Algebra Examples

f (x) = x^{2} - 1

$f (x) = x^{2} - 1$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for

x^{2} - 1

$x^{2} - 1$ .

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Step 1.1.1.1

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 = 0

$b = 0$

c = - 1

$c = - 1$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

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

Step 1.1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.1.1.3.1

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 1}

$d = \frac{0}{2 \cdot 1}$

Step 1.1.1.3.2

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 1.1.1.3.2.1

Factor

2

$2$ out of

0

$0$ .

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

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

Step 1.1.1.3.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 (0)}{2 (1)}

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

Step 1.1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 1.1.1.3.2.2.3

Rewrite the expression.

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

Step 1.1.1.3.2.2.4

Divide

$0$ by

$1$ .

$d = 0$

Step 1.1.1.4

Find the value of

$e$ using the formula

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

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Step 1.1.1.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

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

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raising

$0$ to any positive power yields

$0$ .

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

Step 1.1.1.4.2.1.2

Multiply

$4$ by

$1$ .

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

Step 1.1.1.4.2.1.3

Divide

$0$ by

$4$ .

$e = - 1 - 0$

Step 1.1.1.4.2.1.4

Multiply

$- 1$ by

$0$ .

$e = - 1 + 0$

Step 1.1.1.4.2.2

Add

$- 1$ and

$0$ .

$e = - 1$

Step 1.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x + 0)}^{2} - 1$ .

${(x + 0)}^{2} - 1$

Step 1.1.2

Set

$y$ equal to the new right side.

$y = {(x + 0)}^{2} - 1$

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

$h = 0$

$k = - 1$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

$(0, - 1)$

Step 1.5

Find

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of

$a$ into the formula.

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

Step 1.5.3

Cancel the common factor of

$1$ .

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.6

Find the focus.

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Step 1.6.1

The focus of a parabola can be found by adding

$p$ to the y-coordinate

$k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

Substitute the known values of

$h$ ,

$p$ , and

$k$ into the formula and simplify.

$(0, - \frac{3}{4})$

Step 1.7

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

$x = 0$

Step 1.8

Find the directrix.

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Step 1.8.1

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

$p$ from the y-coordinate

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

$y = k - p$

Step 1.8.2

Substitute the known values of

$p$ and

$k$ into the formula and simplify.

$y = - \frac{5}{4}$

Step 1.9

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

Direction: Opens Up

Vertex:

$(0, - 1)$

Focus:

$(0, - \frac{3}{4})$

Axis of Symmetry:

$x = 0$

Directrix:

$y = - \frac{5}{4}$

Direction: Opens Up

Vertex:

$(0, - 1)$

Focus:

$(0, - \frac{3}{4})$

Axis of Symmetry:

$x = 0$

Directrix:

$y = - \frac{5}{4}$

Step 2

Select a few

$x$ values, and plug them into the equation to find the corresponding

$y$ values. The

$x$ values should be selected around the vertex.

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Step 2.1

Replace the variable

$x$ with

$- 1$ in the expression.

$f (- 1) = {(- 1)}^{2} - 1$

Step 2.2

Simplify the result.

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Step 2.2.1

Raise

$- 1$ to the power of

$2$ .

$f (- 1) = 1 - 1$

Step 2.2.2

Subtract

$1$ from

$1$ .

$f (- 1) = 0$

Step 2.2.3

The final answer is

$0$ .

$0$

Step 2.3

The

$y$ value at

$x = - 1$ is

$0$ .

$y = 0$

Step 2.4

Replace the variable

$x$ with

$- 2$ in the expression.

$f (- 2) = {(- 2)}^{2} - 1$

Step 2.5

Simplify the result.

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Step 2.5.1

Raise

$- 2$ to the power of

$2$ .

$f (- 2) = 4 - 1$

Step 2.5.2

Subtract

$1$ from

$4$ .

$f (- 2) = 3$

Step 2.5.3

The final answer is

$3$ .

$3$

Step 2.6

The

$y$ value at

$x = - 2$ is

$3$ .

$y = 3$

Step 2.7

Replace the variable

$x$ with

$1$ in the expression.

$f (1) = {(1)}^{2} - 1$

Step 2.8

Simplify the result.

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Step 2.8.1

One to any power is one.

$f (1) = 1 - 1$

Step 2.8.2

Subtract

$1$ from

$1$ .

$f (1) = 0$

Step 2.8.3

The final answer is

$0$ .

$0$

Step 2.9

The

$y$ value at

$x = 1$ is

$0$ .

$y = 0$

Step 2.10

Replace the variable

$x$ with

$2$ in the expression.

$f (2) = {(2)}^{2} - 1$

Step 2.11

Simplify the result.

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Step 2.11.1

Raise

$2$ to the power of

$2$ .

$f (2) = 4 - 1$

Step 2.11.2

Subtract

$1$ from

$4$ .

$f (2) = 3$

Step 2.11.3

The final answer is

$3$ .

$3$

Step 2.12

The

$y$ value at

$x = 2$ is

$3$ .

$y = 3$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 0 \\ 0 & - 1 \\ 1 & 0 \\ 2 & 3 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(0, - 1)$

Focus:

$(0, - \frac{3}{4})$

Axis of Symmetry:

$x = 0$

Directrix:

$y = - \frac{5}{4}$

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 0 \\ 0 & - 1 \\ 1 & 0 \\ 2 & 3 \end{array}$

Step 4