Pre-Algebra Examples

$p (x) = 3 x^{3} + 6 x^{2} - 4 x - 3$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = 3 {(- 2)}^{3} + 6 {(- 2)}^{2} - 4 \cdot - 2 - 3$

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

$f (- 2) = 3 \cdot - 8 + 6 {(- 2)}^{2} - 4 \cdot - 2 - 3$

Step 1.2.1.2

Multiply $3$ by $- 8$ .

$f (- 2) = - 24 + 6 {(- 2)}^{2} - 4 \cdot - 2 - 3$

Step 1.2.1.3

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

$f (- 2) = - 24 + 6 \cdot 4 - 4 \cdot - 2 - 3$

Step 1.2.1.4

Multiply $6$ by $4$ .

$f (- 2) = - 24 + 24 - 4 \cdot - 2 - 3$

Step 1.2.1.5

Multiply $- 4$ by $- 2$ .

$f (- 2) = - 24 + 24 + 8 - 3$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $- 24$ and $24$ .

$f (- 2) = 0 + 8 - 3$

Step 1.2.2.2

Add $0$ and $8$ .

$f (- 2) = 8 - 3$

Step 1.2.2.3

Subtract $3$ from $8$ .

$f (- 2) = 5$

Step 1.2.3

The final answer is $5$ .

$5$

Step 1.3

Convert $5$ to decimal.

$y = 5$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = 3 {(- 1)}^{3} + 6 {(- 1)}^{2} - 4 \cdot - 1 - 3$

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

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

$f (- 1) = 3 \cdot - 1 + 6 {(- 1)}^{2} - 4 \cdot - 1 - 3$

Step 2.2.1.2

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 + 6 {(- 1)}^{2} - 4 \cdot - 1 - 3$

Step 2.2.1.3

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

$f (- 1) = - 3 + 6 \cdot 1 - 4 \cdot - 1 - 3$

Step 2.2.1.4

Multiply $6$ by $1$ .

$f (- 1) = - 3 + 6 - 4 \cdot - 1 - 3$

Step 2.2.1.5

Multiply $- 4$ by $- 1$ .

$f (- 1) = - 3 + 6 + 4 - 3$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 3$ and $6$ .

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

Step 2.2.2.2

Add $3$ and $4$ .

$f (- 1) = 7 - 3$

Step 2.2.2.3

Subtract $3$ from $7$ .

$f (- 1) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 2.3

Convert $4$ to decimal.

$y = 4$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = 3 {(0)}^{3} + 6 {(0)}^{2} - 4 \cdot 0 - 3$

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

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

$f (0) = 3 \cdot 0 + 6 {(0)}^{2} - 4 \cdot 0 - 3$

Step 3.2.1.2

Multiply $3$ by $0$ .

$f (0) = 0 + 6 {(0)}^{2} - 4 \cdot 0 - 3$

Step 3.2.1.3

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

$f (0) = 0 + 6 \cdot 0 - 4 \cdot 0 - 3$

Step 3.2.1.4

Multiply $6$ by $0$ .

$f (0) = 0 + 0 - 4 \cdot 0 - 3$

Step 3.2.1.5

Multiply $- 4$ by $0$ .

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

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $0$ and $0$ .

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

Step 3.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 3$

Step 3.2.2.3

Subtract $3$ from $0$ .

$f (0) = - 3$

Step 3.2.3

The final answer is $- 3$ .

$- 3$

Step 3.3

Convert $- 3$ to decimal.

$y = - 3$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = 3 {(1)}^{3} + 6 {(1)}^{2} - 4 \cdot 1 - 3$

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

One to any power is one.

$f (1) = 3 \cdot 1 + 6 {(1)}^{2} - 4 \cdot 1 - 3$

Step 4.2.1.2

Multiply $3$ by $1$ .

$f (1) = 3 + 6 {(1)}^{2} - 4 \cdot 1 - 3$

Step 4.2.1.3

One to any power is one.

$f (1) = 3 + 6 \cdot 1 - 4 \cdot 1 - 3$

Step 4.2.1.4

Multiply $6$ by $1$ .

$f (1) = 3 + 6 - 4 \cdot 1 - 3$

Step 4.2.1.5

Multiply $- 4$ by $1$ .

$f (1) = 3 + 6 - 4 - 3$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $3$ and $6$ .

$f (1) = 9 - 4 - 3$

Step 4.2.2.2

Subtract $4$ from $9$ .

$f (1) = 5 - 3$

Step 4.2.2.3

Subtract $3$ from $5$ .

$f (1) = 2$

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 $x = 2$ .

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Step 5.1

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

$f (2) = 3 {(2)}^{3} + 6 {(2)}^{2} - 4 \cdot 2 - 3$

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 $2$ to the power of $3$ .

$f (2) = 3 \cdot 8 + 6 {(2)}^{2} - 4 \cdot 2 - 3$

Step 5.2.1.2

Multiply $3$ by $8$ .

$f (2) = 24 + 6 {(2)}^{2} - 4 \cdot 2 - 3$

Step 5.2.1.3

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

$f (2) = 24 + 6 \cdot 4 - 4 \cdot 2 - 3$

Step 5.2.1.4

Multiply $6$ by $4$ .

$f (2) = 24 + 24 - 4 \cdot 2 - 3$

Step 5.2.1.5

Multiply $- 4$ by $2$ .

$f (2) = 24 + 24 - 8 - 3$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $24$ and $24$ .

$f (2) = 48 - 8 - 3$

Step 5.2.2.2

Subtract $8$ from $48$ .

$f (2) = 40 - 3$

Step 5.2.2.3

Subtract $3$ from $40$ .

$f (2) = 37$

Step 5.2.3

The final answer is $37$ .

$37$

Step 5.3

Convert $37$ to decimal.

$y = 37$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 5 \\ - 1 & 4 \\ 0 & - 3 \\ 1 & 2 \\ 2 & 37 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 5 \\ - 1 & 4 \\ 0 & - 3 \\ 1 & 2 \\ 2 & 37 \end{array}$

Step 8