Algebra Examples

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

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

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} - 4 x - 2

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

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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 = - 4

$b = - 4$

c = - 2

$c = - 2$

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{- 4}{2 \cdot 1}

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

Step 1.1.1.3.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 1.1.1.3.2.1

Factor

2

$2$ out of

- 4

$- 4$ .

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

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

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 \cdot - 2}{2 (1)}

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

Step 1.1.1.3.2.2.2

Cancel the common factor.

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

Step 1.1.1.3.2.2.3

Rewrite the expression.

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

Step 1.1.1.3.2.2.4

Divide

$- 2$ by

$1$ .

$d = - 2$

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

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

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

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

Cancel the common factor of

${(- 4)}^{2}$ and

$4$ .

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Step 1.1.1.4.2.1.1.1

Rewrite

$- 4$ as

$- 1 (4)$ .

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

Step 1.1.1.4.2.1.1.2

Apply the product rule to

$- 1 (4)$ .

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

Step 1.1.1.4.2.1.1.3

Raise

$- 1$ to the power of

$2$ .

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

Step 1.1.1.4.2.1.1.4

Multiply

$4^{2}$ by

$1$ .

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

Step 1.1.1.4.2.1.1.5

Factor

$4$ out of

$4^{2}$ .

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

Step 1.1.1.4.2.1.1.6

Cancel the common factors.

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Step 1.1.1.4.2.1.1.6.1

Factor

$4$ out of

$4 \cdot 1$ .

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

Step 1.1.1.4.2.1.1.6.2

Cancel the common factor.

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

Step 1.1.1.4.2.1.1.6.3

Rewrite the expression.

$e = - 2 - \frac{4}{1}$

Step 1.1.1.4.2.1.1.6.4

Divide

$4$ by

$1$ .

$e = - 2 - 1 \cdot 4$

Step 1.1.1.4.2.1.2

Multiply

$- 1$ by

$4$ .

$e = - 2 - 4$

Step 1.1.1.4.2.2

Subtract

$4$ from

$- 2$ .

$e = - 6$

Step 1.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x - 2)}^{2} - 6$ .

${(x - 2)}^{2} - 6$

Step 1.1.2

Set

$y$ equal to the new right side.

$y = {(x - 2)}^{2} - 6$

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

$h = 2$

$k = - 6$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

$(2, - 6)$

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.

$(2, - \frac{23}{4})$

Step 1.7

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

$x = 2$

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.

$y = - \frac{25}{4}$

Step 1.9

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

Direction: Opens Up

Vertex:

$(2, - 6)$

Focus:

$(2, - \frac{23}{4})$

Axis of Symmetry:

$x = 2$

Directrix:

$y = - \frac{25}{4}$

Direction: Opens Up

Vertex:

$(2, - 6)$

Focus:

$(2, - \frac{23}{4})$

Axis of Symmetry:

$x = 2$

Directrix:

$y = - \frac{25}{4}$

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.

$f (1) = {(1)}^{2} - 4 \cdot 1 - 2$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

One to any power is one.

$f (1) = 1 - 4 \cdot 1 - 2$

Step 2.2.1.2

Multiply

$- 4$ by

$1$ .

$f (1) = 1 - 4 - 2$

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract

$4$ from

$1$ .

$f (1) = - 3 - 2$

Step 2.2.2.2

Subtract

$2$ from

$- 3$ .

$f (1) = - 5$

Step 2.2.3

The final answer is

$- 5$ .

$- 5$

Step 2.3

The

$y$ value at

$x = 1$ is

$- 5$ .

$y = - 5$

Step 2.4

Replace the variable

$x$ with

$0$ in the expression.

$f (0) = {(0)}^{2} - 4 \cdot 0 - 2$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 0 - 4 \cdot 0 - 2$

Step 2.5.1.2

Multiply

$- 4$ by

$0$ .

$f (0) = 0 + 0 - 2$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Add

$0$ and

$0$ .

$f (0) = 0 - 2$

Step 2.5.2.2

Subtract

$2$ from

$0$ .

$f (0) = - 2$

Step 2.5.3

The final answer is

$- 2$ .

$- 2$

Step 2.6

The

$y$ value at

$x = 0$ is

$- 2$ .

$y = - 2$

Step 2.7

Replace the variable

$x$ with

$3$ in the expression.

$f (3) = {(3)}^{2} - 4 \cdot 3 - 2$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Raise

$3$ to the power of

$2$ .

$f (3) = 9 - 4 \cdot 3 - 2$

Step 2.8.1.2

Multiply

$- 4$ by

$3$ .

$f (3) = 9 - 12 - 2$

Step 2.8.2

Simplify by subtracting numbers.

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Step 2.8.2.1

Subtract

$12$ from

$9$ .

$f (3) = - 3 - 2$

Step 2.8.2.2

Subtract

$2$ from

$- 3$ .

$f (3) = - 5$

Step 2.8.3

The final answer is

$- 5$ .

$- 5$

Step 2.9

The

$y$ value at

$x = 3$ is

$- 5$ .

$y = - 5$

Step 2.10

Replace the variable

$x$ with

$4$ in the expression.

$f (4) = {(4)}^{2} - 4 \cdot 4 - 2$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

Raise

$4$ to the power of

$2$ .

$f (4) = 16 - 4 \cdot 4 - 2$

Step 2.11.1.2

Multiply

$- 4$ by

$4$ .

$f (4) = 16 - 16 - 2$

Step 2.11.2

Simplify by subtracting numbers.

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Step 2.11.2.1

Subtract

$16$ from

$16$ .

$f (4) = 0 - 2$

Step 2.11.2.2

Subtract

$2$ from

$0$ .

$f (4) = - 2$

Step 2.11.3

The final answer is

$- 2$ .

$- 2$

Step 2.12

The

$y$ value at

$x = 4$ is

$- 2$ .

$y = - 2$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 5 \\ 2 & - 6 \\ 3 & - 5 \\ 4 & - 2 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(2, - 6)$

Focus:

$(2, - \frac{23}{4})$

Axis of Symmetry:

$x = 2$

Directrix:

$y = - \frac{25}{4}$

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 5 \\ 2 & - 6 \\ 3 & - 5 \\ 4 & - 2 \end{array}$

Step 4