Algebra Examples

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

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

Step 1

Find the point at

x = - 1

$x = - 1$ .

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Step 1.1

Replace the variable

x

$x$ with

- 1

$- 1$ in the expression.

f (- 1) = {(- 1)}^{3} - {(- 1)}^{2}

$f (- 1) = {(- 1)}^{3} - {(- 1)}^{2}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Raise

- 1

$- 1$ to the power of

3

$3$ .

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

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

Step 1.2.1.2

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

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

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Step 1.2.1.2.1

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

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

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Step 1.2.1.2.1.1

Raise

- 1

$- 1$ to the power of

1

$1$ .

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

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

Step 1.2.1.2.1.2

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)}^{1 + 2}

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

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

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

Step 1.2.1.2.2

Add

1

$1$ and

2

$2$ .

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

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

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

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

Step 1.2.1.3

Raise

- 1

$- 1$ to the power of

3

$3$ .

f (- 1) = - 1 - 1

$f (- 1) = - 1 - 1$

f (- 1) = - 1 - 1

$f (- 1) = - 1 - 1$

Step 1.2.2

Subtract

1

$1$ from

- 1

$- 1$ .

f (- 1) = - 2

$f (- 1) = - 2$

Step 1.2.3

The final answer is

- 2

$- 2$ .

- 2

$- 2$

- 2

$- 2$

Step 1.3

Convert

- 2

$- 2$ to decimal.

y = - 2

$y = - 2$

y = - 2

$y = - 2$

Step 2

Find the point at

x = 0

$x = 0$ .

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Step 2.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f (0) = {(0)}^{3} - {(0)}^{2}

$f (0) = {(0)}^{3} - {(0)}^{2}$

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

Raising

0

$0$ to any positive power yields

0

$0$ .

f (0) = 0 - {(0)}^{2}

$f (0) = 0 - {(0)}^{2}$

Step 2.2.1.2

Raising

0

$0$ to any positive power yields

0

$0$ .

f (0) = 0 - 0

$f (0) = 0 - 0$

Step 2.2.1.3

Multiply

- 1

$- 1$ by

0

$0$ .

f (0) = 0 + 0

$f (0) = 0 + 0$

f (0) = 0 + 0

$f (0) = 0 + 0$

Step 2.2.2

Add

0

$0$ and

0

$0$ .

f (0) = 0

$f (0) = 0$

Step 2.2.3

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.3

Convert

0

$0$ to decimal.

y = 0

$y = 0$

y = 0

$y = 0$

Step 3

Find the point at

x = 2

$x = 2$ .

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Step 3.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = {(2)}^{3} - {(2)}^{2}

$f (2) = {(2)}^{3} - {(2)}^{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise

2

$2$ to the power of

3

$3$ .

f (2) = 8 - {(2)}^{2}

$f (2) = 8 - {(2)}^{2}$

Step 3.2.1.2

Raise

2

$2$ to the power of

2

$2$ .

f (2) = 8 - 1 \cdot 4

$f (2) = 8 - 1 \cdot 4$

Step 3.2.1.3

Multiply

- 1

$- 1$ by

4

$4$ .

f (2) = 8 - 4

$f (2) = 8 - 4$

f (2) = 8 - 4

$f (2) = 8 - 4$

Step 3.2.2

Subtract

4

$4$ from

8

$8$ .

f (2) = 4

$f (2) = 4$

Step 3.2.3

The final answer is

4

$4$ .

4

$4$

4

$4$

Step 3.3

Convert

4

$4$ to decimal.

y = 4

$y = 4$

y = 4

$y = 4$

Step 4

Find the point at

x = 3

$x = 3$ .

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Step 4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = {(3)}^{3} - {(3)}^{2}

$f (3) = {(3)}^{3} - {(3)}^{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise

3

$3$ to the power of

3

$3$ .

f (3) = 27 - {(3)}^{2}

$f (3) = 27 - {(3)}^{2}$

Step 4.2.1.2

Raise

3

$3$ to the power of

2

$2$ .

f (3) = 27 - 1 \cdot 9

$f (3) = 27 - 1 \cdot 9$

Step 4.2.1.3

Multiply

- 1

$- 1$ by

9

$9$ .

f (3) = 27 - 9

$f (3) = 27 - 9$

f (3) = 27 - 9

$f (3) = 27 - 9$

Step 4.2.2

Subtract

9

$9$ from

27

$27$ .

f (3) = 18

$f (3) = 18$

Step 4.2.3

The final answer is

18

$18$ .

18

$18$

18

$18$

Step 4.3

Convert

18

$18$ to decimal.

y = 18

$y = 18$

y = 18

$y = 18$

Step 5

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

\begin{array}{c} x & y \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & 4 \\ 3 & 18 \end{array}

$\begin{array}{c} x & y \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & 4 \\ 3 & 18 \end{array}$

Step 6

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

Falls to the left and rises to the right

\begin{array}{c} x & y \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & 4 \\ 3 & 18 \end{array}

$\begin{array}{c} x & y \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & 4 \\ 3 & 18 \end{array}$

Step 7