Precalculus Examples

$f (x) = - (x^{2} + x - 30)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify $- (x^{2} + x - 30)$ .

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Step 1.1.1

Apply the distributive property.

$y = - x^{2} - x - - 30$

Step 1.1.2

Multiply $- 1$ by $- 30$ .

$y = - x^{2} - x + 30$

Step 1.2

Complete the square for $- x^{2} - x + 30$ .

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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 $c$ .

$a = - 1$

$b = - 1$

$c = 30$

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

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

Step 1.2.3.2

Dividing two negative values results in a positive value.

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

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

$e = 30 - \frac{{(- 1)}^{2}}{4 \cdot - 1}$

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

$e = 30 - \frac{1}{4 \cdot - 1}$

Step 1.2.4.2.1.2

Multiply $4$ by $- 1$ .

$e = 30 - \frac{1}{- 4}$

Step 1.2.4.2.1.3

Move the negative in front of the fraction.

$e = 30 - - \frac{1}{4}$

Step 1.2.4.2.1.4

Multiply $- - \frac{1}{4}$ .

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Step 1.2.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 30 + 1 (\frac{1}{4})$

Step 1.2.4.2.1.4.2

Multiply $\frac{1}{4}$ by $1$ .

$e = 30 + \frac{1}{4}$

Step 1.2.4.2.2

To write $30$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 30 \cdot \frac{4}{4} + \frac{1}{4}$

Step 1.2.4.2.3

Combine $30$ and $\frac{4}{4}$ .

$e = \frac{30 \cdot 4}{4} + \frac{1}{4}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{30 \cdot 4 + 1}{4}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $30$ by $4$ .

$e = \frac{120 + 1}{4}$

Step 1.2.4.2.5.2

Add $120$ and $1$ .

$e = \frac{121}{4}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + \frac{1}{2})}^{2} + \frac{121}{4}$ .

$- {(x + \frac{1}{2})}^{2} + \frac{121}{4}$

Step 1.3

Set $y$ equal to the new right side.

$y = - {(x + \frac{1}{2})}^{2} + \frac{121}{4}$

Step 2

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

$a = - 1$

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

$k = \frac{121}{4}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{1}{2}, \frac{121}{4})$

Step 4