Pre-Algebra Examples

$- x^{3} - 3 x^{2} - 9 x + 4$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

$f (- 3) = - {(- 3)}^{3} - 3 {(- 3)}^{2} - 9 \cdot - 3 + 4$

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

$f (- 3) = 27 - 3 {(- 3)}^{2} - 9 \cdot - 3 + 4$

Step 1.2.1.2

Multiply $- 1$ by $- 27$ .

$f (- 3) = 27 - 3 {(- 3)}^{2} - 9 \cdot - 3 + 4$

Step 1.2.1.3

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

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Step 1.2.1.3.1

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

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Step 1.2.1.3.1.1

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

$f (- 3) = 27 + (- 3) {(- 3)}^{2} - 9 \cdot - 3 + 4$

Step 1.2.1.3.1.2

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

$f (- 3) = 27 + {(- 3)}^{1 + 2} - 9 \cdot - 3 + 4$

Step 1.2.1.3.2

Add $1$ and $2$ .

$f (- 3) = 27 + {(- 3)}^{3} - 9 \cdot - 3 + 4$

Step 1.2.1.4

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

$f (- 3) = 27 - 27 - 9 \cdot - 3 + 4$

Step 1.2.1.5

Multiply $- 9$ by $- 3$ .

$f (- 3) = 27 - 27 + 27 + 4$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $27$ from $27$ .

$f (- 3) = 0 + 27 + 4$

Step 1.2.2.2

Add $0$ and $27$ .

$f (- 3) = 27 + 4$

Step 1.2.2.3

Add $27$ and $4$ .

$f (- 3) = 31$

Step 1.2.3

The final answer is $31$ .

$31$

Step 1.3

Convert $31$ to decimal.

$y = 31$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

$f (- 2) = - {(- 2)}^{3} - 3 {(- 2)}^{2} - 9 \cdot - 2 + 4$

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

$f (- 2) = 8 - 3 {(- 2)}^{2} - 9 \cdot - 2 + 4$

Step 2.2.1.2

Multiply $- 1$ by $- 8$ .

$f (- 2) = 8 - 3 {(- 2)}^{2} - 9 \cdot - 2 + 4$

Step 2.2.1.3

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

$f (- 2) = 8 - 3 \cdot 4 - 9 \cdot - 2 + 4$

Step 2.2.1.4

Multiply $- 3$ by $4$ .

$f (- 2) = 8 - 12 - 9 \cdot - 2 + 4$

Step 2.2.1.5

Multiply $- 9$ by $- 2$ .

$f (- 2) = 8 - 12 + 18 + 4$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $12$ from $8$ .

$f (- 2) = - 4 + 18 + 4$

Step 2.2.2.2

Add $- 4$ and $18$ .

$f (- 2) = 14 + 4$

Step 2.2.2.3

Add $14$ and $4$ .

$f (- 2) = 18$

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 $x = - 1$ .

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Step 3.1

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

$f (- 1) = - {(- 1)}^{3} - 3 {(- 1)}^{2} - 9 \cdot - 1 + 4$

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

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

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Step 3.2.1.1.1

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

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Step 3.2.1.1.1.1

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

$f (- 1) = (- 1) {(- 1)}^{3} - 3 {(- 1)}^{2} - 9 \cdot - 1 + 4$

Step 3.2.1.1.1.2

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

$f (- 1) = {(- 1)}^{1 + 3} - 3 {(- 1)}^{2} - 9 \cdot - 1 + 4$

Step 3.2.1.1.2

Add $1$ and $3$ .

$f (- 1) = {(- 1)}^{4} - 3 {(- 1)}^{2} - 9 \cdot - 1 + 4$

Step 3.2.1.2

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

$f (- 1) = 1 - 3 {(- 1)}^{2} - 9 \cdot - 1 + 4$

Step 3.2.1.3

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

$f (- 1) = 1 - 3 \cdot 1 - 9 \cdot - 1 + 4$

Step 3.2.1.4

Multiply $- 3$ by $1$ .

$f (- 1) = 1 - 3 - 9 \cdot - 1 + 4$

Step 3.2.1.5

Multiply $- 9$ by $- 1$ .

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

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $3$ from $1$ .

$f (- 1) = - 2 + 9 + 4$

Step 3.2.2.2

Add $- 2$ and $9$ .

$f (- 1) = 7 + 4$

Step 3.2.2.3

Add $7$ and $4$ .

$f (- 1) = 11$

Step 3.2.3

The final answer is $11$ .

$11$

Step 3.3

Convert $11$ to decimal.

$y = 11$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = - {(0)}^{3} - 3 {(0)}^{2} - 9 \cdot 0 + 4$

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

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

$f (0) = - 0 - 3 {(0)}^{2} - 9 \cdot 0 + 4$

Step 4.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 - 3 {(0)}^{2} - 9 \cdot 0 + 4$

Step 4.2.1.3

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

$f (0) = 0 - 3 \cdot 0 - 9 \cdot 0 + 4$

Step 4.2.1.4

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - 9 \cdot 0 + 4$

Step 4.2.1.5

Multiply $- 9$ by $0$ .

$f (0) = 0 + 0 + 0 + 4$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 4$

Step 4.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 4$

Step 4.2.2.3

Add $0$ and $4$ .

$f (0) = 4$

Step 4.2.3

The final answer is $4$ .

$4$

Step 4.3

Convert $4$ to decimal.

$y = 4$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

$f (1) = - {(1)}^{3} - 3 {(1)}^{2} - 9 \cdot 1 + 4$

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

One to any power is one.

$f (1) = - 1 \cdot 1 - 3 {(1)}^{2} - 9 \cdot 1 + 4$

Step 5.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 - 3 {(1)}^{2} - 9 \cdot 1 + 4$

Step 5.2.1.3

One to any power is one.

$f (1) = - 1 - 3 \cdot 1 - 9 \cdot 1 + 4$

Step 5.2.1.4

Multiply $- 3$ by $1$ .

$f (1) = - 1 - 3 - 9 \cdot 1 + 4$

Step 5.2.1.5

Multiply $- 9$ by $1$ .

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

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $3$ from $- 1$ .

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

Step 5.2.2.2

Subtract $9$ from $- 4$ .

$f (1) = - 13 + 4$

Step 5.2.2.3

Add $- 13$ and $4$ .

$f (1) = - 9$

Step 5.2.3

The final answer is $- 9$ .

$- 9$

Step 5.3

Convert $- 9$ to decimal.

$y = - 9$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 31 \\ - 2 & 18 \\ - 1 & 11 \\ 0 & 4 \\ 1 & - 9 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 31 \\ - 2 & 18 \\ - 1 & 11 \\ 0 & 4 \\ 1 & - 9 \end{array}$

Step 8