Pre-Algebra Examples

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

Step 1

Find the point at $x = 1$ .

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Step 1.1

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

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

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

Multiply ${((1) - 3)}^{2}$ by $1$ .

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

Step 1.2.1.2

Subtract $3$ from $1$ .

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

Step 1.2.1.3

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

$f (1) = 4 - 2 {((1) - 3)}^{2}$

Step 1.2.1.4

Subtract $3$ from $1$ .

$f (1) = 4 - 2 {(- 2)}^{2}$

Step 1.2.1.5

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

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Step 1.2.1.5.1

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 1.2.1.5.1.1

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

$f (1) = 4 + (- 2) {(- 2)}^{2}$

Step 1.2.1.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (1) = 4 + {(- 2)}^{1 + 2}$

Step 1.2.1.5.2

Add $1$ and $2$ .

$f (1) = 4 + {(- 2)}^{3}$

Step 1.2.1.6

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

$f (1) = 4 - 8$

Step 1.2.2

Subtract $8$ from $4$ .

$f (1) = - 4$

Step 1.2.3

The final answer is $- 4$ .

$- 4$

Step 1.3

Convert $- 4$ to decimal.

$y = - 4$

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

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

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

Subtract $3$ from $2$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Multiply $2$ by $1$ .

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

Step 2.2.1.4

Subtract $3$ from $2$ .

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

Step 2.2.1.5

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

$f (2) = 2 - 2 \cdot 1$

Step 2.2.1.6

Multiply $- 2$ by $1$ .

$f (2) = 2 - 2$

Step 2.2.2

Subtract $2$ from $2$ .

$f (2) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 4$ .

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Step 3.1

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

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

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

Subtract $3$ from $4$ .

$f (4) = 4 \cdot 1^{2} - 2 {((4) - 3)}^{2}$

Step 3.2.1.2

One to any power is one.

$f (4) = 4 \cdot 1 - 2 {((4) - 3)}^{2}$

Step 3.2.1.3

Multiply $4$ by $1$ .

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

Step 3.2.1.4

Subtract $3$ from $4$ .

$f (4) = 4 - 2 \cdot 1^{2}$

Step 3.2.1.5

One to any power is one.

$f (4) = 4 - 2 \cdot 1$

Step 3.2.1.6

Multiply $- 2$ by $1$ .

$f (4) = 4 - 2$

Step 3.2.2

Subtract $2$ from $4$ .

$f (4) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 3.3

Convert $2$ to decimal.

$y = 2$

Step 4

Find the point at $x = 5$ .

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Step 4.1

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

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

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

Subtract $3$ from $5$ .

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

Step 4.2.1.2

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

$f (5) = 5 \cdot 4 - 2 {((5) - 3)}^{2}$

Step 4.2.1.3

Multiply $5$ by $4$ .

$f (5) = 20 - 2 {((5) - 3)}^{2}$

Step 4.2.1.4

Subtract $3$ from $5$ .

$f (5) = 20 - 2 \cdot 2^{2}$

Step 4.2.1.5

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

$f (5) = 20 - 2 \cdot 4$

Step 4.2.1.6

Multiply $- 2$ by $4$ .

$f (5) = 20 - 8$

Step 4.2.2

Subtract $8$ from $20$ .

$f (5) = 12$

Step 4.2.3

The final answer is $12$ .

$12$

Step 4.3

Convert $12$ to decimal.

$y = 12$

Step 5

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

$\begin{array}{c} x & y \\ 1 & - 4 \\ 2 & 0 \\ 3 & 0 \\ 4 & 2 \\ 5 & 12 \end{array}$

Step 6

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ 1 & - 4 \\ 2 & 0 \\ 3 & 0 \\ 4 & 2 \\ 5 & 12 \end{array}$

Step 7