Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

$y = - \frac{x^{2}}{4} + 7$

Step 1.2

Complete the square for $- \frac{x^{2}}{4} + 7$ .

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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 $c$ .

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

$b = 0$

$c = 7$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

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

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.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{1}{4}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 4)$

Step 1.2.3.2.3

Multiply $0 (- 1 \cdot 4)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $4$ .

$d = 0 \cdot - 4$

Step 1.2.3.2.3.2

Multiply $0$ by $- 4$ .

$d = 0$

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

$e = 7 - \frac{0^{2}}{4 (- \frac{1}{4})}$

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

Raising $0$ to any positive power yields $0$ .

$e = 7 - \frac{0}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 7 - \frac{0}{- 4 (\frac{1}{4})}$

Step 1.2.4.2.1.2.2

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

$e = 7 - \frac{0}{\frac{- 4}{4}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $4$ .

$e = 7 - \frac{0}{- 1}$

Step 1.2.4.2.1.4

Divide $0$ by $- 1$ .

$e = 7 - 0$

Step 1.2.4.2.1.5

Multiply $- 1$ by $0$ .

$e = 7 + 0$

Step 1.2.4.2.2

Add $7$ and $0$ .

$e = 7$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x + 0)}^{2} + 7$ .

$- \frac{1}{4} {(x + 0)}^{2} + 7$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x + 0)}^{2} + 7$

Step 2

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

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

$h = 0$

$k = 7$

Step 3

Find the vertex $(h, k)$ .

$(0, 7)$

Step 4