Trigonometry Examples

$g (x) = \frac{1}{2} x^{3} + 4$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = \frac{{(- 2)}^{3}}{2} + 4$

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

Cancel the common factor of ${(- 2)}^{3}$ and $2$ .

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Step 1.2.1.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$f (- 2) = \frac{{(- 1 \cdot 2)}^{3}}{2} + 4$

Step 1.2.1.1.2

Apply the product rule to $- 1 (2)$ .

$f (- 2) = \frac{{(- 1)}^{3} \cdot 2^{3}}{2} + 4$

Step 1.2.1.1.3

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

$f (- 2) = \frac{- 1 \cdot 2^{3}}{2} + 4$

Step 1.2.1.1.4

Factor $2$ out of $- 1 \cdot 2^{3}$ .

$f (- 2) = \frac{2 (- 1 \cdot 2^{2})}{2} + 4$

Step 1.2.1.1.5

Cancel the common factors.

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Step 1.2.1.1.5.1

Factor $2$ out of $2$ .

$f (- 2) = \frac{2 (- 1 \cdot 2^{2})}{2 (1)} + 4$

Step 1.2.1.1.5.2

Cancel the common factor.

$f (- 2) = \frac{2 (- 1 \cdot 2^{2})}{2 \cdot 1} + 4$

Step 1.2.1.1.5.3

Rewrite the expression.

$f (- 2) = \frac{- 1 \cdot 2^{2}}{1} + 4$

Step 1.2.1.1.5.4

Divide $- 1 \cdot 2^{2}$ by $1$ .

$f (- 2) = - 1 \cdot 2^{2} + 4$

Step 1.2.1.2

Rewrite $- 1 \cdot 2^{2}$ as $- 2^{2}$ .

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

Step 1.2.1.3

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

$f (- 2) = - 1 \cdot 4 + 4$

Step 1.2.1.4

Multiply $- 1$ by $4$ .

$f (- 2) = - 4 + 4$

Step 1.2.2

Add $- 4$ and $4$ .

$f (- 2) = 0$

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 $x = 0$ .

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Step 2.1

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

$f (0) = \frac{{(0)}^{3}}{2} + 4$

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

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

$f (0) = \frac{0}{2} + 4$

Step 2.2.1.2

Divide $0$ by $2$ .

$f (0) = 0 + 4$

Step 2.2.2

Add $0$ and $4$ .

$f (0) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 2.3

Convert $4$ to decimal.

$y = 4$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \frac{{(2)}^{3}}{2} + 4$

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

Cancel the common factor of ${(2)}^{3}$ and $2$ .

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Step 3.2.1.1.1

Factor $2$ out of ${(2)}^{3}$ .

$f (2) = \frac{2 \cdot 2^{2}}{2} + 4$

Step 3.2.1.1.2

Cancel the common factors.

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Step 3.2.1.1.2.1

Factor $2$ out of $2$ .

$f (2) = \frac{2 \cdot 2^{2}}{2 (1)} + 4$

Step 3.2.1.1.2.2

Cancel the common factor.

$f (2) = \frac{2 \cdot 2^{2}}{2 \cdot 1} + 4$

Step 3.2.1.1.2.3

Rewrite the expression.

$f (2) = \frac{2^{2}}{1} + 4$

Step 3.2.1.1.2.4

Divide $2^{2}$ by $1$ .

$f (2) = 2^{2} + 4$

Step 3.2.1.2

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

$f (2) = 4 + 4$

Step 3.2.2

Add $4$ and $4$ .

$f (2) = 8$

Step 3.2.3

The final answer is $8$ .

$8$

Step 3.3

Convert $8$ to decimal.

$y = 8$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

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

$f (- 1) = \frac{{(- 1)}^{3}}{2} + 4$

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 $3$ .

$f (- 1) = \frac{- 1}{2} + 4$

Step 4.2.1.2

Move the negative in front of the fraction.

$f (- 1) = - \frac{1}{2} + 4$

Step 4.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 1) = - \frac{1}{2} + 4 \cdot \frac{2}{2}$

Step 4.2.3

Combine $4$ and $\frac{2}{2}$ .

$f (- 1) = - \frac{1}{2} + \frac{4 \cdot 2}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 + 4 \cdot 2}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $4$ by $2$ .

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

Step 4.2.5.2

Add $- 1$ and $8$ .

$f (- 1) = \frac{7}{2}$

Step 4.2.6

The final answer is $\frac{7}{2}$ .

$\frac{7}{2}$

Step 4.3

Convert $\frac{7}{2}$ to decimal.

$y = 3.5$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

$f (1) = \frac{{(1)}^{3}}{2} + 4$

Step 5.2

Simplify the result.

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Step 5.2.1

One to any power is one.

$f (1) = \frac{1}{2} + 4$

Step 5.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (1) = \frac{1}{2} + 4 \cdot \frac{2}{2}$

Step 5.2.3

Combine $4$ and $\frac{2}{2}$ .

$f (1) = \frac{1}{2} + \frac{4 \cdot 2}{2}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{1 + 4 \cdot 2}{2}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $4$ by $2$ .

$f (1) = \frac{1 + 8}{2}$

Step 5.2.5.2

Add $1$ and $8$ .

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

Step 5.2.6

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

Step 5.3

Convert $\frac{9}{2}$ to decimal.

$y = 4.5$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 3.5 \\ 0 & 4 \\ 1 & 4.5 \\ 2 & 8 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 3.5 \\ 0 & 4 \\ 1 & 4.5 \\ 2 & 8 \end{array}$

Step 8