Precalculus Examples

$f (x) = x^{2} (x - 10)$

Step 1

Find the point at $x = 2$ .

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Step 1.1

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

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

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 $3$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $- 10$ by $4$ .

$f (2) = 8 - 40$

Step 1.2.2

Subtract $40$ from $8$ .

$f (2) = - 32$

Step 1.2.3

The final answer is $- 32$ .

$- 32$

Step 1.3

Convert $- 32$ to decimal.

$y = - 32$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

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

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 $3$ to the power of $3$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Multiply $- 10$ by $9$ .

$f (3) = 27 - 90$

Step 2.2.2

Subtract $90$ from $27$ .

$f (3) = - 63$

Step 2.2.3

The final answer is $- 63$ .

$- 63$

Step 2.3

Convert $- 63$ to decimal.

$y = - 63$

Step 3

Find the point at $x = 4$ .

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Step 3.1

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

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

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 $4$ to the power of $3$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply $- 10$ by $16$ .

$f (4) = 64 - 160$

Step 3.2.2

Subtract $160$ from $64$ .

$f (4) = - 96$

Step 3.2.3

The final answer is $- 96$ .

$- 96$

Step 3.3

Convert $- 96$ to decimal.

$y = - 96$

Step 4

Find the point at $x = 5$ .

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Step 4.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{3} - 10 {(5)}^{2}$

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 $5$ to the power of $3$ .

$f (5) = 125 - 10 {(5)}^{2}$

Step 4.2.1.2

Raise $5$ to the power of $2$ .

$f (5) = 125 - 10 \cdot 25$

Step 4.2.1.3

Multiply $- 10$ by $25$ .

$f (5) = 125 - 250$

Step 4.2.2

Subtract $250$ from $125$ .

$f (5) = - 125$

Step 4.2.3

The final answer is $- 125$ .

$- 125$

Step 4.3

Convert $- 125$ to decimal.

$y = - 125$

Step 5

Find the point at $x = 6$ .

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Step 5.1

Replace the variable $x$ with $6$ in the expression.

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

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 $6$ to the power of $3$ .

$f (6) = 216 - 10 {(6)}^{2}$

Step 5.2.1.2

Raise $6$ to the power of $2$ .

$f (6) = 216 - 10 \cdot 36$

Step 5.2.1.3

Multiply $- 10$ by $36$ .

$f (6) = 216 - 360$

Step 5.2.2

Subtract $360$ from $216$ .

$f (6) = - 144$

Step 5.2.3

The final answer is $- 144$ .

$- 144$

Step 5.3

Convert $- 144$ to decimal.

$y = - 144$

Step 6

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

$\begin{array}{c} x & y \\ 2 & - 32 \\ 3 & - 63 \\ 4 & - 96 \\ 5 & - 125 \\ 6 & - 144 \end{array}$

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

$\begin{array}{c} x & y \\ 2 & - 32 \\ 3 & - 63 \\ 4 & - 96 \\ 5 & - 125 \\ 6 & - 144 \end{array}$

Step 8