Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 1.2

Simplify the result.

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Raise $- 1$ to the power of $3$ .

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

Step 1.2.2.2

Raise $- 1$ to the power of $2$ .

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

Step 1.2.2.3

Multiply $- 2$ by $1$ .

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

Step 1.2.3

Simplify by subtracting numbers.

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Step 1.2.3.1

Subtract $2$ from $- 1$ .

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

Step 1.2.3.2

Subtract $1$ from $- 3$ .

$f (- 1) = - 4$

Step 1.2.4

The final answer is $- 4$ .

$- 4$

Step 1.3

Convert $- 4$ to decimal.

$y = - 4$

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Raising $0$ to any positive power yields $0$ .

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

Step 2.2.2.2

Raising $0$ to any positive power yields $0$ .

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

Step 2.2.2.3

Multiply $- 2$ by $0$ .

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

Step 2.2.3

Simplify by adding numbers.

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Step 2.2.3.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$f (0) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Raise $2$ to the power of $3$ .

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

Step 3.2.2.2

Raise $2$ to the power of $2$ .

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

Step 3.2.2.3

Multiply $- 2$ by $4$ .

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

Step 3.2.3

Simplify by adding and subtracting.

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Step 3.2.3.1

Subtract $8$ from $8$ .

$f (2) = 0 + 2$

Step 3.2.3.2

Add $0$ and $2$ .

$f (2) = 2$

Step 3.2.4

The final answer is $2$ .

$2$

Step 3.3

Convert $2$ to decimal.

$y = 2$

Step 4

Find the point at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Raise $3$ to the power of $3$ .

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

Step 4.2.2.2

Raise $3$ to the power of $2$ .

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

Step 4.2.2.3

Multiply $- 2$ by $9$ .

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

Step 4.2.3

Simplify by adding and subtracting.

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Step 4.2.3.1

Subtract $18$ from $27$ .

$f (3) = 9 + 3$

Step 4.2.3.2

Add $9$ and $3$ .

$f (3) = 12$

Step 4.2.4

The final answer is $12$ .

$12$

Step 4.3

Convert $12$ to decimal.

$y = 12$

Step 5

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

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

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

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

Step 7