Precalculus Examples

$y = - \frac{1}{2} x (x - 8)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Remove parentheses.

$y = - \frac{1}{2} \cdot (x (x - 8))$

Step 1.1.2

Remove parentheses.

$y = - \frac{1}{2} \cdot (x (x - 8))$

Step 1.1.3

Simplify $- \frac{1}{2} \cdot (x (x - 8))$ .

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Step 1.1.3.1

Apply the distributive property.

$y = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 8)$

Step 1.1.3.2

Simplify the expression.

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Step 1.1.3.2.1

Multiply $x$ by $x$ .

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

Step 1.1.3.2.2

Move $- 8$ to the left of $x$ .

$y = - \frac{1}{2} \cdot (x^{2} - 8 \cdot x)$

Step 1.1.3.3

Apply the distributive property.

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Cancel the common factor of $2$ .

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Step 1.1.3.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.1.3.5.2

Factor $2$ out of $- 8 x$ .

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

Step 1.1.3.5.3

Cancel the common factor.

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

Step 1.1.3.5.4

Rewrite the expression.

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

Step 1.1.3.6

Multiply $- 4$ by $- 1$ .

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

Step 1.2

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

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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 $c$ .

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

$b = 4$

$c = 0$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $4$ .

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

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.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 2 (- 1 \cdot 2)$

Step 1.2.3.2.3

Multiply $2 (- 1 \cdot 2)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $2$ .

$d = 2 \cdot - 2$

Step 1.2.3.2.3.2

Multiply $2$ by $- 2$ .

$d = - 4$

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

$e = 0 - \frac{4^{2}}{4 (- \frac{1}{2})}$

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.2.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.2

Cancel the common factors.

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Step 1.2.4.2.1.1.2.1

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.2.2

Rewrite the expression.

$e = 0 - \frac{4}{- \frac{1}{2}}$

Step 1.2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (4 (- 1 \cdot 2))$

Step 1.2.4.2.1.3

Multiply $- (4 (- 1 \cdot 2))$ .

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Step 1.2.4.2.1.3.1

Multiply $- 1$ by $2$ .

$e = 0 - (4 \cdot - 2)$

Step 1.2.4.2.1.3.2

Multiply $4$ by $- 2$ .

$e = 0 - - 8$

Step 1.2.4.2.1.3.3

Multiply $- 1$ by $- 8$ .

$e = 0 + 8$

Step 1.2.4.2.2

Add $0$ and $8$ .

$e = 8$

Step 1.2.5

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 4$

$k = 8$

Step 3

Find the vertex $(h, k)$ .

$(4, 8)$

Step 4