Calculus Examples

$f (x) = x^{3} - 12 x + 1$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = {(- 2)}^{3} - 12 \cdot - 2 + 1$

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 $3$ .

$f (- 2) = - 8 - 12 \cdot - 2 + 1$

Step 1.2.1.2

Multiply $- 12$ by $- 2$ .

$f (- 2) = - 8 + 24 + 1$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $- 8$ and $24$ .

$f (- 2) = 16 + 1$

Step 1.2.2.2

Add $16$ and $1$ .

$f (- 2) = 17$

Step 1.2.3

The final answer is $17$ .

$17$

Step 1.3

Convert $17$ to decimal.

$y = 17$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = {(- 1)}^{3} - 12 \cdot - 1 + 1$

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

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

$f (- 1) = - 1 - 12 \cdot - 1 + 1$

Step 2.2.1.2

Multiply $- 12$ by $- 1$ .

$f (- 1) = - 1 + 12 + 1$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $- 1$ and $12$ .

$f (- 1) = 11 + 1$

Step 2.2.2.2

Add $11$ and $1$ .

$f (- 1) = 12$

Step 2.2.3

The final answer is $12$ .

$12$

Step 2.3

Convert $12$ to decimal.

$y = 12$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = {(0)}^{3} - 12 \cdot 0 + 1$

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

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

$f (0) = 0 - 12 \cdot 0 + 1$

Step 3.2.1.2

Multiply $- 12$ by $0$ .

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

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 1$

Step 3.2.2.2

Add $0$ and $1$ .

$f (0) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 3.3

Convert $1$ to decimal.

$y = 1$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = {(1)}^{3} - 12 \cdot 1 + 1$

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

One to any power is one.

$f (1) = 1 - 12 \cdot 1 + 1$

Step 4.2.1.2

Multiply $- 12$ by $1$ .

$f (1) = 1 - 12 + 1$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $12$ from $1$ .

$f (1) = - 11 + 1$

Step 4.2.2.2

Add $- 11$ and $1$ .

$f (1) = - 10$

Step 4.2.3

The final answer is $- 10$ .

$- 10$

Step 4.3

Convert $- 10$ to decimal.

$y = - 10$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = {(2)}^{3} - 12 \cdot 2 + 1$

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 $3$ .

$f (2) = 8 - 12 \cdot 2 + 1$

Step 5.2.1.2

Multiply $- 12$ by $2$ .

$f (2) = 8 - 24 + 1$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $24$ from $8$ .

$f (2) = - 16 + 1$

Step 5.2.2.2

Add $- 16$ and $1$ .

$f (2) = - 15$

Step 5.2.3

The final answer is $- 15$ .

$- 15$

Step 5.3

Convert $- 15$ to decimal.

$y = - 15$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 17 \\ - 1 & 12 \\ 0 & 1 \\ 1 & - 10 \\ 2 & - 15 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 17 \\ - 1 & 12 \\ 0 & 1 \\ 1 & - 10 \\ 2 & - 15 \end{array}$

Step 8