Algebra Examples

x^{3} - 2 x^{2} + x

$x^{3} - 2 x^{2} + x$

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Remove parentheses.

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Raise

- 1

$- 1$ to the power of

3

$3$ .

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

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

Step 1.2.2.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 1.2.2.3

Multiply

- 2

$- 2$ by

1

$1$ .

f (- 1) = - 1 - 2 - 1

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

f (- 1) = - 1 - 2 - 1

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

Step 1.2.3

Simplify by subtracting numbers.

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Step 1.2.3.1

Subtract

2

$2$ from

- 1

$- 1$ .

f (- 1) = - 3 - 1

$f (- 1) = - 3 - 1$

Step 1.2.3.2

Subtract

1

$1$ from

- 3

$- 3$ .

f (- 1) = - 4

$f (- 1) = - 4$

f (- 1) = - 4

$f (- 1) = - 4$

Step 1.2.4

The final answer is

- 4

$- 4$ .

- 4

$- 4$

- 4

$- 4$

Step 1.3

Convert

- 4

$- 4$ to decimal.

y = - 4

$y = - 4$

y = - 4

$y = - 4$

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} - 2 {(0)}^{2} + 0

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

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

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Raising

0

$0$ to any positive power yields

0

$0$ .

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

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

Step 2.2.2.2

Raising

0

$0$ to any positive power yields

0

$0$ .

f (0) = 0 - 2 \cdot 0 + 0

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

Step 2.2.2.3

Multiply

- 2

$- 2$ by

0

$0$ .

f (0) = 0 + 0 + 0

$f (0) = 0 + 0 + 0$

f (0) = 0 + 0 + 0

$f (0) = 0 + 0 + 0$

Step 2.2.3

Simplify by adding numbers.

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Step 2.2.3.1

Add

0

$0$ and

0

$0$ .

f (0) = 0 + 0

$f (0) = 0 + 0$

Step 2.2.3.2

Add

0

$0$ and

0

$0$ .

f (0) = 0

$f (0) = 0$

f (0) = 0

$f (0) = 0$

Step 2.2.4

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)}^{2} + 2

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

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

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

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Raise

2

$2$ to the power of

3

$3$ .

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

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

Step 3.2.2.2

Raise

2

$2$ to the power of

2

$2$ .

f (2) = 8 - 2 \cdot 4 + 2

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

Step 3.2.2.3

Multiply

- 2

$- 2$ by

4

$4$ .

f (2) = 8 - 8 + 2

$f (2) = 8 - 8 + 2$

f (2) = 8 - 8 + 2

$f (2) = 8 - 8 + 2$

Step 3.2.3

Simplify by adding and subtracting.

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Step 3.2.3.1

Subtract

8

$8$ from

8

$8$ .

f (2) = 0 + 2

$f (2) = 0 + 2$

Step 3.2.3.2

Add

0

$0$ and

2

$2$ .

f (2) = 2

$f (2) = 2$

f (2) = 2

$f (2) = 2$

Step 3.2.4

The final answer is

2

$2$ .

2

$2$

2

$2$

Step 3.3

Convert

2

$2$ to decimal.

y = 2

$y = 2$

y = 2

$y = 2$

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} - 2 {(3)}^{2} + 3

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

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

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

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Raise

3

$3$ to the power of

3

$3$ .

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

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

Step 4.2.2.2

Raise

3

$3$ to the power of

2

$2$ .

f (3) = 27 - 2 \cdot 9 + 3

$f (3) = 27 - 2 \cdot 9 + 3$

Step 4.2.2.3

Multiply

- 2

$- 2$ by

9

$9$ .

f (3) = 27 - 18 + 3

$f (3) = 27 - 18 + 3$

f (3) = 27 - 18 + 3

$f (3) = 27 - 18 + 3$

Step 4.2.3

Simplify by adding and subtracting.

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Step 4.2.3.1

Subtract

18

$18$ from

27

$27$ .

f (3) = 9 + 3

$f (3) = 9 + 3$

Step 4.2.3.2

Add

9

$9$ and

3

$3$ .

f (3) = 12

$f (3) = 12$

f (3) = 12

$f (3) = 12$

Step 4.2.4

The final answer is

12

$12$ .

12

$12$

12

$12$

Step 4.3

Convert

12

$12$ to decimal.

y = 12

$y = 12$

y = 12

$y = 12$

Step 5

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

\begin{matrix} x & y - 1 & - 4 0 & 0 1 & 0 2 & 2 3 & 12 \end{matrix}

$\begin{array}{c} x & y \\ - 1 & - 4 \\ 0 & 0 \\ 1 & 0 \\ 2 & 2 \\ 3 & 12 \end{array}$

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{matrix} x & y - 1 & - 4 0 & 0 1 & 0 2 & 2 3 & 12 \end{matrix}

$\begin{array}{c} x & y \\ - 1 & - 4 \\ 0 & 0 \\ 1 & 0 \\ 2 & 2 \\ 3 & 12 \end{array}$

Step 7