Trigonometry Examples

$f (x) = 2 x (x - 2) (x - 1)$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = 2 {(- 1)}^{3} - 6 {(- 1)}^{2} + 4 (- 1)$

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

$f (- 1) = 2 \cdot - 1 - 6 {(- 1)}^{2} + 4 (- 1)$

Step 1.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = - 2 - 6 {(- 1)}^{2} + 4 (- 1)$

Step 1.2.1.3

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

$f (- 1) = - 2 - 6 \cdot 1 + 4 (- 1)$

Step 1.2.1.4

Multiply $- 6$ by $1$ .

$f (- 1) = - 2 - 6 + 4 (- 1)$

Step 1.2.1.5

Multiply $4$ by $- 1$ .

$f (- 1) = - 2 - 6 - 4$

Step 1.2.2

Simplify by subtracting numbers.

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Step 1.2.2.1

Subtract $6$ from $- 2$ .

$f (- 1) = - 8 - 4$

Step 1.2.2.2

Subtract $4$ from $- 8$ .

$f (- 1) = - 12$

Step 1.2.3

The final answer is $- 12$ .

$- 12$

Step 1.3

Convert $- 12$ to decimal.

$y = - 12$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = 2 {(0)}^{3} - 6 {(0)}^{2} + 4 (0)$

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 $0$ .

$f (0) = 2 \cdot 0 - 6 {(0)}^{2} + 4 (0)$

Step 2.2.1.2

Multiply $2$ by $0$ .

$f (0) = 0 - 6 {(0)}^{2} + 4 (0)$

Step 2.2.1.3

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

$f (0) = 0 - 6 \cdot 0 + 4 (0)$

Step 2.2.1.4

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0 + 4 (0)$

Step 2.2.1.5

Multiply $4$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

$f (3) = 2 {(3)}^{3} - 6 {(3)}^{2} + 4 (3)$

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 $3$ to the power of $3$ .

$f (3) = 2 \cdot 27 - 6 {(3)}^{2} + 4 (3)$

Step 3.2.1.2

Multiply $2$ by $27$ .

$f (3) = 54 - 6 {(3)}^{2} + 4 (3)$

Step 3.2.1.3

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

$f (3) = 54 - 6 \cdot 9 + 4 (3)$

Step 3.2.1.4

Multiply $- 6$ by $9$ .

$f (3) = 54 - 54 + 4 (3)$

Step 3.2.1.5

Multiply $4$ by $3$ .

$f (3) = 54 - 54 + 12$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $54$ from $54$ .

$f (3) = 0 + 12$

Step 3.2.2.2

Add $0$ and $12$ .

$f (3) = 12$

Step 3.2.3

The final answer is $12$ .

$12$

Step 3.3

Convert $12$ to decimal.

$y = 12$

Step 4

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

$\begin{array}{c} x & y \\ - 1 & - 12 \\ 0 & 0 \\ 1 & 0 \\ 2 & 0 \\ 3 & 12 \end{array}$

Step 5

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 1 & - 12 \\ 0 & 0 \\ 1 & 0 \\ 2 & 0 \\ 3 & 12 \end{array}$

Step 6