Trigonometry Examples

$f (x) = \frac{1}{3} \cdot {(x - 2)}^{3} - 7$

Step 1

Find the point at $x = 2$ .

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Step 1.1

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

$f (2) = \frac{{((2) - 2)}^{3} - 21}{3}$

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

Subtract $2$ from $2$ .

$f (2) = \frac{0^{3} - 21}{3}$

Step 1.2.1.2

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

$f (2) = \frac{0 - 21}{3}$

Step 1.2.1.3

Subtract $21$ from $0$ .

$f (2) = \frac{- 21}{3}$

Step 1.2.2

Divide $- 21$ by $3$ .

$f (2) = - 7$

Step 1.2.3

The final answer is $- 7$ .

$- 7$

Step 1.3

Convert $- 7$ to decimal.

$y = - 7$

Step 2

Find the point at $x = 5$ .

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Step 2.1

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

$f (5) = \frac{{((5) - 2)}^{3} - 21}{3}$

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

Subtract $2$ from $5$ .

$f (5) = \frac{3^{3} - 21}{3}$

Step 2.2.1.2

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

$f (5) = \frac{27 - 21}{3}$

Step 2.2.1.3

Subtract $21$ from $27$ .

$f (5) = \frac{6}{3}$

Step 2.2.2

Divide $6$ by $3$ .

$f (5) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 2.3

Convert $2$ to decimal.

$y = 2$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = \frac{{((0) - 2)}^{3} - 21}{3}$

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

Subtract $2$ from $0$ .

$f (0) = \frac{{(- 2)}^{3} - 21}{3}$

Step 3.2.1.2

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

$f (0) = \frac{- 8 - 21}{3}$

Step 3.2.1.3

Subtract $21$ from $- 8$ .

$f (0) = \frac{- 29}{3}$

Step 3.2.2

Move the negative in front of the fraction.

$f (0) = - \frac{29}{3}$

Step 3.2.3

The final answer is $- \frac{29}{3}$ .

$- \frac{29}{3}$

Step 3.3

Convert $- \frac{29}{3}$ to decimal.

$y = - 9.\overline{6}$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = \frac{{((1) - 2)}^{3} - 21}{3}$

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

Subtract $2$ from $1$ .

$f (1) = \frac{{(- 1)}^{3} - 21}{3}$

Step 4.2.1.2

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

$f (1) = \frac{- 1 - 21}{3}$

Step 4.2.1.3

Subtract $21$ from $- 1$ .

$f (1) = \frac{- 22}{3}$

Step 4.2.2

Move the negative in front of the fraction.

$f (1) = - \frac{22}{3}$

Step 4.2.3

The final answer is $- \frac{22}{3}$ .

$- \frac{22}{3}$

Step 4.3

Convert $- \frac{22}{3}$ to decimal.

$y = - 7.\overline{3}$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

$f (3) = \frac{{((3) - 2)}^{3} - 21}{3}$

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

Subtract $2$ from $3$ .

$f (3) = \frac{1^{3} - 21}{3}$

Step 5.2.1.2

One to any power is one.

$f (3) = \frac{1 - 21}{3}$

Step 5.2.1.3

Subtract $21$ from $1$ .

$f (3) = \frac{- 20}{3}$

Step 5.2.2

Move the negative in front of the fraction.

$f (3) = - \frac{20}{3}$

Step 5.2.3

The final answer is $- \frac{20}{3}$ .

$- \frac{20}{3}$

Step 5.3

Convert $- \frac{20}{3}$ to decimal.

$y = - 6.\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ 0 & - 9.667 \\ 1 & - 7.333 \\ 2 & - 7 \\ 3 & - 6.667 \\ 5 & 2 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & - 9.667 \\ 1 & - 7.333 \\ 2 & - 7 \\ 3 & - 6.667 \\ 5 & 2 \end{array}$

Step 8