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Precalculus Examples

y = x^{2} - 8 x - 4

$y = x^{2} - 8 x - 4$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

x^{2} - 8 x - 4

$x^{2} - 8 x - 4$ .

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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 = 1

$a = 1$

b = - 8

$b = - 8$

c = - 4

$c = - 4$

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{- 8}{2 \cdot 1}

$d = \frac{- 8}{2 \cdot 1}$

Step 1.1.3.2

Cancel the common factor of

- 8

$- 8$ and

2

$2$ .

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Step 1.1.3.2.1

Factor

2

$2$ out of

- 8

$- 8$ .

d = \frac{2 \cdot - 4}{2 \cdot 1}

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

Step 1.1.3.2.2

Cancel the common factors.

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Step 1.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 1.1.3.2.2.2

Cancel the common factor.

d = \frac{2 \cdot - 4}{2 \cdot 1}

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

Step 1.1.3.2.2.3

Rewrite the expression.

d = \frac{- 4}{1}

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

Step 1.1.3.2.2.4

Divide

- 4

$- 4$ by

1

$1$ .

d = - 4

$d = - 4$

d = - 4

$d = - 4$

d = - 4

$d = - 4$

d = - 4

$d = - 4$

Step 1.1.4

Find the value of

e

$e$ using the formula

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

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

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Step 1.1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

e = - 4 - \frac{{(- 8)}^{2}}{4 \cdot 1}

$e = - 4 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

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

Raise

- 8

$- 8$ to the power of

2

$2$ .

e = - 4 - \frac{64}{4 \cdot 1}

$e = - 4 - \frac{64}{4 \cdot 1}$

Step 1.1.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

e = - 4 - \frac{64}{4}

$e = - 4 - \frac{64}{4}$

Step 1.1.4.2.1.3

Divide

64

$64$ by

4

$4$ .

e = - 4 - 1 \cdot 16

$e = - 4 - 1 \cdot 16$

Step 1.1.4.2.1.4

Multiply

- 1

$- 1$ by

16

$16$ .

e = - 4 - 16

$e = - 4 - 16$

e = - 4 - 16

$e = - 4 - 16$

Step 1.1.4.2.2

Subtract

16

$16$ from

- 4

$- 4$ .

e = - 20

$e = - 20$

e = - 20

$e = - 20$

e = - 20

$e = - 20$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(x - 4)}^{2} - 20

${(x - 4)}^{2} - 20$ .

{(x - 4)}^{2} - 20

${(x - 4)}^{2} - 20$

{(x - 4)}^{2} - 20

${(x - 4)}^{2} - 20$

Step 1.2

Set

y

$y$ equal to the new right side.

y = {(x - 4)}^{2} - 20

$y = {(x - 4)}^{2} - 20$

y = {(x - 4)}^{2} - 20

$y = {(x - 4)}^{2} - 20$

Step 2

Use the vertex form,

y = a {(x - h)}^{2} + k

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = 1

$a = 1$

h = 4

$h = 4$

k = - 20

$k = - 20$

Step 3

Find the vertex

(h, k)

$(h, k)$ .

(4, - 20)

$(4, - 20)$

Step 4

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

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