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Precalculus Examples

f (x) = - 2 x^{2} - 8

$f (x) = - 2 x^{2} - 8$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

- 2 x^{2} - 8

$- 2 x^{2} - 8$ .

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Step 1.1.1

Use the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , to find the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

a = - 2

$a = - 2$

b = 0

$b = 0$

c = - 8

$c = - 8$

Step 1.1.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{0}{2 \cdot - 2}

$d = \frac{0}{2 \cdot - 2}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 1.1.3.2.1.1

Factor

2

$2$ out of

0

$0$ .

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

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

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor

2

$2$ out of

2 \cdot - 2

$2 \cdot - 2$ .

d = \frac{2 (0)}{2 (- 2)}

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

Step 1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot - 2}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{0}{- 2}$

$d = \frac{0}{- 2}$

$d = \frac{0}{- 2}$

Step 1.1.3.2.2

Cancel the common factor of

$0$ and

$- 2$ .

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Step 1.1.3.2.2.1

Factor

$2$ out of

$0$ .

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

Step 1.1.3.2.2.2

Move the negative one from the denominator of

$\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

$d = - 1 \cdot 0$

Step 1.1.3.2.3

Rewrite

$- 1 \cdot 0$ as

$- 0$ .

$d = - 0$

Step 1.1.3.2.4

Multiply

$- 1$ by

$0$ .

$d = 0$

$d = 0$

$d = 0$

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

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

$e = - 8 - \frac{0^{2}}{4 \cdot - 2}$

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

Raising

$0$ to any positive power yields

$0$ .

$e = - 8 - \frac{0}{4 \cdot - 2}$

Step 1.1.4.2.1.2

Multiply

$4$ by

$- 2$ .

$e = - 8 - \frac{0}{- 8}$

Step 1.1.4.2.1.3

Divide

$0$ by

$- 8$ .

$e = - 8 - 0$

Step 1.1.4.2.1.4

Multiply

$- 1$ by

$0$ .

$e = - 8 + 0$

$e = - 8 + 0$

Step 1.1.4.2.2

Add

$- 8$ and

$0$ .

$e = - 8$

$e = - 8$

$e = - 8$

Step 1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$- 2 {(x + 0)}^{2} - 8$ .

$- 2 {(x + 0)}^{2} - 8$

$- 2 {(x + 0)}^{2} - 8$

Step 1.2

Set

$y$ equal to the new right side.

$y = - 2 {(x + 0)}^{2} - 8$

$y = - 2 {(x + 0)}^{2} - 8$

Step 2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = - 2$

$h = 0$

$k = - 8$

Step 3

Find the vertex

$(h, k)$ .

$(0, - 8)$

Step 4

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$