Trigonometry Examples

$f (x) = \frac{π}{3} x^{3} - 5 π x^{2} + \frac{500 π d}{3}$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = \frac{π {(- 2)}^{3}}{3} - 5 π {(- 2)}^{2} + \frac{500 π d}{3}$

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

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

$f (- 2) = \frac{π \cdot - 8}{3} - 5 π {(- 2)}^{2} + \frac{500 π d}{3}$

Step 1.2.1.2

Move $- 8$ to the left of $π$ .

$f (- 2) = \frac{- 8 \cdot π}{3} - 5 π {(- 2)}^{2} + \frac{500 π d}{3}$

Step 1.2.1.3

Move the negative in front of the fraction.

$f (- 2) = - \frac{8 \cdot π}{3} - 5 π {(- 2)}^{2} + \frac{500 π d}{3}$

Step 1.2.1.4

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

$f (- 2) = - \frac{8 π}{3} - 5 π \cdot 4 + \frac{500 π d}{3}$

Step 1.2.1.5

Multiply $4$ by $- 5$ .

$f (- 2) = - \frac{8 π}{3} - 20 π + \frac{500 π d}{3}$

Step 1.2.2

To write $- 20 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 2) = - \frac{8 π}{3} - 20 π \cdot \frac{3}{3} + \frac{500 π d}{3}$

Step 1.2.3

Combine $- 20 π$ and $\frac{3}{3}$ .

$f (- 2) = - \frac{8 π}{3} + \frac{- 20 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 1.2.4

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 8 π - 20 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 1.2.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 8 π - 20 π \cdot 3 + 500 π d}{3}$

Step 1.2.6

Multiply $3$ by $- 20$ .

$f (- 2) = \frac{- 8 π - 60 π + 500 π d}{3}$

Step 1.2.7

Subtract $60 π$ from $- 8 π$ .

$f (- 2) = \frac{- 68 π + 500 π d}{3}$

Step 1.2.8

Factor $- 1$ out of $- 68 π$ .

$f (- 2) = \frac{- (68 π) + 500 π d}{3}$

Step 1.2.9

Factor $- 1$ out of $500 π d$ .

$f (- 2) = \frac{- (68 π) - (- 500 π d)}{3}$

Step 1.2.10

Factor $- 1$ out of $- (68 π) - (- 500 π d)$ .

$f (- 2) = \frac{- (68 π - 500 π d)}{3}$

Step 1.2.11

Simplify the expression.

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Step 1.2.11.1

Rewrite $- (68 π - 500 π d)$ as $- 1 (68 π - 500 π d)$ .

$f (- 2) = \frac{- 1 (68 π - 500 π d)}{3}$

Step 1.2.11.2

Move the negative in front of the fraction.

$f (- 2) = - \frac{68 π - 500 π d}{3}$

Step 1.2.12

The final answer is $- \frac{68 π - 500 π d}{3}$ .

$- \frac{68 π - 500 π d}{3}$

Step 1.3

Convert $- \frac{68 π - 500 π d}{3}$ to decimal.

$y = - \frac{68 π - 500 π d}{3}$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = \frac{π {(- 1)}^{3}}{3} - 5 π {(- 1)}^{2} + \frac{500 π d}{3}$

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

$f (- 1) = \frac{π \cdot - 1}{3} - 5 π {(- 1)}^{2} + \frac{500 π d}{3}$

Step 2.2.1.2

Move $- 1$ to the left of $π$ .

$f (- 1) = \frac{- 1 \cdot π}{3} - 5 π {(- 1)}^{2} + \frac{500 π d}{3}$

Step 2.2.1.3

Move the negative in front of the fraction.

$f (- 1) = - \frac{π}{3} - 5 π {(- 1)}^{2} + \frac{500 π d}{3}$

Step 2.2.1.4

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

$f (- 1) = - \frac{π}{3} - 5 π \cdot 1 + \frac{500 π d}{3}$

Step 2.2.1.5

Multiply $- 5$ by $1$ .

$f (- 1) = - \frac{π}{3} - 5 π + \frac{500 π d}{3}$

Step 2.2.2

To write $- 5 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 1) = - \frac{π}{3} - 5 π \cdot \frac{3}{3} + \frac{500 π d}{3}$

Step 2.2.3

Combine $- 5 π$ and $\frac{3}{3}$ .

$f (- 1) = - \frac{π}{3} + \frac{- 5 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 2.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{- π - 5 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 2.2.5

Combine the numerators over the common denominator.

$f (- 1) = \frac{- π - 5 π \cdot 3 + 500 π d}{3}$

Step 2.2.6

Multiply $3$ by $- 5$ .

$f (- 1) = \frac{- π - 15 π + 500 π d}{3}$

Step 2.2.7

Subtract $15 π$ from $- π$ .

$f (- 1) = \frac{- 16 π + 500 π d}{3}$

Step 2.2.8

Factor $- 1$ out of $- 16 π$ .

$f (- 1) = \frac{- (16 π) + 500 π d}{3}$

Step 2.2.9

Factor $- 1$ out of $500 π d$ .

$f (- 1) = \frac{- (16 π) - (- 500 π d)}{3}$

Step 2.2.10

Factor $- 1$ out of $- (16 π) - (- 500 π d)$ .

$f (- 1) = \frac{- (16 π - 500 π d)}{3}$

Step 2.2.11

Simplify the expression.

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Step 2.2.11.1

Rewrite $- (16 π - 500 π d)$ as $- 1 (16 π - 500 π d)$ .

$f (- 1) = \frac{- 1 (16 π - 500 π d)}{3}$

Step 2.2.11.2

Move the negative in front of the fraction.

$f (- 1) = - \frac{16 π - 500 π d}{3}$

Step 2.2.12

The final answer is $- \frac{16 π - 500 π d}{3}$ .

$- \frac{16 π - 500 π d}{3}$

Step 2.3

Convert $- \frac{16 π - 500 π d}{3}$ to decimal.

$y = - \frac{16 π - 500 π d}{3}$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = \frac{π {(0)}^{3}}{3} - 5 π {(0)}^{2} + \frac{500 π d}{3}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (0) = - 5 π {(0)}^{2} + \frac{π {(0)}^{3} + 500 π d}{3}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (0) = - 5 π {(0)}^{2} + \frac{π \cdot 0 + 500 π d}{3}$

Step 3.2.2.2

Multiply $π$ by $0$ .

$f (0) = - 5 π {(0)}^{2} + \frac{0 + 500 π d}{3}$

Step 3.2.3

Add $0$ and $500 π d$ .

$f (0) = - 5 π {(0)}^{2} + \frac{500 π d}{3}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

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

$f (0) = - 5 π \cdot 0 + \frac{500 π d}{3}$

Step 3.2.4.2

Multiply $- 5 π \cdot 0$ .

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Step 3.2.4.2.1

Multiply $0$ by $- 5$ .

$f (0) = 0 π + \frac{500 π d}{3}$

Step 3.2.4.2.2

Multiply $0$ by $π$ .

$f (0) = 0 + \frac{500 π d}{3}$

Step 3.2.5

Add $0$ and $\frac{500 π d}{3}$ .

$f (0) = \frac{500 π d}{3}$

Step 3.2.6

The final answer is $\frac{500 π d}{3}$ .

$\frac{500 π d}{3}$

Step 3.3

Convert $\frac{500 π d}{3}$ to decimal.

$y = \frac{500 π d}{3}$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = \frac{π {(1)}^{3}}{3} - 5 π {(1)}^{2} + \frac{500 π d}{3}$

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

One to any power is one.

$f (1) = \frac{π \cdot 1}{3} - 5 π {(1)}^{2} + \frac{500 π d}{3}$

Step 4.2.1.2

Multiply $π$ by $1$ .

$f (1) = \frac{π}{3} - 5 π {(1)}^{2} + \frac{500 π d}{3}$

Step 4.2.1.3

One to any power is one.

$f (1) = \frac{π}{3} - 5 π \cdot 1 + \frac{500 π d}{3}$

Step 4.2.1.4

Multiply $- 5$ by $1$ .

$f (1) = \frac{π}{3} - 5 π + \frac{500 π d}{3}$

Step 4.2.2

To write $- 5 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (1) = \frac{π}{3} - 5 π \cdot \frac{3}{3} + \frac{500 π d}{3}$

Step 4.2.3

Combine $- 5 π$ and $\frac{3}{3}$ .

$f (1) = \frac{π}{3} + \frac{- 5 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{π - 5 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 4.2.5

Combine the numerators over the common denominator.

$f (1) = \frac{π - 5 π \cdot 3 + 500 π d}{3}$

Step 4.2.6

Multiply $3$ by $- 5$ .

$f (1) = \frac{π - 15 π + 500 π d}{3}$

Step 4.2.7

Subtract $15 π$ from $π$ .

$f (1) = \frac{- 14 π + 500 π d}{3}$

Step 4.2.8

Factor $- 1$ out of $- 14 π$ .

$f (1) = \frac{- (14 π) + 500 π d}{3}$

Step 4.2.9

Factor $- 1$ out of $500 π d$ .

$f (1) = \frac{- (14 π) - (- 500 π d)}{3}$

Step 4.2.10

Factor $- 1$ out of $- (14 π) - (- 500 π d)$ .

$f (1) = \frac{- (14 π - 500 π d)}{3}$

Step 4.2.11

Simplify the expression.

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Step 4.2.11.1

Rewrite $- (14 π - 500 π d)$ as $- 1 (14 π - 500 π d)$ .

$f (1) = \frac{- 1 (14 π - 500 π d)}{3}$

Step 4.2.11.2

Move the negative in front of the fraction.

$f (1) = - \frac{14 π - 500 π d}{3}$

Step 4.2.12

The final answer is $- \frac{14 π - 500 π d}{3}$ .

$- \frac{14 π - 500 π d}{3}$

Step 4.3

Convert $- \frac{14 π - 500 π d}{3}$ to decimal.

$y = - \frac{14 π - 500 π d}{3}$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = \frac{π {(2)}^{3}}{3} - 5 π {(2)}^{2} + \frac{500 π d}{3}$

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 $3$ .

$f (2) = \frac{π \cdot 8}{3} - 5 π {(2)}^{2} + \frac{500 π d}{3}$

Step 5.2.1.2

Move $8$ to the left of $π$ .

$f (2) = \frac{8 \cdot π}{3} - 5 π {(2)}^{2} + \frac{500 π d}{3}$

Step 5.2.1.3

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

$f (2) = \frac{8 π}{3} - 5 π \cdot 4 + \frac{500 π d}{3}$

Step 5.2.1.4

Multiply $4$ by $- 5$ .

$f (2) = \frac{8 π}{3} - 20 π + \frac{500 π d}{3}$

Step 5.2.2

To write $- 20 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (2) = \frac{8 π}{3} - 20 π \cdot \frac{3}{3} + \frac{500 π d}{3}$

Step 5.2.3

Combine $- 20 π$ and $\frac{3}{3}$ .

$f (2) = \frac{8 π}{3} + \frac{- 20 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{8 π - 20 π \cdot 3}{3} + \frac{500 π d}{3}$

Step 5.2.5

Combine the numerators over the common denominator.

$f (2) = \frac{8 π - 20 π \cdot 3 + 500 π d}{3}$

Step 5.2.6

Multiply $3$ by $- 20$ .

$f (2) = \frac{8 π - 60 π + 500 π d}{3}$

Step 5.2.7

Subtract $60 π$ from $8 π$ .

$f (2) = \frac{- 52 π + 500 π d}{3}$

Step 5.2.8

Factor $- 1$ out of $- 52 π$ .

$f (2) = \frac{- (52 π) + 500 π d}{3}$

Step 5.2.9

Factor $- 1$ out of $500 π d$ .

$f (2) = \frac{- (52 π) - (- 500 π d)}{3}$

Step 5.2.10

Factor $- 1$ out of $- (52 π) - (- 500 π d)$ .

$f (2) = \frac{- (52 π - 500 π d)}{3}$

Step 5.2.11

Simplify the expression.

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Step 5.2.11.1

Rewrite $- (52 π - 500 π d)$ as $- 1 (52 π - 500 π d)$ .

$f (2) = \frac{- 1 (52 π - 500 π d)}{3}$

Step 5.2.11.2

Move the negative in front of the fraction.

$f (2) = - \frac{52 π - 500 π d}{3}$

Step 5.2.12

The final answer is $- \frac{52 π - 500 π d}{3}$ .

$- \frac{52 π - 500 π d}{3}$

Step 5.3

Convert $- \frac{52 π - 500 π d}{3}$ to decimal.

$y = - \frac{52 π - 500 π d}{3}$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & - \frac{68 π - 500 π d}{3} \\ - 1 & - \frac{16 π - 500 π d}{3} \\ 0 & \frac{500 π d}{3} \\ 1 & - \frac{14 π - 500 π d}{3} \\ 2 & - \frac{52 π - 500 π d}{3} \end{array}$

Step 7

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

The leading coefficient test requires one variable only

Step 8