Algebra Examples

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

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

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 $3$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $- 8$ and $4$ .

$f (- 2) = - 4 + 4 - 1$

Step 1.2.2.2

Add $- 4$ and $4$ .

$f (- 2) = 0 - 1$

Step 1.2.2.3

Subtract $1$ from $0$ .

$f (- 2) = - 1$

Step 1.2.3

The final answer is $- 1$ .

$- 1$

Step 1.3

Convert $- 1$ to decimal.

$y = - 1$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

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

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

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Multiply $- 2$ by $- 1$ .

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

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 1$ and $1$ .

$f (- 1) = 0 + 2 - 1$

Step 2.2.2.2

Add $0$ and $2$ .

$f (- 1) = 2 - 1$

Step 2.2.2.3

Subtract $1$ from $2$ .

$f (- 1) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = {(0)}^{3} + {(0)}^{2} - 2 \cdot 0 - 1$

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 $0$ .

$f (0) = 0 + {(0)}^{2} - 2 \cdot 0 - 1$

Step 3.2.1.2

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

$f (0) = 0 + 0 - 2 \cdot 0 - 1$

Step 3.2.1.3

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0 + 0 - 1$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 - 1$

Step 3.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 1$

Step 3.2.2.3

Subtract $1$ from $0$ .

$f (0) = - 1$

Step 3.2.3

The final answer is $- 1$ .

$- 1$

Step 3.3

Convert $- 1$ to decimal.

$y = - 1$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

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

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 $3$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $- 2$ by $2$ .

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

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $8$ and $4$ .

$f (2) = 12 - 4 - 1$

Step 4.2.2.2

Subtract $4$ from $12$ .

$f (2) = 8 - 1$

Step 4.2.2.3

Subtract $1$ from $8$ .

$f (2) = 7$

Step 4.2.3

The final answer is $7$ .

$7$

Step 4.3

Convert $7$ to decimal.

$y = 7$

Step 5

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

$\begin{array}{c} x & y \\ - 2 & - 1 \\ - 1 & 1 \\ 0 & - 1 \\ 1 & - 1 \\ 2 & 7 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & - 1 \\ - 1 & 1 \\ 0 & - 1 \\ 1 & - 1 \\ 2 & 7 \end{array}$

Step 7