Pre-Algebra Examples

$f (x) = x^{3} - 3 x^{2} - 4 x - 12$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = {(- 1)}^{3} - 3 {(- 1)}^{2} - 4 \cdot - 1 - 12$

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 $3$ .

$f (- 1) = - 1 - 3 {(- 1)}^{2} - 4 \cdot - 1 - 12$

Step 1.2.1.2

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

$f (- 1) = - 1 - 3 \cdot 1 - 4 \cdot - 1 - 12$

Step 1.2.1.3

Multiply $- 3$ by $1$ .

$f (- 1) = - 1 - 3 - 4 \cdot - 1 - 12$

Step 1.2.1.4

Multiply $- 4$ by $- 1$ .

$f (- 1) = - 1 - 3 + 4 - 12$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $3$ from $- 1$ .

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

Step 1.2.2.2

Add $- 4$ and $4$ .

$f (- 1) = 0 - 12$

Step 1.2.2.3

Subtract $12$ from $0$ .

$f (- 1) = - 12$

Step 1.2.3

The final answer is $- 12$ .

$- 12$

Step 1.3

Convert $- 12$ to decimal.

$y = - 12$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = {(1)}^{3} - 3 {(1)}^{2} - 4 \cdot 1 - 12$

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

One to any power is one.

$f (1) = 1 - 3 {(1)}^{2} - 4 \cdot 1 - 12$

Step 2.2.1.2

One to any power is one.

$f (1) = 1 - 3 \cdot 1 - 4 \cdot 1 - 12$

Step 2.2.1.3

Multiply $- 3$ by $1$ .

$f (1) = 1 - 3 - 4 \cdot 1 - 12$

Step 2.2.1.4

Multiply $- 4$ by $1$ .

$f (1) = 1 - 3 - 4 - 12$

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract $3$ from $1$ .

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

Step 2.2.2.2

Subtract $4$ from $- 2$ .

$f (1) = - 6 - 12$

Step 2.2.2.3

Subtract $12$ from $- 6$ .

$f (1) = - 18$

Step 2.2.3

The final answer is $- 18$ .

$- 18$

Step 2.3

Convert $- 18$ to decimal.

$y = - 18$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = {(2)}^{3} - 3 {(2)}^{2} - 4 \cdot 2 - 12$

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 $3$ .

$f (2) = 8 - 3 {(2)}^{2} - 4 \cdot 2 - 12$

Step 3.2.1.2

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

$f (2) = 8 - 3 \cdot 4 - 4 \cdot 2 - 12$

Step 3.2.1.3

Multiply $- 3$ by $4$ .

$f (2) = 8 - 12 - 4 \cdot 2 - 12$

Step 3.2.1.4

Multiply $- 4$ by $2$ .

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

Step 3.2.2

Simplify by subtracting numbers.

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Step 3.2.2.1

Subtract $12$ from $8$ .

$f (2) = - 4 - 8 - 12$

Step 3.2.2.2

Subtract $8$ from $- 4$ .

$f (2) = - 12 - 12$

Step 3.2.2.3

Subtract $12$ from $- 12$ .

$f (2) = - 24$

Step 3.2.3

The final answer is $- 24$ .

$- 24$

Step 3.3

Convert $- 24$ to decimal.

$y = - 24$

Step 4

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

$\begin{array}{c} x & y \\ - 1 & - 12 \\ 0 & - 12 \\ 1 & - 18 \\ 2 & - 24 \\ 3 & - 24 \end{array}$

Step 5

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 1 & - 12 \\ 0 & - 12 \\ 1 & - 18 \\ 2 & - 24 \\ 3 & - 24 \end{array}$

Step 6