Pre-Algebra Examples

$f (x) = x^{3} + 10 x^{2} + 33 x + 36$

Step 1

Find the point at $x = - 5$ .

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Step 1.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = {(- 5)}^{3} + 10 {(- 5)}^{2} + 33 (- 5) + 36$

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 $- 5$ to the power of $3$ .

$f (- 5) = - 125 + 10 {(- 5)}^{2} + 33 (- 5) + 36$

Step 1.2.1.2

Raise $- 5$ to the power of $2$ .

$f (- 5) = - 125 + 10 \cdot 25 + 33 (- 5) + 36$

Step 1.2.1.3

Multiply $10$ by $25$ .

$f (- 5) = - 125 + 250 + 33 (- 5) + 36$

Step 1.2.1.4

Multiply $33$ by $- 5$ .

$f (- 5) = - 125 + 250 - 165 + 36$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $- 125$ and $250$ .

$f (- 5) = 125 - 165 + 36$

Step 1.2.2.2

Subtract $165$ from $125$ .

$f (- 5) = - 40 + 36$

Step 1.2.2.3

Add $- 40$ and $36$ .

$f (- 5) = - 4$

Step 1.2.3

The final answer is $- 4$ .

$- 4$

Step 1.3

Convert $- 4$ to decimal.

$y = - 4$

Step 2

Find the point at $x = - 4$ .

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Step 2.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = {(- 4)}^{3} + 10 {(- 4)}^{2} + 33 (- 4) + 36$

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 $- 4$ to the power of $3$ .

$f (- 4) = - 64 + 10 {(- 4)}^{2} + 33 (- 4) + 36$

Step 2.2.1.2

Raise $- 4$ to the power of $2$ .

$f (- 4) = - 64 + 10 \cdot 16 + 33 (- 4) + 36$

Step 2.2.1.3

Multiply $10$ by $16$ .

$f (- 4) = - 64 + 160 + 33 (- 4) + 36$

Step 2.2.1.4

Multiply $33$ by $- 4$ .

$f (- 4) = - 64 + 160 - 132 + 36$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 64$ and $160$ .

$f (- 4) = 96 - 132 + 36$

Step 2.2.2.2

Subtract $132$ from $96$ .

$f (- 4) = - 36 + 36$

Step 2.2.2.3

Add $- 36$ and $36$ .

$f (- 4) = 0$

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 $x = - 2$ .

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Step 3.1

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

$f (- 2) = {(- 2)}^{3} + 10 {(- 2)}^{2} + 33 (- 2) + 36$

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 $3$ .

$f (- 2) = - 8 + 10 {(- 2)}^{2} + 33 (- 2) + 36$

Step 3.2.1.2

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

$f (- 2) = - 8 + 10 \cdot 4 + 33 (- 2) + 36$

Step 3.2.1.3

Multiply $10$ by $4$ .

$f (- 2) = - 8 + 40 + 33 (- 2) + 36$

Step 3.2.1.4

Multiply $33$ by $- 2$ .

$f (- 2) = - 8 + 40 - 66 + 36$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 8$ and $40$ .

$f (- 2) = 32 - 66 + 36$

Step 3.2.2.2

Subtract $66$ from $32$ .

$f (- 2) = - 34 + 36$

Step 3.2.2.3

Add $- 34$ and $36$ .

$f (- 2) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 3.3

Convert $2$ to decimal.

$y = 2$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

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

$f (- 1) = {(- 1)}^{3} + 10 {(- 1)}^{2} + 33 (- 1) + 36$

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

$f (- 1) = - 1 + 10 {(- 1)}^{2} + 33 (- 1) + 36$

Step 4.2.1.2

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

$f (- 1) = - 1 + 10 \cdot 1 + 33 (- 1) + 36$

Step 4.2.1.3

Multiply $10$ by $1$ .

$f (- 1) = - 1 + 10 + 33 (- 1) + 36$

Step 4.2.1.4

Multiply $33$ by $- 1$ .

$f (- 1) = - 1 + 10 - 33 + 36$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $- 1$ and $10$ .

$f (- 1) = 9 - 33 + 36$

Step 4.2.2.2

Subtract $33$ from $9$ .

$f (- 1) = - 24 + 36$

Step 4.2.2.3

Add $- 24$ and $36$ .

$f (- 1) = 12$

Step 4.2.3

The final answer is $12$ .

$12$

Step 4.3

Convert $12$ to decimal.

$y = 12$

Step 5

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

$\begin{array}{c} x & y \\ - 5 & - 4 \\ - 4 & 0 \\ - 3 & 0 \\ - 2 & 2 \\ - 1 & 12 \end{array}$

Step 6

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 5 & - 4 \\ - 4 & 0 \\ - 3 & 0 \\ - 2 & 2 \\ - 1 & 12 \end{array}$

Step 7