Pre-Algebra Examples

$4 x^{3} - 42 x^{2} + 110 x$

Step 1

Find the point at $x = 2$ .

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Step 1.1

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

$f (2) = 4 {(2)}^{3} - 42 {(2)}^{2} + 110 (2)$

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 $3$ .

$f (2) = 4 \cdot 8 - 42 {(2)}^{2} + 110 (2)$

Step 1.2.1.2

Multiply $4$ by $8$ .

$f (2) = 32 - 42 {(2)}^{2} + 110 (2)$

Step 1.2.1.3

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

$f (2) = 32 - 42 \cdot 4 + 110 (2)$

Step 1.2.1.4

Multiply $- 42$ by $4$ .

$f (2) = 32 - 168 + 110 (2)$

Step 1.2.1.5

Multiply $110$ by $2$ .

$f (2) = 32 - 168 + 220$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $168$ from $32$ .

$f (2) = - 136 + 220$

Step 1.2.2.2

Add $- 136$ and $220$ .

$f (2) = 84$

Step 1.2.3

The final answer is $84$ .

$84$

Step 1.3

Convert $84$ to decimal.

$y = 84$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = 4 {(3)}^{3} - 42 {(3)}^{2} + 110 (3)$

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 $3$ to the power of $3$ .

$f (3) = 4 \cdot 27 - 42 {(3)}^{2} + 110 (3)$

Step 2.2.1.2

Multiply $4$ by $27$ .

$f (3) = 108 - 42 {(3)}^{2} + 110 (3)$

Step 2.2.1.3

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

$f (3) = 108 - 42 \cdot 9 + 110 (3)$

Step 2.2.1.4

Multiply $- 42$ by $9$ .

$f (3) = 108 - 378 + 110 (3)$

Step 2.2.1.5

Multiply $110$ by $3$ .

$f (3) = 108 - 378 + 330$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $378$ from $108$ .

$f (3) = - 270 + 330$

Step 2.2.2.2

Add $- 270$ and $330$ .

$f (3) = 60$

Step 2.2.3

The final answer is $60$ .

$60$

Step 2.3

Convert $60$ to decimal.

$y = 60$

Step 3

Find the point at $x = 4$ .

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Step 3.1

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

$f (4) = 4 {(4)}^{3} - 42 {(4)}^{2} + 110 (4)$

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 $4$ by ${(4)}^{3}$ by adding the exponents.

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Step 3.2.1.1.1

Multiply $4$ by ${(4)}^{3}$ .

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Step 3.2.1.1.1.1

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

$f (4) = 4 {(4)}^{3} - 42 {(4)}^{2} + 110 (4)$

Step 3.2.1.1.1.2

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

$f (4) = 4^{1 + 3} - 42 {(4)}^{2} + 110 (4)$

Step 3.2.1.1.2

Add $1$ and $3$ .

$f (4) = 4^{4} - 42 {(4)}^{2} + 110 (4)$

Step 3.2.1.2

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

$f (4) = 256 - 42 {(4)}^{2} + 110 (4)$

Step 3.2.1.3

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

$f (4) = 256 - 42 \cdot 16 + 110 (4)$

Step 3.2.1.4

Multiply $- 42$ by $16$ .

$f (4) = 256 - 672 + 110 (4)$

Step 3.2.1.5

Multiply $110$ by $4$ .

$f (4) = 256 - 672 + 440$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $672$ from $256$ .

$f (4) = - 416 + 440$

Step 3.2.2.2

Add $- 416$ and $440$ .

$f (4) = 24$

Step 3.2.3

The final answer is $24$ .

$24$

Step 3.3

Convert $24$ to decimal.

$y = 24$

Step 4

Find the point at $x = 5$ .

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Step 4.1

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

$f (5) = 4 {(5)}^{3} - 42 {(5)}^{2} + 110 (5)$

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 $5$ to the power of $3$ .

$f (5) = 4 \cdot 125 - 42 {(5)}^{2} + 110 (5)$

Step 4.2.1.2

Multiply $4$ by $125$ .

$f (5) = 500 - 42 {(5)}^{2} + 110 (5)$

Step 4.2.1.3

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

$f (5) = 500 - 42 \cdot 25 + 110 (5)$

Step 4.2.1.4

Multiply $- 42$ by $25$ .

$f (5) = 500 - 1050 + 110 (5)$

Step 4.2.1.5

Multiply $110$ by $5$ .

$f (5) = 500 - 1050 + 550$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $1050$ from $500$ .

$f (5) = - 550 + 550$

Step 4.2.2.2

Add $- 550$ and $550$ .

$f (5) = 0$

Step 4.2.3

The final answer is $0$ .

$0$

Step 4.3

Convert $0$ to decimal.

$y = 0$

Step 5

Find the point at $x = 6$ .

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Step 5.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = 4 {(6)}^{3} - 42 {(6)}^{2} + 110 (6)$

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 $6$ to the power of $3$ .

$f (6) = 4 \cdot 216 - 42 {(6)}^{2} + 110 (6)$

Step 5.2.1.2

Multiply $4$ by $216$ .

$f (6) = 864 - 42 {(6)}^{2} + 110 (6)$

Step 5.2.1.3

Raise $6$ to the power of $2$ .

$f (6) = 864 - 42 \cdot 36 + 110 (6)$

Step 5.2.1.4

Multiply $- 42$ by $36$ .

$f (6) = 864 - 1512 + 110 (6)$

Step 5.2.1.5

Multiply $110$ by $6$ .

$f (6) = 864 - 1512 + 660$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $1512$ from $864$ .

$f (6) = - 648 + 660$

Step 5.2.2.2

Add $- 648$ and $660$ .

$f (6) = 12$

Step 5.2.3

The final answer is $12$ .

$12$

Step 5.3

Convert $12$ to decimal.

$y = 12$

Step 6

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

$\begin{array}{c} x & y \\ 2 & 84 \\ 3 & 60 \\ 4 & 24 \\ 5 & 0 \\ 6 & 12 \end{array}$

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

$\begin{array}{c} x & y \\ 2 & 84 \\ 3 & 60 \\ 4 & 24 \\ 5 & 0 \\ 6 & 12 \end{array}$

Step 8