Algebra Examples

{(x^{2} + y^{2} - 1)}^{3} = x^{2}

${(x^{2} + y^{2} - 1)}^{3} = x^{2}$

y^{3}

$y^{3}$

Step 1

Graph

y^{3}

$y^{3}$ .

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Step 1.1

Find the point at

x = - 1

$x = - 1$ .

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Step 1.1.1

Replace the variable

x

$x$ with

- 1

$- 1$ in the expression.

f (- 1) = \sqrt[3]{- 1}

$f (- 1) = \sqrt[3]{- 1}$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Remove parentheses.

f (- 1) = \sqrt[3]{- 1}

$f (- 1) = \sqrt[3]{- 1}$

Step 1.1.2.2

Rewrite

- 1

$- 1$ as

{(- 1)}^{3}

${(- 1)}^{3}$ .

f (- 1) = \sqrt[3]{{(- 1)}^{3}}

$f (- 1) = \sqrt[3]{{(- 1)}^{3}}$

Step 1.1.2.3

Pull terms out from under the radical, assuming real numbers.

f (- 1) = - 1

$f (- 1) = - 1$

Step 1.1.2.4

The final answer is

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

Step 1.1.3

Convert

- 1

$- 1$ to decimal.

y = - 1

$y = - 1$

y = - 1

$y = - 1$

Step 1.2

Find the point at

x = 0

$x = 0$ .

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Step 1.2.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f (0) = \sqrt[3]{0}

$f (0) = \sqrt[3]{0}$

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

Remove parentheses.

f (0) = \sqrt[3]{0}

$f (0) = \sqrt[3]{0}$

Step 1.2.2.2

Rewrite

0

$0$ as

0^{3}

$0^{3}$ .

f (0) = \sqrt[3]{0^{3}}

$f (0) = \sqrt[3]{0^{3}}$

Step 1.2.2.3

Pull terms out from under the radical, assuming real numbers.

f (0) = 0

$f (0) = 0$

Step 1.2.2.4

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 1.2.3

Convert

0

$0$ to decimal.

y = 0

$y = 0$

y = 0

$y = 0$

Step 1.3

Find the point at

x = 1

$x = 1$ .

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Step 1.3.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

f (1) = \sqrt[3]{1}

$f (1) = \sqrt[3]{1}$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Remove parentheses.

f (1) = \sqrt[3]{1}

$f (1) = \sqrt[3]{1}$

Step 1.3.2.2

Any root of

1

$1$ is

1

$1$ .

f (1) = 1

$f (1) = 1$

Step 1.3.2.3

The final answer is

1

$1$ .

1

$1$

1

$1$

Step 1.3.3

Convert

1

$1$ to decimal.

y = 1

$y = 1$

y = 1

$y = 1$

Step 1.4

Find the point at

x = - 2

$x = - 2$ .

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Step 1.4.1

Replace the variable

x

$x$ with

- 2

$- 2$ in the expression.

f (- 2) = \sqrt[3]{- 2}

$f (- 2) = \sqrt[3]{- 2}$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

Remove parentheses.

f (- 2) = \sqrt[3]{- 2}

$f (- 2) = \sqrt[3]{- 2}$

Step 1.4.2.2

Rewrite

- 2

$- 2$ as

{(- 1)}^{3} \cdot 2

${(- 1)}^{3} \cdot 2$ .

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Step 1.4.2.2.1

Rewrite

- 2

$- 2$ as

- 1 (2)

$- 1 (2)$ .

f (- 2) = \sqrt[3]{- 1 \cdot 2}

$f (- 2) = \sqrt[3]{- 1 \cdot 2}$

Step 1.4.2.2.2

Rewrite

- 1

$- 1$ as

{(- 1)}^{3}

${(- 1)}^{3}$ .

f (- 2) = \sqrt[3]{{(- 1)}^{3} \cdot 2}

$f (- 2) = \sqrt[3]{{(- 1)}^{3} \cdot 2}$

f (- 2) = \sqrt[3]{{(- 1)}^{3} \cdot 2}

$f (- 2) = \sqrt[3]{{(- 1)}^{3} \cdot 2}$

Step 1.4.2.3

Pull terms out from under the radical.

f (- 2) = - 1 \sqrt[3]{2}

$f (- 2) = - 1 \sqrt[3]{2}$

Step 1.4.2.4

Rewrite

- 1 \sqrt[3]{2}

$- 1 \sqrt[3]{2}$ as

- \sqrt[3]{2}

$- \sqrt[3]{2}$ .

f (- 2) = - \sqrt[3]{2}

$f (- 2) = - \sqrt[3]{2}$

Step 1.4.2.5

The final answer is

- \sqrt[3]{2}

$- \sqrt[3]{2}$ .

- \sqrt[3]{2}

$- \sqrt[3]{2}$

- \sqrt[3]{2}

$- \sqrt[3]{2}$

Step 1.4.3

Convert

- \sqrt[3]{2}

$- \sqrt[3]{2}$ to decimal.

y = - 1.25992104

$y = - 1.25992104$

y = - 1.25992104

$y = - 1.25992104$

Step 1.5

Find the point at

x = 2

$x = 2$ .

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Step 1.5.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = \sqrt[3]{2}

$f (2) = \sqrt[3]{2}$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Remove parentheses.

f (2) = \sqrt[3]{2}

$f (2) = \sqrt[3]{2}$

Step 1.5.2.2

The final answer is

\sqrt[3]{2}

$\sqrt[3]{2}$ .

\sqrt[3]{2}

$\sqrt[3]{2}$

\sqrt[3]{2}

$\sqrt[3]{2}$

Step 1.5.3

Convert

\sqrt[3]{2}

$\sqrt[3]{2}$ to decimal.

y = 1.25992104

$y = 1.25992104$

y = 1.25992104

$y = 1.25992104$

Step 1.6

The cubic function can be graphed using the function behavior and the points.

\begin{matrix} x & y - 2 & - 1.26 - 1 & - 1 0 & 0 1 & 1 2 & 1.26 \end{matrix}

$\begin{array}{c} x & y \\ - 2 & - 1.26 \\ - 1 & - 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.26 \end{array}$

Step 1.7

The cubic function can be graphed using the function behavior and the selected points.

Not a polynomial

\begin{matrix} x & y - 2 & - 1.26 - 1 & - 1 0 & 0 1 & 1 2 & 1.26 \end{matrix}

$\begin{array}{c} x & y \\ - 2 & - 1.26 \\ - 1 & - 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.26 \end{array}$

Not a polynomial

\begin{matrix} x & y - 2 & - 1.26 - 1 & - 1 0 & 0 1 & 1 2 & 1.26 \end{matrix}

$\begin{array}{c} x & y \\ - 2 & - 1.26 \\ - 1 & - 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.26 \end{array}$

Step 2

Plot each graph on the same coordinate system.

{(x^{2} + y^{2} - 1)}^{3} = x^{2}

${(x^{2} + y^{2} - 1)}^{3} = x^{2}$

y^{3}

$y^{3}$

Step 3