Calculus Examples

$y = 2 x^{3} - 24 x - 6$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = 2 {(- 2)}^{3} - 24 \cdot - 2 - 6$

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 $3$ .

$f (- 2) = 2 \cdot - 8 - 24 \cdot - 2 - 6$

Step 1.2.1.2

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 - 24 \cdot - 2 - 6$

Step 1.2.1.3

Multiply $- 24$ by $- 2$ .

$f (- 2) = - 16 + 48 - 6$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $- 16$ and $48$ .

$f (- 2) = 32 - 6$

Step 1.2.2.2

Subtract $6$ from $32$ .

$f (- 2) = 26$

Step 1.2.3

The final answer is $26$ .

$26$

Step 1.3

Convert $26$ to decimal.

$y = 26$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = 2 {(- 1)}^{3} - 24 \cdot - 1 - 6$

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 $3$ .

$f (- 1) = 2 \cdot - 1 - 24 \cdot - 1 - 6$

Step 2.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = - 2 - 24 \cdot - 1 - 6$

Step 2.2.1.3

Multiply $- 24$ by $- 1$ .

$f (- 1) = - 2 + 24 - 6$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 2$ and $24$ .

$f (- 1) = 22 - 6$

Step 2.2.2.2

Subtract $6$ from $22$ .

$f (- 1) = 16$

Step 2.2.3

The final answer is $16$ .

$16$

Step 2.3

Convert $16$ to decimal.

$y = 16$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = 2 {(0)}^{3} - 24 \cdot 0 - 6$

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 $0$ .

$f (0) = 2 \cdot 0 - 24 \cdot 0 - 6$

Step 3.2.1.2

Multiply $2$ by $0$ .

$f (0) = 0 - 24 \cdot 0 - 6$

Step 3.2.1.3

Multiply $- 24$ by $0$ .

$f (0) = 0 + 0 - 6$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 - 6$

Step 3.2.2.2

Subtract $6$ from $0$ .

$f (0) = - 6$

Step 3.2.3

The final answer is $- 6$ .

$- 6$

Step 3.3

Convert $- 6$ to decimal.

$y = - 6$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = 2 {(1)}^{3} - 24 \cdot 1 - 6$

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.

$f (1) = 2 \cdot 1 - 24 \cdot 1 - 6$

Step 4.2.1.2

Multiply $2$ by $1$ .

$f (1) = 2 - 24 \cdot 1 - 6$

Step 4.2.1.3

Multiply $- 24$ by $1$ .

$f (1) = 2 - 24 - 6$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $24$ from $2$ .

$f (1) = - 22 - 6$

Step 4.2.2.2

Subtract $6$ from $- 22$ .

$f (1) = - 28$

Step 4.2.3

The final answer is $- 28$ .

$- 28$

Step 4.3

Convert $- 28$ to decimal.

$y = - 28$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = 2 {(2)}^{3} - 24 \cdot 2 - 6$

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

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

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Step 5.2.1.1.1

Multiply $2$ by ${(2)}^{3}$ .

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Step 5.2.1.1.1.1

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

$f (2) = 2 {(2)}^{3} - 24 \cdot 2 - 6$

Step 5.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (2) = 2^{1 + 3} - 24 \cdot 2 - 6$

Step 5.2.1.1.2

Add $1$ and $3$ .

$f (2) = 2^{4} - 24 \cdot 2 - 6$

Step 5.2.1.2

Raise $2$ to the power of $4$ .

$f (2) = 16 - 24 \cdot 2 - 6$

Step 5.2.1.3

Multiply $- 24$ by $2$ .

$f (2) = 16 - 48 - 6$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $48$ from $16$ .

$f (2) = - 32 - 6$

Step 5.2.2.2

Subtract $6$ from $- 32$ .

$f (2) = - 38$

Step 5.2.3

The final answer is $- 38$ .

$- 38$

Step 5.3

Convert $- 38$ to decimal.

$y = - 38$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 26 \\ - 1 & 16 \\ 0 & - 6 \\ 1 & - 28 \\ 2 & - 38 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 26 \\ - 1 & 16 \\ 0 & - 6 \\ 1 & - 28 \\ 2 & - 38 \end{array}$

Step 8