Calculus Examples

$y = x^{3} - 6 x^{2} + 3 x + 10$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = {(0)}^{3} - 6 {(0)}^{2} + 3 (0) + 10$

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 $0$ .

$f (0) = 0 - 6 {(0)}^{2} + 3 (0) + 10$

Step 1.2.1.2

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

$f (0) = 0 - 6 \cdot 0 + 3 (0) + 10$

Step 1.2.1.3

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0 + 3 (0) + 10$

Step 1.2.1.4

Multiply $3$ by $0$ .

$f (0) = 0 + 0 + 0 + 10$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 10$

Step 1.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 10$

Step 1.2.2.3

Add $0$ and $10$ .

$f (0) = 10$

Step 1.2.3

The final answer is $10$ .

$10$

Step 1.3

Convert $10$ to decimal.

$y = 10$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = {(1)}^{3} - 6 {(1)}^{2} + 3 (1) + 10$

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.

$f (1) = 1 - 6 {(1)}^{2} + 3 (1) + 10$

Step 2.2.1.2

One to any power is one.

$f (1) = 1 - 6 \cdot 1 + 3 (1) + 10$

Step 2.2.1.3

Multiply $- 6$ by $1$ .

$f (1) = 1 - 6 + 3 (1) + 10$

Step 2.2.1.4

Multiply $3$ by $1$ .

$f (1) = 1 - 6 + 3 + 10$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $6$ from $1$ .

$f (1) = - 5 + 3 + 10$

Step 2.2.2.2

Add $- 5$ and $3$ .

$f (1) = - 2 + 10$

Step 2.2.2.3

Add $- 2$ and $10$ .

$f (1) = 8$

Step 2.2.3

The final answer is $8$ .

$8$

Step 2.3

Convert $8$ to decimal.

$y = 8$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = {(2)}^{3} - 6 {(2)}^{2} + 3 (2) + 10$

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 $3$ .

$f (2) = 8 - 6 {(2)}^{2} + 3 (2) + 10$

Step 3.2.1.2

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

$f (2) = 8 - 6 \cdot 4 + 3 (2) + 10$

Step 3.2.1.3

Multiply $- 6$ by $4$ .

$f (2) = 8 - 24 + 3 (2) + 10$

Step 3.2.1.4

Multiply $3$ by $2$ .

$f (2) = 8 - 24 + 6 + 10$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $24$ from $8$ .

$f (2) = - 16 + 6 + 10$

Step 3.2.2.2

Add $- 16$ and $6$ .

$f (2) = - 10 + 10$

Step 3.2.2.3

Add $- 10$ and $10$ .

$f (2) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = {(3)}^{3} - 6 {(3)}^{2} + 3 (3) + 10$

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 $3$ .

$f (3) = 27 - 6 {(3)}^{2} + 3 (3) + 10$

Step 4.2.1.2

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

$f (3) = 27 - 6 \cdot 9 + 3 (3) + 10$

Step 4.2.1.3

Multiply $- 6$ by $9$ .

$f (3) = 27 - 54 + 3 (3) + 10$

Step 4.2.1.4

Multiply $3$ by $3$ .

$f (3) = 27 - 54 + 9 + 10$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $54$ from $27$ .

$f (3) = - 27 + 9 + 10$

Step 4.2.2.2

Add $- 27$ and $9$ .

$f (3) = - 18 + 10$

Step 4.2.2.3

Add $- 18$ and $10$ .

$f (3) = - 8$

Step 4.2.3

The final answer is $- 8$ .

$- 8$

Step 4.3

Convert $- 8$ to decimal.

$y = - 8$

Step 5

Find the point at $x = 4$ .

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Step 5.1

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

$f (4) = {(4)}^{3} - 6 {(4)}^{2} + 3 (4) + 10$

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 $3$ .

$f (4) = 64 - 6 {(4)}^{2} + 3 (4) + 10$

Step 5.2.1.2

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

$f (4) = 64 - 6 \cdot 16 + 3 (4) + 10$

Step 5.2.1.3

Multiply $- 6$ by $16$ .

$f (4) = 64 - 96 + 3 (4) + 10$

Step 5.2.1.4

Multiply $3$ by $4$ .

$f (4) = 64 - 96 + 12 + 10$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $96$ from $64$ .

$f (4) = - 32 + 12 + 10$

Step 5.2.2.2

Add $- 32$ and $12$ .

$f (4) = - 20 + 10$

Step 5.2.2.3

Add $- 20$ and $10$ .

$f (4) = - 10$

Step 5.2.3

The final answer is $- 10$ .

$- 10$

Step 5.3

Convert $- 10$ to decimal.

$y = - 10$

Step 6

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

$\begin{array}{c} x & y \\ 0 & 10 \\ 1 & 8 \\ 2 & 0 \\ 3 & - 8 \\ 4 & - 10 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & 10 \\ 1 & 8 \\ 2 & 0 \\ 3 & - 8 \\ 4 & - 10 \end{array}$

Step 8