Calculus Examples

$y = x^{3} - 6 x^{2} - 96 x$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = {(0)}^{3} - 6 {(0)}^{2} - 96 \cdot 0$

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

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

$f (0) = 0 - 6 {(0)}^{2} - 96 \cdot 0$

Step 1.2.1.2

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

$f (0) = 0 - 6 \cdot 0 - 96 \cdot 0$

Step 1.2.1.3

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0 - 96 \cdot 0$

Step 1.2.1.4

Multiply $- 96$ by $0$ .

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

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = {(1)}^{3} - 6 {(1)}^{2} - 96 \cdot 1$

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

One to any power is one.

$f (1) = 1 - 6 {(1)}^{2} - 96 \cdot 1$

Step 2.2.1.2

One to any power is one.

$f (1) = 1 - 6 \cdot 1 - 96 \cdot 1$

Step 2.2.1.3

Multiply $- 6$ by $1$ .

$f (1) = 1 - 6 - 96 \cdot 1$

Step 2.2.1.4

Multiply $- 96$ by $1$ .

$f (1) = 1 - 6 - 96$

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract $6$ from $1$ .

$f (1) = - 5 - 96$

Step 2.2.2.2

Subtract $96$ from $- 5$ .

$f (1) = - 101$

Step 2.2.3

The final answer is $- 101$ .

$- 101$

Step 2.3

Convert $- 101$ to decimal.

$y = - 101$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = {(2)}^{3} - 6 {(2)}^{2} - 96 \cdot 2$

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 $3$ .

$f (2) = 8 - 6 {(2)}^{2} - 96 \cdot 2$

Step 3.2.1.2

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

$f (2) = 8 - 6 \cdot 4 - 96 \cdot 2$

Step 3.2.1.3

Multiply $- 6$ by $4$ .

$f (2) = 8 - 24 - 96 \cdot 2$

Step 3.2.1.4

Multiply $- 96$ by $2$ .

$f (2) = 8 - 24 - 192$

Step 3.2.2

Simplify by subtracting numbers.

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Step 3.2.2.1

Subtract $24$ from $8$ .

$f (2) = - 16 - 192$

Step 3.2.2.2

Subtract $192$ from $- 16$ .

$f (2) = - 208$

Step 3.2.3

The final answer is $- 208$ .

$- 208$

Step 3.3

Convert $- 208$ to decimal.

$y = - 208$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = {(3)}^{3} - 6 {(3)}^{2} - 96 \cdot 3$

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

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

$f (3) = 27 - 6 {(3)}^{2} - 96 \cdot 3$

Step 4.2.1.2

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

$f (3) = 27 - 6 \cdot 9 - 96 \cdot 3$

Step 4.2.1.3

Multiply $- 6$ by $9$ .

$f (3) = 27 - 54 - 96 \cdot 3$

Step 4.2.1.4

Multiply $- 96$ by $3$ .

$f (3) = 27 - 54 - 288$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $54$ from $27$ .

$f (3) = - 27 - 288$

Step 4.2.2.2

Subtract $288$ from $- 27$ .

$f (3) = - 315$

Step 4.2.3

The final answer is $- 315$ .

$- 315$

Step 4.3

Convert $- 315$ to decimal.

$y = - 315$

Step 5

Find the point at $x = 4$ .

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Step 5.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{3} - 6 {(4)}^{2} - 96 \cdot 4$

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 $4$ to the power of $3$ .

$f (4) = 64 - 6 {(4)}^{2} - 96 \cdot 4$

Step 5.2.1.2

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

$f (4) = 64 - 6 \cdot 16 - 96 \cdot 4$

Step 5.2.1.3

Multiply $- 6$ by $16$ .

$f (4) = 64 - 96 - 96 \cdot 4$

Step 5.2.1.4

Multiply $- 96$ by $4$ .

$f (4) = 64 - 96 - 384$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $96$ from $64$ .

$f (4) = - 32 - 384$

Step 5.2.2.2

Subtract $384$ from $- 32$ .

$f (4) = - 416$

Step 5.2.3

The final answer is $- 416$ .

$- 416$

Step 5.3

Convert $- 416$ to decimal.

$y = - 416$

Step 6

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & - 101 \\ 2 & - 208 \\ 3 & - 315 \\ 4 & - 416 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & - 101 \\ 2 & - 208 \\ 3 & - 315 \\ 4 & - 416 \end{array}$

Step 8