Pre-Algebra Examples

$f (x) = x^{3} - 2 x^{2} - 24 x$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = {(- 1)}^{3} - 2 {(- 1)}^{2} - 24 \cdot - 1$

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

$f (- 1) = - 1 - 2 {(- 1)}^{2} - 24 \cdot - 1$

Step 1.2.1.2

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

$f (- 1) = - 1 - 2 \cdot 1 - 24 \cdot - 1$

Step 1.2.1.3

Multiply $- 2$ by $1$ .

$f (- 1) = - 1 - 2 - 24 \cdot - 1$

Step 1.2.1.4

Multiply $- 24$ by $- 1$ .

$f (- 1) = - 1 - 2 + 24$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $2$ from $- 1$ .

$f (- 1) = - 3 + 24$

Step 1.2.2.2

Add $- 3$ and $24$ .

$f (- 1) = 21$

Step 1.2.3

The final answer is $21$ .

$21$

Step 1.3

Convert $21$ to decimal.

$y = 21$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = {(0)}^{3} - 2 {(0)}^{2} - 24 \cdot 0$

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 $0$ .

$f (0) = 0 - 2 {(0)}^{2} - 24 \cdot 0$

Step 2.2.1.2

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

$f (0) = 0 - 2 \cdot 0 - 24 \cdot 0$

Step 2.2.1.3

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0 - 24 \cdot 0$

Step 2.2.1.4

Multiply $- 24$ by $0$ .

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

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = {(1)}^{3} - 2 {(1)}^{2} - 24 \cdot 1$

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

One to any power is one.

$f (1) = 1 - 2 {(1)}^{2} - 24 \cdot 1$

Step 3.2.1.2

One to any power is one.

$f (1) = 1 - 2 \cdot 1 - 24 \cdot 1$

Step 3.2.1.3

Multiply $- 2$ by $1$ .

$f (1) = 1 - 2 - 24 \cdot 1$

Step 3.2.1.4

Multiply $- 24$ by $1$ .

$f (1) = 1 - 2 - 24$

Step 3.2.2

Simplify by subtracting numbers.

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Step 3.2.2.1

Subtract $2$ from $1$ .

$f (1) = - 1 - 24$

Step 3.2.2.2

Subtract $24$ from $- 1$ .

$f (1) = - 25$

Step 3.2.3

The final answer is $- 25$ .

$- 25$

Step 3.3

Convert $- 25$ to decimal.

$y = - 25$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = {(2)}^{3} - 2 {(2)}^{2} - 24 \cdot 2$

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 $3$ .

$f (2) = 8 - 2 {(2)}^{2} - 24 \cdot 2$

Step 4.2.1.2

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

$f (2) = 8 - 2 \cdot 4 - 24 \cdot 2$

Step 4.2.1.3

Multiply $- 2$ by $4$ .

$f (2) = 8 - 8 - 24 \cdot 2$

Step 4.2.1.4

Multiply $- 24$ by $2$ .

$f (2) = 8 - 8 - 48$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $8$ from $8$ .

$f (2) = 0 - 48$

Step 4.2.2.2

Subtract $48$ from $0$ .

$f (2) = - 48$

Step 4.2.3

The final answer is $- 48$ .

$- 48$

Step 4.3

Convert $- 48$ to decimal.

$y = - 48$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

$f (3) = {(3)}^{3} - 2 {(3)}^{2} - 24 \cdot 3$

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 $3$ .

$f (3) = 27 - 2 {(3)}^{2} - 24 \cdot 3$

Step 5.2.1.2

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

$f (3) = 27 - 2 \cdot 9 - 24 \cdot 3$

Step 5.2.1.3

Multiply $- 2$ by $9$ .

$f (3) = 27 - 18 - 24 \cdot 3$

Step 5.2.1.4

Multiply $- 24$ by $3$ .

$f (3) = 27 - 18 - 72$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $18$ from $27$ .

$f (3) = 9 - 72$

Step 5.2.2.2

Subtract $72$ from $9$ .

$f (3) = - 63$

Step 5.2.3

The final answer is $- 63$ .

$- 63$

Step 5.3

Convert $- 63$ to decimal.

$y = - 63$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 21 \\ 0 & 0 \\ 1 & - 25 \\ 2 & - 48 \\ 3 & - 63 \end{array}$

Step 7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 1 & 21 \\ 0 & 0 \\ 1 & - 25 \\ 2 & - 48 \\ 3 & - 63 \end{array}$

Step 8