Algebra Examples

x^{2} - 4 x

$x^{2} - 4 x$

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

$x^{2} - 4 x$ .

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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 = 0

$c = 0$

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}

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

Step 1.1.1.3.2.2.3

Rewrite the expression.

d = \frac{- 2}{1}

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

Step 1.1.1.3.2.2.4

Divide

- 2

$- 2$ by

1

$1$ .

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

Step 1.1.1.4

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 1.1.1.4.1

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 = 0 - \frac{{(- 4)}^{2}}{4 \cdot 1}

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

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}

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

4

$4$ .

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Step 1.1.1.4.2.1.1.1

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

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

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

Step 1.1.1.4.2.1.1.2

Apply the product rule to

- 1 (4)

$- 1 (4)$ .

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

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

Step 1.1.1.4.2.1.1.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 1.1.1.4.2.1.1.4

Multiply

4^{2}

$4^{2}$ by

1

$1$ .

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

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

Step 1.1.1.4.2.1.1.5

Factor

4

$4$ out of

4^{2}

$4^{2}$ .

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

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

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

$4$ out of

4 \cdot 1

$4 \cdot 1$ .

e = 0 - \frac{4 \cdot 4}{4 (1)}

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

Step 1.1.1.4.2.1.1.6.2

Cancel the common factor.

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

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

Step 1.1.1.4.2.1.1.6.3

Rewrite the expression.

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

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

Step 1.1.1.4.2.1.1.6.4

Divide

4

$4$ by

1

$1$ .

e = 0 - 1 \cdot 4

$e = 0 - 1 \cdot 4$

e = 0 - 1 \cdot 4

$e = 0 - 1 \cdot 4$

e = 0 - 1 \cdot 4

$e = 0 - 1 \cdot 4$

Step 1.1.1.4.2.1.2

Multiply

- 1

$- 1$ by

4

$4$ .

e = 0 - 4

$e = 0 - 4$

e = 0 - 4

$e = 0 - 4$

Step 1.1.1.4.2.2

Subtract

4

$4$ from

0

$0$ .

e = - 4

$e = - 4$

e = - 4

$e = - 4$

e = - 4

$e = - 4$

Step 1.1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

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

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

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

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

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

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

Step 1.1.2

Set

y

$y$ equal to the new right side.

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

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

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

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

Step 1.2

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 = 2

$h = 2$

k = - 4

$k = - 4$

Step 1.3

Since the value of

a

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

(h, k)

$(h, k)$ .

(2, - 4)

$(2, - 4)$

Step 1.5

Find

p

$p$ , 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}

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot 1}

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

Step 1.5.3

Cancel the common factor of

1

$1$ .

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Step 1.5.3.1

Cancel the common factor.

\frac{1}{4 \cdot 1}

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

Step 1.5.3.2

Rewrite the expression.

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

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

$p$ to the y-coordinate

k

$k$ if the parabola opens up or down.

(h, k + p)

$(h, k + p)$

Step 1.6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(2, - \frac{15}{4})

$(2, - \frac{15}{4})$

(2, - \frac{15}{4})

$(2, - \frac{15}{4})$

Step 1.7

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

x = 2

$x = 2$

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

$p$ from the y-coordinate

k

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

y = k - p

$y = k - p$

Step 1.8.2

Substitute the known values of

p

$p$ and

k

$k$ into the formula and simplify.

y = - \frac{17}{4}

$y = - \frac{17}{4}$

y = - \frac{17}{4}

$y = - \frac{17}{4}$

Step 1.9

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

Direction: Opens Up

Vertex:

(2, - 4)

$(2, - 4)$

Focus:

(2, - \frac{15}{4})

$(2, - \frac{15}{4})$

Axis of Symmetry:

x = 2

$x = 2$

Directrix:

y = - \frac{17}{4}

$y = - \frac{17}{4}$

Direction: Opens Up

Vertex:

(2, - 4)

$(2, - 4)$

Focus:

(2, - \frac{15}{4})

$(2, - \frac{15}{4})$

Axis of Symmetry:

x = 2

$x = 2$

Directrix:

y = - \frac{17}{4}

$y = - \frac{17}{4}$

Step 2

Select a few

x

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

y

$y$ values. The

x

$x$ values should be selected around the vertex.

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Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

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

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

Step 2.2.1.2

Multiply

- 4

$- 4$ by

1

$1$ .

f (1) = 1 - 4

$f (1) = 1 - 4$

f (1) = 1 - 4

$f (1) = 1 - 4$

Step 2.2.2

Subtract

4

$4$ from

1

$1$ .

f (1) = - 3

$f (1) = - 3$

Step 2.2.3

The final answer is

- 3

$- 3$ .

- 3

$- 3$

- 3

$- 3$

Step 2.3

The

y

$y$ value at

x = 1

$x = 1$ is

- 3

$- 3$ .

y = - 3

$y = - 3$

Step 2.4

Replace the variable

x

$x$ with

0

$0$ in the expression.

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

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

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

$0$ to any positive power yields

0

$0$ .

f (0) = 0 - 4 \cdot 0

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

Step 2.5.1.2

Multiply

- 4

$- 4$ by

0

$0$ .

f (0) = 0 + 0

$f (0) = 0 + 0$

f (0) = 0 + 0

$f (0) = 0 + 0$

Step 2.5.2

Add

0

$0$ and

0

$0$ .

f (0) = 0

$f (0) = 0$

Step 2.5.3

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.6

The

y

$y$ value at

x = 0

$x = 0$ is

0

$0$ .

y = 0

$y = 0$

Step 2.7

Replace the variable

x

$x$ with

3

$3$ in the expression.

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

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

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

$3$ to the power of

2

$2$ .

f (3) = 9 - 4 \cdot 3

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

Step 2.8.1.2

Multiply

- 4

$- 4$ by

3

$3$ .

f (3) = 9 - 12

$f (3) = 9 - 12$

f (3) = 9 - 12

$f (3) = 9 - 12$

Step 2.8.2

Subtract

12

$12$ from

9

$9$ .

f (3) = - 3

$f (3) = - 3$

Step 2.8.3

The final answer is

- 3

$- 3$ .

- 3

$- 3$

- 3

$- 3$

Step 2.9

The

y

$y$ value at

x = 3

$x = 3$ is

- 3

$- 3$ .

y = - 3

$y = - 3$

Step 2.10

Replace the variable

x

$x$ with

4

$4$ in the expression.

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

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

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

$4$ to the power of

2

$2$ .

f (4) = 16 - 4 \cdot 4

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

Step 2.11.1.2

Multiply

- 4

$- 4$ by

4

$4$ .

f (4) = 16 - 16

$f (4) = 16 - 16$

f (4) = 16 - 16

$f (4) = 16 - 16$

Step 2.11.2

Subtract

16

$16$ from

16

$16$ .

f (4) = 0

$f (4) = 0$

Step 2.11.3

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.12

The

y

$y$ value at

x = 4

$x = 4$ is

0

$0$ .

y = 0

$y = 0$

Step 2.13

Graph the parabola using its properties and the selected points.

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

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

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

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

(2, - 4)

$(2, - 4)$

Focus:

(2, - \frac{15}{4})

$(2, - \frac{15}{4})$

Axis of Symmetry:

x = 2

$x = 2$

Directrix:

y = - \frac{17}{4}

$y = - \frac{17}{4}$

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

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

Step 4