Algebra Examples

$v (t) = \frac{1}{10} {(3 t - 8)}^{3} + 2$

Step 1

Find the point at $x = 6$ .

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Step 1.1

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

$f (6) = \frac{{(3 (6) - 8)}^{3} + 20}{10}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Multiply $3$ by $6$ .

$f (6) = \frac{{(18 - 8)}^{3} + 20}{10}$

Step 1.2.1.2

Subtract $8$ from $18$ .

$f (6) = \frac{10^{3} + 20}{10}$

Step 1.2.1.3

Raise $10$ to the power of $3$ .

$f (6) = \frac{1000 + 20}{10}$

Step 1.2.1.4

Add $1000$ and $20$ .

$f (6) = \frac{1020}{10}$

Step 1.2.2

Divide $1020$ by $10$ .

$f (6) = 102$

Step 1.2.3

The final answer is $102$ .

$102$

Step 1.3

Convert $102$ to decimal.

$y = 102$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \frac{{(3 (1) - 8)}^{3} + 20}{10}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $3$ by $1$ .

$f (1) = \frac{{(3 - 8)}^{3} + 20}{10}$

Step 2.2.1.2

Subtract $8$ from $3$ .

$f (1) = \frac{{(- 5)}^{3} + 20}{10}$

Step 2.2.1.3

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

$f (1) = \frac{- 125 + 20}{10}$

Step 2.2.1.4

Add $- 125$ and $20$ .

$f (1) = \frac{- 105}{10}$

Step 2.2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.2.1

Cancel the common factor of $- 105$ and $10$ .

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Step 2.2.2.1.1

Factor $5$ out of $- 105$ .

$f (1) = \frac{5 (- 21)}{10}$

Step 2.2.2.1.2

Cancel the common factors.

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Step 2.2.2.1.2.1

Factor $5$ out of $10$ .

$f (1) = \frac{5 \cdot - 21}{5 \cdot 2}$

Step 2.2.2.1.2.2

Cancel the common factor.

$f (1) = \frac{5 \cdot - 21}{5 \cdot 2}$

Step 2.2.2.1.2.3

Rewrite the expression.

$f (1) = \frac{- 21}{2}$

Step 2.2.2.2

Move the negative in front of the fraction.

$f (1) = - \frac{21}{2}$

Step 2.2.3

The final answer is $- \frac{21}{2}$ .

$- \frac{21}{2}$

Step 2.3

Convert $- \frac{21}{2}$ to decimal.

$y = - 10.5$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \frac{{(3 (2) - 8)}^{3} + 20}{10}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Multiply $3$ by $2$ .

$f (2) = \frac{{(6 - 8)}^{3} + 20}{10}$

Step 3.2.1.2

Subtract $8$ from $6$ .

$f (2) = \frac{{(- 2)}^{3} + 20}{10}$

Step 3.2.1.3

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

$f (2) = \frac{- 8 + 20}{10}$

Step 3.2.1.4

Add $- 8$ and $20$ .

$f (2) = \frac{12}{10}$

Step 3.2.2

Cancel the common factor of $12$ and $10$ .

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Step 3.2.2.1

Factor $2$ out of $12$ .

$f (2) = \frac{2 (6)}{10}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $2$ out of $10$ .

$f (2) = \frac{2 \cdot 6}{2 \cdot 5}$

Step 3.2.2.2.2

Cancel the common factor.

$f (2) = \frac{2 \cdot 6}{2 \cdot 5}$

Step 3.2.2.2.3

Rewrite the expression.

$f (2) = \frac{6}{5}$

Step 3.2.3

The final answer is $\frac{6}{5}$ .

$\frac{6}{5}$

Step 3.3

Convert $\frac{6}{5}$ to decimal.

$y = 1.2$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \frac{{(3 (3) - 8)}^{3} + 20}{10}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Multiply $3$ by $3$ .

$f (3) = \frac{{(9 - 8)}^{3} + 20}{10}$

Step 4.2.1.2

Subtract $8$ from $9$ .

$f (3) = \frac{1^{3} + 20}{10}$

Step 4.2.1.3

One to any power is one.

$f (3) = \frac{1 + 20}{10}$

Step 4.2.1.4

Add $1$ and $20$ .

$f (3) = \frac{21}{10}$

Step 4.2.2

The final answer is $\frac{21}{10}$ .

$\frac{21}{10}$

Step 4.3

Convert $\frac{21}{10}$ to decimal.

$y = 2.1$

Step 5

Find the point at $x = 4$ .

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Step 5.1

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

$f (4) = \frac{{(3 (4) - 8)}^{3} + 20}{10}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $3$ by $4$ .

$f (4) = \frac{{(12 - 8)}^{3} + 20}{10}$

Step 5.2.1.2

Subtract $8$ from $12$ .

$f (4) = \frac{4^{3} + 20}{10}$

Step 5.2.1.3

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

$f (4) = \frac{64 + 20}{10}$

Step 5.2.1.4

Add $64$ and $20$ .

$f (4) = \frac{84}{10}$

Step 5.2.2

Cancel the common factor of $84$ and $10$ .

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Step 5.2.2.1

Factor $2$ out of $84$ .

$f (4) = \frac{2 (42)}{10}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $2$ out of $10$ .

$f (4) = \frac{2 \cdot 42}{2 \cdot 5}$

Step 5.2.2.2.2

Cancel the common factor.

$f (4) = \frac{2 \cdot 42}{2 \cdot 5}$

Step 5.2.2.2.3

Rewrite the expression.

$f (4) = \frac{42}{5}$

Step 5.2.3

The final answer is $\frac{42}{5}$ .

$\frac{42}{5}$

Step 5.3

Convert $\frac{42}{5}$ to decimal.

$y = 8.4$

Step 6

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

$\begin{array}{c} x & y \\ 1 & - 10.5 \\ 2 & 1.2 \\ 3 & 2.1 \\ 4 & 8.4 \\ 6 & 102 \end{array}$

Step 7

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

Rises to the left and rises to the right

$\begin{array}{c} x & y \\ 1 & - 10.5 \\ 2 & 1.2 \\ 3 & 2.1 \\ 4 & 8.4 \\ 6 & 102 \end{array}$

Step 8