Trigonometry Examples

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

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

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

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

$f (- 2) = \frac{- 8}{4} + {(- 2)}^{2} - 1$

Step 1.2.1.2

Divide $- 8$ by $4$ .

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

Step 1.2.1.3

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

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

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $- 2$ and $4$ .

$f (- 2) = 2 - 1$

Step 1.2.2.2

Subtract $1$ from $2$ .

$f (- 2) = 1$

Step 1.2.3

The final answer is $1$ .

$1$

Step 1.3

Convert $1$ to decimal.

$y = 1$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = \frac{{(0)}^{3}}{4} + {(0)}^{2} - 1$

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 $0$ .

$f (0) = \frac{0}{4} + {(0)}^{2} - 1$

Step 2.2.1.2

Divide $0$ by $4$ .

$f (0) = 0 + {(0)}^{2} - 1$

Step 2.2.1.3

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

$f (0) = 0 + 0 - 1$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 - 1$

Step 2.2.2.2

Subtract $1$ from $0$ .

$f (0) = - 1$

Step 2.2.3

The final answer is $- 1$ .

$- 1$

Step 2.3

Convert $- 1$ to decimal.

$y = - 1$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

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 $3$ .

$f (2) = \frac{8}{4} + {(2)}^{2} - 1$

Step 3.2.1.2

Divide $8$ by $4$ .

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

Step 3.2.1.3

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

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

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $2$ and $4$ .

$f (2) = 6 - 1$

Step 3.2.2.2

Subtract $1$ from $6$ .

$f (2) = 5$

Step 3.2.3

The final answer is $5$ .

$5$

Step 3.3

Convert $5$ to decimal.

$y = 5$

Step 4

Find the point at $x = - 3$ .

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Step 4.1

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

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

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

$f (- 3) = \frac{- 27}{4} + {(- 3)}^{2} - 1$

Step 4.2.1.2

Move the negative in front of the fraction.

$f (- 3) = - \frac{27}{4} + {(- 3)}^{2} - 1$

Step 4.2.1.3

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

$f (- 3) = - \frac{27}{4} + 9 - 1$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $9$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{27}{4} + \frac{9}{1} - 1$

Step 4.2.2.2

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (- 3) = - \frac{27}{4} + \frac{9}{1} \cdot \frac{4}{4} - 1$

Step 4.2.2.3

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (- 3) = - \frac{27}{4} + \frac{9 \cdot 4}{4} - 1$

Step 4.2.2.4

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

$f (- 3) = - \frac{27}{4} + \frac{9 \cdot 4}{4} + \frac{- 1}{1}$

Step 4.2.2.5

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

$f (- 3) = - \frac{27}{4} + \frac{9 \cdot 4}{4} + \frac{- 1}{1} \cdot \frac{4}{4}$

Step 4.2.2.6

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

$f (- 3) = - \frac{27}{4} + \frac{9 \cdot 4}{4} + \frac{- 1 \cdot 4}{4}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 27 + 9 \cdot 4 - 1 \cdot 4}{4}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $9$ by $4$ .

$f (- 3) = \frac{- 27 + 36 - 1 \cdot 4}{4}$

Step 4.2.4.2

Multiply $- 1$ by $4$ .

$f (- 3) = \frac{- 27 + 36 - 4}{4}$

Step 4.2.5

Simplify by adding and subtracting.

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Step 4.2.5.1

Add $- 27$ and $36$ .

$f (- 3) = \frac{9 - 4}{4}$

Step 4.2.5.2

Subtract $4$ from $9$ .

$f (- 3) = \frac{5}{4}$

Step 4.2.6

The final answer is $\frac{5}{4}$ .

$\frac{5}{4}$

Step 4.3

Convert $\frac{5}{4}$ to decimal.

$y = 1.25$

Step 5

Find the point at $x = - 1$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Find the common denominator.

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Step 5.2.1.1

Write ${(- 1)}^{2}$ as a fraction with denominator $1$ .

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

Step 5.2.1.2

Multiply $\frac{{(- 1)}^{2}}{1}$ by $\frac{4}{4}$ .

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

Step 5.2.1.3

Multiply $\frac{{(- 1)}^{2}}{1}$ by $\frac{4}{4}$ .

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

Step 5.2.1.4

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

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

Step 5.2.1.5

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

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

Step 5.2.1.6

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

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

Step 5.2.2

Combine the numerators over the common denominator.

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

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

Step 5.2.3.2

Raise $- 1$ to the power of $2$ .

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

Step 5.2.3.3

Multiply $4$ by $1$ .

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

Step 5.2.3.4

Multiply $- 1$ by $4$ .

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

Step 5.2.4

Simplify the expression.

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Step 5.2.4.1

Add $- 1$ and $4$ .

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

Step 5.2.4.2

Subtract $4$ from $3$ .

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

Step 5.2.4.3

Move the negative in front of the fraction.

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

Step 5.2.5

The final answer is $- \frac{1}{4}$ .

$- \frac{1}{4}$

Step 5.3

Convert $- \frac{1}{4}$ to decimal.

$y = - 0.25$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 1.25 \\ - 2 & 1 \\ - 1 & - 0.25 \\ 0 & - 1 \\ 2 & 5 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 1.25 \\ - 2 & 1 \\ - 1 & - 0.25 \\ 0 & - 1 \\ 2 & 5 \end{array}$

Step 8