Trigonometry Examples

$f (x) = - \frac{1}{9} x^{3} - 3$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = - \frac{{(0)}^{3}}{9} - 3$

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 $0$ .

$f (0) = - \frac{0}{9} - 3$

Step 1.2.1.2

Divide $0$ by $9$ .

$f (0) = - 0 - 3$

Step 1.2.1.3

Multiply $- 1$ by $0$ .

$f (0) = 0 - 3$

Step 1.2.2

Subtract $3$ from $0$ .

$f (0) = - 3$

Step 1.2.3

The final answer is $- 3$ .

$- 3$

Step 1.3

Convert $- 3$ to decimal.

$y = - 3$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = - \frac{{(3)}^{3}}{9} - 3$

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 $3$ .

$f (3) = - \frac{27}{9} - 3$

Step 2.2.1.2

Divide $27$ by $9$ .

$f (3) = - 1 \cdot 3 - 3$

Step 2.2.1.3

Multiply $- 1$ by $3$ .

$f (3) = - 3 - 3$

Step 2.2.2

Subtract $3$ from $- 3$ .

$f (3) = - 6$

Step 2.2.3

The final answer is $- 6$ .

$- 6$

Step 2.3

Convert $- 6$ to decimal.

$y = - 6$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

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

$f (- 2) = - \frac{{(- 2)}^{3}}{9} - 3$

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 $3$ .

$f (- 2) = - \frac{- 8}{9} - 3$

Step 3.2.1.2

Move the negative in front of the fraction.

$f (- 2) = \frac{8}{9} - 3$

Step 3.2.1.3

Multiply $- - \frac{8}{9}$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 2) = 1 (\frac{8}{9}) - 3$

Step 3.2.1.3.2

Multiply $\frac{8}{9}$ by $1$ .

$f (- 2) = \frac{8}{9} - 3$

Step 3.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- 2) = \frac{8}{9} - 3 \cdot \frac{9}{9}$

Step 3.2.3

Combine $- 3$ and $\frac{9}{9}$ .

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

Step 3.2.4

Combine the numerators over the common denominator.

$f (- 2) = \frac{8 - 3 \cdot 9}{9}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 3$ by $9$ .

$f (- 2) = \frac{8 - 27}{9}$

Step 3.2.5.2

Subtract $27$ from $8$ .

$f (- 2) = \frac{- 19}{9}$

Step 3.2.6

Move the negative in front of the fraction.

$f (- 2) = - \frac{19}{9}$

Step 3.2.7

The final answer is $- \frac{19}{9}$ .

$- \frac{19}{9}$

Step 3.3

Convert $- \frac{19}{9}$ to decimal.

$y = - 2.\overline{1}$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

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

$f (- 1) = - \frac{{(- 1)}^{3}}{9} - 3$

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 $3$ .

$f (- 1) = - \frac{- 1}{9} - 3$

Step 4.2.1.2

Move the negative in front of the fraction.

$f (- 1) = \frac{1}{9} - 3$

Step 4.2.1.3

Multiply $- - \frac{1}{9}$ .

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Step 4.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = 1 (\frac{1}{9}) - 3$

Step 4.2.1.3.2

Multiply $\frac{1}{9}$ by $1$ .

$f (- 1) = \frac{1}{9} - 3$

Step 4.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- 1) = \frac{1}{9} - 3 \cdot \frac{9}{9}$

Step 4.2.3

Combine $- 3$ and $\frac{9}{9}$ .

$f (- 1) = \frac{1}{9} + \frac{- 3 \cdot 9}{9}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{1 - 3 \cdot 9}{9}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 3$ by $9$ .

$f (- 1) = \frac{1 - 27}{9}$

Step 4.2.5.2

Subtract $27$ from $1$ .

$f (- 1) = \frac{- 26}{9}$

Step 4.2.6

Move the negative in front of the fraction.

$f (- 1) = - \frac{26}{9}$

Step 4.2.7

The final answer is $- \frac{26}{9}$ .

$- \frac{26}{9}$

Step 4.3

Convert $- \frac{26}{9}$ to decimal.

$y = - 2.\overline{8}$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

$f (1) = - \frac{{(1)}^{3}}{9} - 3$

Step 5.2

Simplify the result.

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Step 5.2.1

One to any power is one.

$f (1) = - \frac{1}{9} - 3$

Step 5.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (1) = - \frac{1}{9} - 3 \cdot \frac{9}{9}$

Step 5.2.3

Combine $- 3$ and $\frac{9}{9}$ .

$f (1) = - \frac{1}{9} + \frac{- 3 \cdot 9}{9}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{- 1 - 3 \cdot 9}{9}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 3$ by $9$ .

$f (1) = \frac{- 1 - 27}{9}$

Step 5.2.5.2

Subtract $27$ from $- 1$ .

$f (1) = \frac{- 28}{9}$

Step 5.2.6

Move the negative in front of the fraction.

$f (1) = - \frac{28}{9}$

Step 5.2.7

The final answer is $- \frac{28}{9}$ .

$- \frac{28}{9}$

Step 5.3

Convert $- \frac{28}{9}$ to decimal.

$y = - 3.\overline{1}$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & - 2.111 \\ - 1 & - 2.889 \\ 0 & - 3 \\ 1 & - 3.111 \\ 3 & - 6 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & - 2.111 \\ - 1 & - 2.889 \\ 0 & - 3 \\ 1 & - 3.111 \\ 3 & - 6 \end{array}$

Step 8