Trigonometry Examples

$y = x - \frac{1}{60} x^{3}$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = (0) - \frac{{(0)}^{3}}{60}$

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

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

$f (0) = 0 - \frac{0}{60}$

Step 1.2.1.2

Divide $0$ by $60$ .

$f (0) = 0 - 0$

Step 1.2.1.3

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0$

Step 1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

$f (- 2) = (- 2) - \frac{{(- 2)}^{3}}{60}$

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 $3$ .

$f (- 2) = - 2 - \frac{- 8}{60}$

Step 2.2.1.2

Cancel the common factor of $- 8$ and $60$ .

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Step 2.2.1.2.1

Factor $4$ out of $- 8$ .

$f (- 2) = - 2 - \frac{4 (- 2)}{60}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $4$ out of $60$ .

$f (- 2) = - 2 - \frac{4 \cdot - 2}{4 \cdot 15}$

Step 2.2.1.2.2.2

Cancel the common factor.

$f (- 2) = - 2 - \frac{4 \cdot - 2}{4 \cdot 15}$

Step 2.2.1.2.2.3

Rewrite the expression.

$f (- 2) = - 2 - \frac{- 2}{15}$

Step 2.2.1.3

Move the negative in front of the fraction.

$f (- 2) = - 2 + \frac{2}{15}$

Step 2.2.1.4

Multiply $- - \frac{2}{15}$ .

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Step 2.2.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- 2) = - 2 + 1 (\frac{2}{15})$

Step 2.2.1.4.2

Multiply $\frac{2}{15}$ by $1$ .

$f (- 2) = - 2 + \frac{2}{15}$

Step 2.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$f (- 2) = - 2 \cdot \frac{15}{15} + \frac{2}{15}$

Step 2.2.3

Combine $- 2$ and $\frac{15}{15}$ .

$f (- 2) = \frac{- 2 \cdot 15}{15} + \frac{2}{15}$

Step 2.2.4

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 2 \cdot 15 + 2}{15}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 2$ by $15$ .

$f (- 2) = \frac{- 30 + 2}{15}$

Step 2.2.5.2

Add $- 30$ and $2$ .

$f (- 2) = \frac{- 28}{15}$

Step 2.2.6

Move the negative in front of the fraction.

$f (- 2) = - \frac{28}{15}$

Step 2.2.7

The final answer is $- \frac{28}{15}$ .

$- \frac{28}{15}$

Step 2.3

Convert $- \frac{28}{15}$ to decimal.

$y = - 1.8\overline{6}$

Step 3

Find the point at $x = - 1$ .

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Step 3.1

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

$f (- 1) = (- 1) - \frac{{(- 1)}^{3}}{60}$

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

$f (- 1) = - 1 - \frac{- 1}{60}$

Step 3.2.1.2

Move the negative in front of the fraction.

$f (- 1) = - 1 + \frac{1}{60}$

Step 3.2.1.3

Multiply $- - \frac{1}{60}$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = - 1 + 1 (\frac{1}{60})$

Step 3.2.1.3.2

Multiply $\frac{1}{60}$ by $1$ .

$f (- 1) = - 1 + \frac{1}{60}$

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{60}{60}$ .

$f (- 1) = - 1 \cdot \frac{60}{60} + \frac{1}{60}$

Step 3.2.3

Combine $- 1$ and $\frac{60}{60}$ .

$f (- 1) = \frac{- 1 \cdot 60}{60} + \frac{1}{60}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 \cdot 60 + 1}{60}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 1$ by $60$ .

$f (- 1) = \frac{- 60 + 1}{60}$

Step 3.2.5.2

Add $- 60$ and $1$ .

$f (- 1) = \frac{- 59}{60}$

Step 3.2.6

Move the negative in front of the fraction.

$f (- 1) = - \frac{59}{60}$

Step 3.2.7

The final answer is $- \frac{59}{60}$ .

$- \frac{59}{60}$

Step 3.3

Convert $- \frac{59}{60}$ to decimal.

$y = - 0.98\overline{3}$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = (1) - \frac{{(1)}^{3}}{60}$

Step 4.2

Simplify the result.

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Step 4.2.1

One to any power is one.

$f (1) = 1 - \frac{1}{60}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Write $1$ as a fraction with a common denominator.

$f (1) = \frac{60}{60} - \frac{1}{60}$

Step 4.2.2.2

Combine the numerators over the common denominator.

$f (1) = \frac{60 - 1}{60}$

Step 4.2.2.3

Subtract $1$ from $60$ .

$f (1) = \frac{59}{60}$

Step 4.2.3

The final answer is $\frac{59}{60}$ .

$\frac{59}{60}$

Step 4.3

Convert $\frac{59}{60}$ to decimal.

$y = 0.98\overline{3}$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = (2) - \frac{{(2)}^{3}}{60}$

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 $3$ .

$f (2) = 2 - \frac{8}{60}$

Step 5.2.1.2

Cancel the common factor of $8$ and $60$ .

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Step 5.2.1.2.1

Factor $4$ out of $8$ .

$f (2) = 2 - \frac{4 (2)}{60}$

Step 5.2.1.2.2

Cancel the common factors.

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Step 5.2.1.2.2.1

Factor $4$ out of $60$ .

$f (2) = 2 - \frac{4 \cdot 2}{4 \cdot 15}$

Step 5.2.1.2.2.2

Cancel the common factor.

$f (2) = 2 - \frac{4 \cdot 2}{4 \cdot 15}$

Step 5.2.1.2.2.3

Rewrite the expression.

$f (2) = 2 - \frac{2}{15}$

Step 5.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$f (2) = 2 \cdot \frac{15}{15} - \frac{2}{15}$

Step 5.2.3

Combine $2$ and $\frac{15}{15}$ .

$f (2) = \frac{2 \cdot 15}{15} - \frac{2}{15}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{2 \cdot 15 - 2}{15}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $2$ by $15$ .

$f (2) = \frac{30 - 2}{15}$

Step 5.2.5.2

Subtract $2$ from $30$ .

$f (2) = \frac{28}{15}$

Step 5.2.6

The final answer is $\frac{28}{15}$ .

$\frac{28}{15}$

Step 5.3

Convert $\frac{28}{15}$ to decimal.

$y = 1.8\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & - 1.867 \\ - 1 & - 0.983 \\ 0 & 0 \\ 1 & 0.983 \\ 2 & 1.867 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & - 1.867 \\ - 1 & - 0.983 \\ 0 & 0 \\ 1 & 0.983 \\ 2 & 1.867 \end{array}$

Step 8