Algebra Examples

f (x) = - x^{2}

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

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}

$- x^{2}$ .

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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 = 0

$b = 0$

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{0}{2 \cdot - 1}

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

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 1.1.1.3.2.1.1

Factor

2

$2$ out of

0

$0$ .

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

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

Step 1.1.1.3.2.1.2

Move the negative one from the denominator of

\frac{0}{- 1}

$\frac{0}{- 1}$ .

d = - 1 \cdot 0

$d = - 1 \cdot 0$

d = - 1 \cdot 0

$d = - 1 \cdot 0$

Step 1.1.1.3.2.2

Rewrite

- 1 \cdot 0

$- 1 \cdot 0$ as

- 0

$- 0$ .

d = - 0

$d = - 0$

Step 1.1.1.3.2.3

Multiply

- 1

$- 1$ by

0

$0$ .

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

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{0^{2}}{4 \cdot - 1}

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

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

$0$ to any positive power yields

0

$0$ .

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

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

Step 1.1.1.4.2.1.2

Multiply

4

$4$ by

- 1

$- 1$ .

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

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

Step 1.1.1.4.2.1.3

Divide

0

$0$ by

- 4

$- 4$ .

e = 0 - 0

$e = 0 - 0$

Step 1.1.1.4.2.1.4

Multiply

- 1

$- 1$ by

0

$0$ .

e = 0 + 0

$e = 0 + 0$

e = 0 + 0

$e = 0 + 0$

Step 1.1.1.4.2.2

Add

0

$0$ and

0

$0$ .

e = 0

$e = 0$

e = 0

$e = 0$

e = 0

$e = 0$

Step 1.1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

- x^{2}

$- x^{2}$ .

- x^{2}

$- x^{2}$

- x^{2}

$- x^{2}$

Step 1.1.2

Set

y

$y$ equal to the new right side.

y = - x^{2}

$y = - x^{2}$

y = - x^{2}

$y = - x^{2}$

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 = 0

$h = 0$

k = 0

$k = 0$

Step 1.3

Since the value of

a

$a$ is negative, the parabola opens down.

Opens Down

Step 1.4

Find the vertex

(h, k)

$(h, k)$ .

(0, 0)

$(0, 0)$

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

$1$ and

- 1

$- 1$ .

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Step 1.5.3.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

\frac{- 1 (- 1)}{4 \cdot - 1}

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Step 1.5.3.2

Move the negative in front of the fraction.

- \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.

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

$(0, - \frac{1}{4})$

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

$(0, - \frac{1}{4})$

Step 1.7

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

x = 0

$x = 0$

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

$p$ 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{1}{4}

$y = \frac{1}{4}$

y = \frac{1}{4}

$y = \frac{1}{4}$

Step 1.9

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

Direction: Opens Down

Vertex:

(0, 0)

$(0, 0)$

Focus:

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

$(0, - \frac{1}{4})$

Axis of Symmetry:

x = 0

$x = 0$

Directrix:

y = \frac{1}{4}

$y = \frac{1}{4}$

Direction: Opens Down

Vertex:

(0, 0)

$(0, 0)$

Focus:

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

$(0, - \frac{1}{4})$

Axis of Symmetry:

x = 0

$x = 0$

Directrix:

y = \frac{1}{4}

$y = \frac{1}{4}$

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}

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

${(- 1)}^{2}$ by adding the exponents.

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Step 2.2.1.1

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

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

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Step 2.2.1.1.1

Raise

- 1

$- 1$ to the power of

1

$1$ .

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

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

Step 2.2.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 2.2.1.2

Add

1

$1$ and

2

$2$ .

f (- 1) = {(- 1)}^{3}

$f (- 1) = {(- 1)}^{3}$

f (- 1) = {(- 1)}^{3}

$f (- 1) = {(- 1)}^{3}$

Step 2.2.2

Raise

- 1

$- 1$ to the power of

3

$3$ .

f (- 1) = - 1

$f (- 1) = - 1$

Step 2.2.3

The final answer is

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

Step 2.3

The

y

$y$ value at

x = - 1

$x = - 1$ is

- 1

$- 1$ .

y = - 1

$y = - 1$

Step 2.4

Replace the variable

x

$x$ with

- 2

$- 2$ in the expression.

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

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

Step 2.5

Simplify the result.

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Step 2.5.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

f (- 2) = - 1 \cdot 4

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

Step 2.5.2

Multiply

- 1

$- 1$ by

4

$4$ .

f (- 2) = - 4

$f (- 2) = - 4$

Step 2.5.3

The final answer is

- 4

$- 4$ .

- 4

$- 4$

- 4

$- 4$

Step 2.6

The

y

$y$ value at

x = - 2

$x = - 2$ is

- 4

$- 4$ .

y = - 4

$y = - 4$

Step 2.7

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 2.8

Simplify the result.

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Step 2.8.1

One to any power is one.

f (1) = - 1 \cdot 1

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

Step 2.8.2

Multiply

- 1

$- 1$ by

1

$1$ .

f (1) = - 1

$f (1) = - 1$

Step 2.8.3

The final answer is

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

Step 2.9

The

y

$y$ value at

x = 1

$x = 1$ is

- 1

$- 1$ .

y = - 1

$y = - 1$

Step 2.10

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

Step 2.11

Simplify the result.

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Step 2.11.1

Raise

2

$2$ to the power of

2

$2$ .

f (2) = - 1 \cdot 4

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

Step 2.11.2

Multiply

- 1

$- 1$ by

4

$4$ .

f (2) = - 4

$f (2) = - 4$

Step 2.11.3

The final answer is

- 4

$- 4$ .

- 4

$- 4$

- 4

$- 4$

Step 2.12

The

y

$y$ value at

x = 2

$x = 2$ is

- 4

$- 4$ .

y = - 4

$y = - 4$

Step 2.13

Graph the parabola using its properties and the selected points.

\begin{matrix} x & y - 2 & - 4 - 1 & - 1 0 & 0 1 & - 1 2 & - 4 \end{matrix}

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

\begin{matrix} x & y - 2 & - 4 - 1 & - 1 0 & 0 1 & - 1 2 & - 4 \end{matrix}

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex:

(0, 0)

$(0, 0)$

Focus:

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

$(0, - \frac{1}{4})$

Axis of Symmetry:

x = 0

$x = 0$

Directrix:

y = \frac{1}{4}

$y = \frac{1}{4}$

\begin{matrix} x & y - 2 & - 4 - 1 & - 1 0 & 0 1 & - 1 2 & - 4 \end{matrix}

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

Step 4