Calculus Examples

$y = 2 x^{3} + 9 x^{2} - 24 x - 10$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

$f (- 3) = 2 {(- 3)}^{3} + 9 {(- 3)}^{2} - 24 \cdot - 3 - 10$

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

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

$f (- 3) = 2 \cdot - 27 + 9 {(- 3)}^{2} - 24 \cdot - 3 - 10$

Step 1.2.1.2

Multiply $2$ by $- 27$ .

$f (- 3) = - 54 + 9 {(- 3)}^{2} - 24 \cdot - 3 - 10$

Step 1.2.1.3

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

$f (- 3) = - 54 + 9 \cdot 9 - 24 \cdot - 3 - 10$

Step 1.2.1.4

Multiply $9$ by $9$ .

$f (- 3) = - 54 + 81 - 24 \cdot - 3 - 10$

Step 1.2.1.5

Multiply $- 24$ by $- 3$ .

$f (- 3) = - 54 + 81 + 72 - 10$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $- 54$ and $81$ .

$f (- 3) = 27 + 72 - 10$

Step 1.2.2.2

Add $27$ and $72$ .

$f (- 3) = 99 - 10$

Step 1.2.2.3

Subtract $10$ from $99$ .

$f (- 3) = 89$

Step 1.2.3

The final answer is $89$ .

$89$

Step 1.3

Convert $89$ to decimal.

$y = 89$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

$f (- 2) = 2 {(- 2)}^{3} + 9 {(- 2)}^{2} - 24 \cdot - 2 - 10$

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

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

$f (- 2) = 2 \cdot - 8 + 9 {(- 2)}^{2} - 24 \cdot - 2 - 10$

Step 2.2.1.2

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 + 9 {(- 2)}^{2} - 24 \cdot - 2 - 10$

Step 2.2.1.3

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

$f (- 2) = - 16 + 9 \cdot 4 - 24 \cdot - 2 - 10$

Step 2.2.1.4

Multiply $9$ by $4$ .

$f (- 2) = - 16 + 36 - 24 \cdot - 2 - 10$

Step 2.2.1.5

Multiply $- 24$ by $- 2$ .

$f (- 2) = - 16 + 36 + 48 - 10$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 16$ and $36$ .

$f (- 2) = 20 + 48 - 10$

Step 2.2.2.2

Add $20$ and $48$ .

$f (- 2) = 68 - 10$

Step 2.2.2.3

Subtract $10$ from $68$ .

$f (- 2) = 58$

Step 2.2.3

The final answer is $58$ .

$58$

Step 2.3

Convert $58$ to decimal.

$y = 58$

Step 3

Find the point at $x = - 1$ .

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Step 3.1

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

$f (- 1) = 2 {(- 1)}^{3} + 9 {(- 1)}^{2} - 24 \cdot - 1 - 10$

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

$f (- 1) = 2 \cdot - 1 + 9 {(- 1)}^{2} - 24 \cdot - 1 - 10$

Step 3.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = - 2 + 9 {(- 1)}^{2} - 24 \cdot - 1 - 10$

Step 3.2.1.3

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

$f (- 1) = - 2 + 9 \cdot 1 - 24 \cdot - 1 - 10$

Step 3.2.1.4

Multiply $9$ by $1$ .

$f (- 1) = - 2 + 9 - 24 \cdot - 1 - 10$

Step 3.2.1.5

Multiply $- 24$ by $- 1$ .

$f (- 1) = - 2 + 9 + 24 - 10$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 2$ and $9$ .

$f (- 1) = 7 + 24 - 10$

Step 3.2.2.2

Add $7$ and $24$ .

$f (- 1) = 31 - 10$

Step 3.2.2.3

Subtract $10$ from $31$ .

$f (- 1) = 21$

Step 3.2.3

The final answer is $21$ .

$21$

Step 3.3

Convert $21$ to decimal.

$y = 21$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = 2 {(0)}^{3} + 9 {(0)}^{2} - 24 \cdot 0 - 10$

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

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

$f (0) = 2 \cdot 0 + 9 {(0)}^{2} - 24 \cdot 0 - 10$

Step 4.2.1.2

Multiply $2$ by $0$ .

$f (0) = 0 + 9 {(0)}^{2} - 24 \cdot 0 - 10$

Step 4.2.1.3

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

$f (0) = 0 + 9 \cdot 0 - 24 \cdot 0 - 10$

Step 4.2.1.4

Multiply $9$ by $0$ .

$f (0) = 0 + 0 - 24 \cdot 0 - 10$

Step 4.2.1.5

Multiply $- 24$ by $0$ .

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

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $0$ and $0$ .

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

Step 4.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 10$

Step 4.2.2.3

Subtract $10$ from $0$ .

$f (0) = - 10$

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 $x = 1$ .

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Step 5.1

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

$f (1) = 2 {(1)}^{3} + 9 {(1)}^{2} - 24 \cdot 1 - 10$

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

One to any power is one.

$f (1) = 2 \cdot 1 + 9 {(1)}^{2} - 24 \cdot 1 - 10$

Step 5.2.1.2

Multiply $2$ by $1$ .

$f (1) = 2 + 9 {(1)}^{2} - 24 \cdot 1 - 10$

Step 5.2.1.3

One to any power is one.

$f (1) = 2 + 9 \cdot 1 - 24 \cdot 1 - 10$

Step 5.2.1.4

Multiply $9$ by $1$ .

$f (1) = 2 + 9 - 24 \cdot 1 - 10$

Step 5.2.1.5

Multiply $- 24$ by $1$ .

$f (1) = 2 + 9 - 24 - 10$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $2$ and $9$ .

$f (1) = 11 - 24 - 10$

Step 5.2.2.2

Subtract $24$ from $11$ .

$f (1) = - 13 - 10$

Step 5.2.2.3

Subtract $10$ from $- 13$ .

$f (1) = - 23$

Step 5.2.3

The final answer is $- 23$ .

$- 23$

Step 5.3

Convert $- 23$ to decimal.

$y = - 23$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 89 \\ - 2 & 58 \\ - 1 & 21 \\ 0 & - 10 \\ 1 & - 23 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 89 \\ - 2 & 58 \\ - 1 & 21 \\ 0 & - 10 \\ 1 & - 23 \end{array}$

Step 8