Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

$y = \frac{x^{2}}{2} + x - 1$

Step 1.2

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

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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 $c$ .

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

$b = 1$

$c = - 1$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Combine $2$ and $\frac{1}{2}$ .

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

Step 1.2.3.2.2

Divide $2$ by $2$ .

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

Step 1.2.3.2.3

Divide $1$ by $1$ .

$d = 1$

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

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

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

One to any power is one.

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

Step 1.2.4.2.1.2

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

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

Step 1.2.4.2.1.3

Divide $4$ by $2$ .

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

Step 1.2.4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2.4.2.3

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.2.4.2.4

Combine the numerators over the common denominator.

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

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $- 1$ by $2$ .

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

Step 1.2.4.2.5.2

Subtract $1$ from $- 2$ .

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

Step 1.2.4.2.6

Move the negative in front of the fraction.

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

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x + 1)}^{2} - \frac{3}{2}$ .

$\frac{1}{2} {(x + 1)}^{2} - \frac{3}{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x + 1)}^{2} - \frac{3}{2}$

Step 2

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

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

$h = - 1$

$k = - \frac{3}{2}$

Step 3

Find the vertex $(h, k)$ .

$(- 1, - \frac{3}{2})$

Step 4