Algebra Examples

$y = - x (x - 1) (x + 2)$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

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

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

Step 1.2.1.2

Multiply $- 1$ by $- 8$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Multiply $- 1$ by $4$ .

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

Step 1.2.1.5

Multiply $2$ by $- 2$ .

$f (- 2) = 8 - 4 - 4$

Step 1.2.2

Simplify by subtracting numbers.

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Step 1.2.2.1

Subtract $4$ from $8$ .

$f (- 2) = 4 - 4$

Step 1.2.2.2

Subtract $4$ from $4$ .

$f (- 2) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

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

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

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 2.2.1.1.1

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

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Step 2.2.1.1.1.1

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

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

Step 2.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.1.1.2

Add $1$ and $3$ .

$f (- 1) = {(- 1)}^{4} - {(- 1)}^{2} + 2 (- 1)$

Step 2.2.1.2

Raise $- 1$ to the power of $4$ .

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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Step 2.2.1.3.1.1

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

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

Step 2.2.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.1.3.2

Add $1$ and $2$ .

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

Step 2.2.1.4

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

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

Step 2.2.1.5

Multiply $2$ by $- 1$ .

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

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract $1$ from $1$ .

$f (- 1) = 0 - 2$

Step 2.2.2.2

Subtract $2$ from $0$ .

$f (- 1) = - 2$

Step 2.2.3

The final answer is $- 2$ .

$- 2$

Step 2.3

Convert $- 2$ to decimal.

$y = - 2$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

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

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

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

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

Step 3.2.1.2

Multiply $- 1$ by $0$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0 + 2 (0)$

Step 3.2.1.5

Multiply $2$ by $0$ .

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

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 3.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

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

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 $2$ to the power of $3$ .

$f (2) = - 1 \cdot 8 - {(2)}^{2} + 2 (2)$

Step 4.2.1.2

Multiply $- 1$ by $8$ .

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Multiply $- 1$ by $4$ .

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

Step 4.2.1.5

Multiply $2$ by $2$ .

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

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $4$ from $- 8$ .

$f (2) = - 12 + 4$

Step 4.2.2.2

Add $- 12$ and $4$ .

$f (2) = - 8$

Step 4.2.3

The final answer is $- 8$ .

$- 8$

Step 4.3

Convert $- 8$ to decimal.

$y = - 8$

Step 5

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & - 8 \end{array}$

Step 6

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & - 8 \end{array}$

Step 7