Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

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

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

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Step 1.2.1.1.1

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

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Step 1.2.1.1.1.1

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

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

Step 1.2.1.1.1.2

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

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

Step 1.2.1.1.2

Add $1$ and $3$ .

$f (- 1) = {(- 1)}^{4} + 3 {(- 1)}^{2} - 2$

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Multiply $3$ by $1$ .

$f (- 1) = 1 + 3 - 2$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $1$ and $3$ .

$f (- 1) = 4 - 2$

Step 1.2.2.2

Subtract $2$ from $4$ .

$f (- 1) = 2$

Step 1.2.3

The final answer is $2$ .

$2$

Step 1.3

Convert $2$ to decimal.

$y = 2$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

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

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

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

Step 2.2.1.2

Multiply $- 1$ by $0$ .

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

Step 2.2.1.3

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

$f (0) = 0 + 3 \cdot 0 - 2$

Step 2.2.1.4

Multiply $3$ by $0$ .

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

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 - 2$

Step 2.2.2.2

Subtract $2$ from $0$ .

$f (0) = - 2$

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 $x = 1$ .

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Step 3.1

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

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

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

One to any power is one.

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

Step 3.2.1.2

Multiply $- 1$ by $1$ .

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

Step 3.2.1.3

One to any power is one.

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

Step 3.2.1.4

Multiply $3$ by $1$ .

$f (1) = - 1 + 3 - 2$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 1$ and $3$ .

$f (1) = 2 - 2$

Step 3.2.2.2

Subtract $2$ from $2$ .

$f (1) = 0$

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.

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

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 $3$ .

$f (2) = - 1 \cdot 8 + 3 {(2)}^{2} - 2$

Step 4.2.1.2

Multiply $- 1$ by $8$ .

$f (2) = - 8 + 3 {(2)}^{2} - 2$

Step 4.2.1.3

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

$f (2) = - 8 + 3 \cdot 4 - 2$

Step 4.2.1.4

Multiply $3$ by $4$ .

$f (2) = - 8 + 12 - 2$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $- 8$ and $12$ .

$f (2) = 4 - 2$

Step 4.2.2.2

Subtract $2$ from $4$ .

$f (2) = 2$

Step 4.2.3

The final answer is $2$ .

$2$

Step 4.3

Convert $2$ to decimal.

$y = 2$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (3) = - 1 \cdot 27 + 3 {(3)}^{2} - 2$

Step 5.2.1.2

Multiply $- 1$ by $27$ .

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

Step 5.2.1.3

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 5.2.1.3.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 5.2.1.3.1.1

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

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

Step 5.2.1.3.1.2

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

$f (3) = - 27 + 3^{1 + 2} - 2$

Step 5.2.1.3.2

Add $1$ and $2$ .

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

Step 5.2.1.4

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

$f (3) = - 27 + 27 - 2$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 27$ and $27$ .

$f (3) = 0 - 2$

Step 5.2.2.2

Subtract $2$ from $0$ .

$f (3) = - 2$

Step 5.2.3

The final answer is $- 2$ .

$- 2$

Step 5.3

Convert $- 2$ to decimal.

$y = - 2$

Step 6

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

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

Step 7

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

Rises to the left and falls to the right

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

Step 8