Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

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

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

Step 1.2.1.2

Multiply $- 2$ by $- 1$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Multiply $3$ by $1$ .

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

Step 1.2.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $2$ and $3$ .

$f (- 1) = 5 + 1 - 3$

Step 1.2.2.2

Add $5$ and $1$ .

$f (- 1) = 6 - 3$

Step 1.2.2.3

Subtract $3$ from $6$ .

$f (- 1) = 3$

Step 1.2.3

The final answer is $3$ .

$3$

Step 1.3

Convert $3$ to decimal.

$y = 3$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

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

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 $0$ .

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

Step 2.2.1.2

Multiply $- 2$ by $0$ .

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Multiply $3$ by $0$ .

$f (0) = 0 + 0 - (0) - 3$

Step 2.2.1.5

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0 + 0 - 3$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 - 3$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 3$

Step 2.2.2.3

Subtract $3$ from $0$ .

$f (0) = - 3$

Step 2.2.3

The final answer is $- 3$ .

$- 3$

Step 2.3

Convert $- 3$ to decimal.

$y = - 3$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

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

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

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

Step 3.2.1.2

Multiply $- 2$ by $8$ .

$f (2) = - 16 + 3 {(2)}^{2} - (2) - 3$

Step 3.2.1.3

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

$f (2) = - 16 + 3 \cdot 4 - (2) - 3$

Step 3.2.1.4

Multiply $3$ by $4$ .

$f (2) = - 16 + 12 - (2) - 3$

Step 3.2.1.5

Multiply $- 1$ by $2$ .

$f (2) = - 16 + 12 - 2 - 3$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 16$ and $12$ .

$f (2) = - 4 - 2 - 3$

Step 3.2.2.2

Subtract $2$ from $- 4$ .

$f (2) = - 6 - 3$

Step 3.2.2.3

Subtract $3$ from $- 6$ .

$f (2) = - 9$

Step 3.2.3

The final answer is $- 9$ .

$- 9$

Step 3.3

Convert $- 9$ to decimal.

$y = - 9$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

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 $3$ to the power of $3$ .

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

Step 4.2.1.2

Multiply $- 2$ by $27$ .

$f (3) = - 54 + 3 {(3)}^{2} - (3) - 3$

Step 4.2.1.3

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

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Step 4.2.1.3.1

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

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Step 4.2.1.3.1.1

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

$f (3) = - 54 + 3 {(3)}^{2} - (3) - 3$

Step 4.2.1.3.1.2

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

$f (3) = - 54 + 3^{1 + 2} - (3) - 3$

Step 4.2.1.3.2

Add $1$ and $2$ .

$f (3) = - 54 + 3^{3} - (3) - 3$

Step 4.2.1.4

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

$f (3) = - 54 + 27 - (3) - 3$

Step 4.2.1.5

Multiply $- 1$ by $3$ .

$f (3) = - 54 + 27 - 3 - 3$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $- 54$ and $27$ .

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

Step 4.2.2.2

Subtract $3$ from $- 27$ .

$f (3) = - 30 - 3$

Step 4.2.2.3

Subtract $3$ from $- 30$ .

$f (3) = - 33$

Step 4.2.3

The final answer is $- 33$ .

$- 33$

Step 4.3

Convert $- 33$ to decimal.

$y = - 33$

Step 5

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

$\begin{array}{c} x & y \\ - 1 & 3 \\ 0 & - 3 \\ 1 & - 3 \\ 2 & - 9 \\ 3 & - 33 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & 3 \\ 0 & - 3 \\ 1 & - 3 \\ 2 & - 9 \\ 3 & - 33 \end{array}$

Step 7