$ by $2$ .

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

Step 1.1.3.2.2

Move the negative in front of the fraction.

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

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

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

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

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

Step 1.1.4.2.1.2

Multiply $4$ by $2$ .

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

Step 1.1.4.2.2

Write $1$ as a fraction with a common denominator.

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

Step 1.1.4.2.3

Combine the numerators over the common denominator.

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

Step 1.1.4.2.4

Subtract $1$ from $8$ .

$e = \frac{7}{8}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - \frac{1}{4})}^{2} + \frac{7}{8}$ .

$2 {(x - \frac{1}{4})}^{2} + \frac{7}{8}$

Step 1.2

Set $y$ equal to the new right side.

$y = 2 {(x - \frac{1}{4})}^{2} + \frac{7}{8}$

Step 2

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

$a = 2$

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

$k = \frac{7}{8}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{4}, \frac{7}{8})$

Step 4