Pre-Algebra Examples

$(3 x - 1) {(x - 2)}^{2}$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

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

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

Multiply $3$ by $0$ .

$f (0) = 0 {((0) - 2)}^{2} - {((0) - 2)}^{2}$

Step 1.2.1.2

Subtract $2$ from $0$ .

$f (0) = 0 {(- 2)}^{2} - {((0) - 2)}^{2}$

Step 1.2.1.3

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

$f (0) = 0 \cdot 4 - {((0) - 2)}^{2}$

Step 1.2.1.4

Multiply $0$ by $4$ .

$f (0) = 0 - {((0) - 2)}^{2}$

Step 1.2.1.5

Subtract $2$ from $0$ .

$f (0) = 0 - {(- 2)}^{2}$

Step 1.2.1.6

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

$f (0) = 0 - 1 \cdot 4$

Step 1.2.1.7

Multiply $- 1$ by $4$ .

$f (0) = 0 - 4$

Step 1.2.2

Subtract $4$ from $0$ .

$f (0) = - 4$

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 $x = 1$ .

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Step 2.1

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

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

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

Multiply $3$ by $1$ .

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

Step 2.2.1.2

Subtract $2$ from $1$ .

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Multiply $3$ by $1$ .

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

Step 2.2.1.5

Subtract $2$ from $1$ .

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

Step 2.2.1.6

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.2.1.6.1

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

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Step 2.2.1.6.1.1

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

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

Step 2.2.1.6.1.2

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

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

Step 2.2.1.6.2

Add $1$ and $2$ .

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

Step 2.2.1.7

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

$f (1) = 3 - 1$

Step 2.2.2

Subtract $1$ from $3$ .

$f (1) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 2.3

Convert $2$ to decimal.

$y = 2$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

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

Multiply $3$ by $2$ .

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

Step 3.2.1.2

Subtract $2$ from $2$ .

$f (2) = 6 \cdot 0^{2} - {((2) - 2)}^{2}$

Step 3.2.1.3

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

$f (2) = 6 \cdot 0 - {((2) - 2)}^{2}$

Step 3.2.1.4

Multiply $6$ by $0$ .

$f (2) = 0 - {((2) - 2)}^{2}$

Step 3.2.1.5

Subtract $2$ from $2$ .

$f (2) = 0 - 0^{2}$

Step 3.2.1.6

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

$f (2) = 0 - 0$

Step 3.2.1.7

Multiply $- 1$ by $0$ .

$f (2) = 0 + 0$

Step 3.2.2

Add $0$ and $0$ .

$f (2) = 0$

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.

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

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

Multiply $3$ by $3$ .

$f (3) = 9 {((3) - 2)}^{2} - {((3) - 2)}^{2}$

Step 4.2.1.2

Subtract $2$ from $3$ .

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

Step 4.2.1.3

One to any power is one.

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

Step 4.2.1.4

Multiply $9$ by $1$ .

$f (3) = 9 - {((3) - 2)}^{2}$

Step 4.2.1.5

Subtract $2$ from $3$ .

$f (3) = 9 - 1^{2}$

Step 4.2.1.6

One to any power is one.

$f (3) = 9 - 1 \cdot 1$

Step 4.2.1.7

Multiply $- 1$ by $1$ .

$f (3) = 9 - 1$

Step 4.2.2

Subtract $1$ from $9$ .

$f (3) = 8$

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.

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

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

Multiply $3$ by $4$ .

$f (4) = 12 {((4) - 2)}^{2} - {((4) - 2)}^{2}$

Step 5.2.1.2

Subtract $2$ from $4$ .

$f (4) = 12 \cdot 2^{2} - {((4) - 2)}^{2}$

Step 5.2.1.3

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

$f (4) = 12 \cdot 4 - {((4) - 2)}^{2}$

Step 5.2.1.4

Multiply $12$ by $4$ .

$f (4) = 48 - {((4) - 2)}^{2}$

Step 5.2.1.5

Subtract $2$ from $4$ .

$f (4) = 48 - 2^{2}$

Step 5.2.1.6

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

$f (4) = 48 - 1 \cdot 4$

Step 5.2.1.7

Multiply $- 1$ by $4$ .

$f (4) = 48 - 4$

Step 5.2.2

Subtract $4$ from $48$ .

$f (4) = 44$

Step 5.2.3

The final answer is $44$ .

$44$

Step 5.3

Convert $44$ to decimal.

$y = 44$

Step 6

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

$\begin{array}{c} x & y \\ 0 & - 4 \\ 1 & 2 \\ 2 & 0 \\ 3 & 8 \\ 4 & 44 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & - 4 \\ 1 & 2 \\ 2 & 0 \\ 3 & 8 \\ 4 & 44 \end{array}$

Step 8