Calculus Examples

$f (x) = x {(x - 9)}^{2}$

Step 1

Find the point at $x = 4$ .

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Step 1.1

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

$f (4) = (4) {((4) - 9)}^{2}$

Step 1.2

Simplify the result.

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Step 1.2.1

Subtract $9$ from $4$ .

$f (4) = 4 {(- 5)}^{2}$

Step 1.2.2

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

$f (4) = 4 \cdot 25$

Step 1.2.3

Multiply $4$ by $25$ .

$f (4) = 100$

Step 1.2.4

The final answer is $100$ .

$100$

Step 1.3

Convert $100$ to decimal.

$y = 100$

Step 2

Find the point at $x = 5$ .

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Step 2.1

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

$f (5) = (5) {((5) - 9)}^{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $9$ from $5$ .

$f (5) = 5 {(- 4)}^{2}$

Step 2.2.2

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

$f (5) = 5 \cdot 16$

Step 2.2.3

Multiply $5$ by $16$ .

$f (5) = 80$

Step 2.2.4

The final answer is $80$ .

$80$

Step 2.3

Convert $80$ to decimal.

$y = 80$

Step 3

Find the point at $x = 6$ .

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Step 3.1

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

$f (6) = (6) {((6) - 9)}^{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $9$ from $6$ .

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

Step 3.2.2

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

$f (6) = 6 \cdot 9$

Step 3.2.3

Multiply $6$ by $9$ .

$f (6) = 54$

Step 3.2.4

The final answer is $54$ .

$54$

Step 3.3

Convert $54$ to decimal.

$y = 54$

Step 4

Find the point at $x = 7$ .

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Step 4.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = (7) {((7) - 9)}^{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $9$ from $7$ .

$f (7) = 7 {(- 2)}^{2}$

Step 4.2.2

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

$f (7) = 7 \cdot 4$

Step 4.2.3

Multiply $7$ by $4$ .

$f (7) = 28$

Step 4.2.4

The final answer is $28$ .

$28$

Step 4.3

Convert $28$ to decimal.

$y = 28$

Step 5

Find the point at $x = 8$ .

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Step 5.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = (8) {((8) - 9)}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Subtract $9$ from $8$ .

$f (8) = 8 {(- 1)}^{2}$

Step 5.2.2

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

$f (8) = 8 \cdot 1$

Step 5.2.3

Multiply $8$ by $1$ .

$f (8) = 8$

Step 5.2.4

The final answer is $8$ .

$8$

Step 5.3

Convert $8$ to decimal.

$y = 8$

Step 6

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

$\begin{array}{c} x & y \\ 4 & 100 \\ 5 & 80 \\ 6 & 54 \\ 7 & 28 \\ 8 & 8 \end{array}$

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

$\begin{array}{c} x & y \\ 4 & 100 \\ 5 & 80 \\ 6 & 54 \\ 7 & 28 \\ 8 & 8 \end{array}$

Step 8