Precalculus Examples

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

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

Step 1

Write

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

$f (x) = x^{2} - 2 x - 8$ as an equation.

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

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

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for

x^{2} - 2 x - 8

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

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Step 2.1.1

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 = - 2

$b = - 2$

c = - 8

$c = - 8$

Step 2.1.2

Consider the vertex form of a parabola.

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

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

Step 2.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 2.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

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

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

Step 2.1.3.2

Cancel the common factor of

- 2

$- 2$ and

2

$2$ .

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Step 2.1.3.2.1

Factor

2

$2$ out of

- 2

$- 2$ .

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

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

Step 2.1.3.2.2

Cancel the common factors.

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Step 2.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 2.1.3.2.2.2

Cancel the common factor.

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

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

Step 2.1.3.2.2.3

Rewrite the expression.

d = \frac{- 1}{1}

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

Step 2.1.3.2.2.4

Divide

- 1

$- 1$ by

1

$1$ .

d = - 1

$d = - 1$

d = - 1

$d = - 1$

d = - 1

$d = - 1$

d = - 1

$d = - 1$

Step 2.1.4

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 2.1.4.1

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

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

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Simplify each term.

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Step 2.1.4.2.1.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

e = - 8 - \frac{4}{4 \cdot 1}

$e = - 8 - \frac{4}{4 \cdot 1}$

Step 2.1.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

e = - 8 - \frac{4}{4}

$e = - 8 - \frac{4}{4}$

Step 2.1.4.2.1.3

Cancel the common factor of

4

$4$ .

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Step 2.1.4.2.1.3.1

Cancel the common factor.

e = - 8 - \frac{4}{4}

$e = - 8 - \frac{4}{4}$

Step 2.1.4.2.1.3.2

Rewrite the expression.

e = - 8 - 1 \cdot 1

$e = - 8 - 1 \cdot 1$

e = - 8 - 1 \cdot 1

$e = - 8 - 1 \cdot 1$

Step 2.1.4.2.1.4

Multiply

- 1

$- 1$ by

1

$1$ .

e = - 8 - 1

$e = - 8 - 1$

e = - 8 - 1

$e = - 8 - 1$

Step 2.1.4.2.2

Subtract

1

$1$ from

- 8

$- 8$ .

e = - 9

$e = - 9$

e = - 9

$e = - 9$

e = - 9

$e = - 9$

Step 2.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(x - 1)}^{2} - 9

${(x - 1)}^{2} - 9$ .

{(x - 1)}^{2} - 9

${(x - 1)}^{2} - 9$

{(x - 1)}^{2} - 9

${(x - 1)}^{2} - 9$

Step 2.2

Set

y

$y$ equal to the new right side.

y = {(x - 1)}^{2} - 9

$y = {(x - 1)}^{2} - 9$

y = {(x - 1)}^{2} - 9

$y = {(x - 1)}^{2} - 9$

Step 3

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 = 1

$h = 1$

k = - 9

$k = - 9$

Step 4

Since the value of

a

$a$ is positive, the parabola opens up.

Opens Up

Step 5

Find the vertex

(h, k)

$(h, k)$ .

(1, - 9)

$(1, - 9)$

Step 6

Find

p

$p$ , the distance from the vertex to the focus.

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Step 6.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

\frac{1}{4 a}

$\frac{1}{4 a}$

Step 6.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot 1}

$\frac{1}{4 \cdot 1}$

Step 6.3

Cancel the common factor of

1

$1$ .

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Step 6.3.1

Cancel the common factor.

\frac{1}{4 \cdot 1}

$\frac{1}{4 \cdot 1}$

Step 6.3.2

Rewrite the expression.

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

Step 7

Find the focus.

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Step 7.1

The focus of a parabola can be found by adding

p

$p$ to the y-coordinate

k

$k$ if the parabola opens up or down.

(h, k + p)

$(h, k + p)$

Step 7.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(1, - \frac{35}{4})

$(1, - \frac{35}{4})$

(1, - \frac{35}{4})

$(1, - \frac{35}{4})$

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

x = 1

$x = 1$

Step 9

Find the directrix.

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Step 9.1

The directrix of a parabola is the horizontal line found by subtracting

p

$p$ from the y-coordinate

k

$k$ of the vertex if the parabola opens up or down.

y = k - p

$y = k - p$

Step 9.2

Substitute the known values of

p

$p$ and

k

$k$ into the formula and simplify.

y = - \frac{37}{4}

$y = - \frac{37}{4}$

y = - \frac{37}{4}

$y = - \frac{37}{4}$

Step 10

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex:

(1, - 9)

$(1, - 9)$

Focus:

(1, - \frac{35}{4})

$(1, - \frac{35}{4})$

Axis of Symmetry:

x = 1

$x = 1$

Directrix:

y = - \frac{37}{4}

$y = - \frac{37}{4}$

Step 11

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