Pre-Algebra Examples

$2 x^{3} - 9 x^{2} + 12 x - 1$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = 2 {(0)}^{3} - 9 {(0)}^{2} + 12 (0) - 1$

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

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

$f (0) = 2 \cdot 0 - 9 {(0)}^{2} + 12 (0) - 1$

Step 1.2.1.2

Multiply $2$ by $0$ .

$f (0) = 0 - 9 {(0)}^{2} + 12 (0) - 1$

Step 1.2.1.3

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

$f (0) = 0 - 9 \cdot 0 + 12 (0) - 1$

Step 1.2.1.4

Multiply $- 9$ by $0$ .

$f (0) = 0 + 0 + 12 (0) - 1$

Step 1.2.1.5

Multiply $12$ by $0$ .

$f (0) = 0 + 0 + 0 - 1$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 - 1$

Step 1.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 1$

Step 1.2.2.3

Subtract $1$ from $0$ .

$f (0) = - 1$

Step 1.2.3

The final answer is $- 1$ .

$- 1$

Step 1.3

Convert $- 1$ to decimal.

$y = - 1$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = 2 {(1)}^{3} - 9 {(1)}^{2} + 12 (1) - 1$

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

One to any power is one.

$f (1) = 2 \cdot 1 - 9 {(1)}^{2} + 12 (1) - 1$

Step 2.2.1.2

Multiply $2$ by $1$ .

$f (1) = 2 - 9 {(1)}^{2} + 12 (1) - 1$

Step 2.2.1.3

One to any power is one.

$f (1) = 2 - 9 \cdot 1 + 12 (1) - 1$

Step 2.2.1.4

Multiply $- 9$ by $1$ .

$f (1) = 2 - 9 + 12 (1) - 1$

Step 2.2.1.5

Multiply $12$ by $1$ .

$f (1) = 2 - 9 + 12 - 1$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $9$ from $2$ .

$f (1) = - 7 + 12 - 1$

Step 2.2.2.2

Add $- 7$ and $12$ .

$f (1) = 5 - 1$

Step 2.2.2.3

Subtract $1$ from $5$ .

$f (1) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 2.3

Convert $4$ to decimal.

$y = 4$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = 2 {(2)}^{3} - 9 {(2)}^{2} + 12 (2) - 1$

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 $2$ by ${(2)}^{3}$ by adding the exponents.

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Step 3.2.1.1.1

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

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Step 3.2.1.1.1.1

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

$f (2) = 2 {(2)}^{3} - 9 {(2)}^{2} + 12 (2) - 1$

Step 3.2.1.1.1.2

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

$f (2) = 2^{1 + 3} - 9 {(2)}^{2} + 12 (2) - 1$

Step 3.2.1.1.2

Add $1$ and $3$ .

$f (2) = 2^{4} - 9 {(2)}^{2} + 12 (2) - 1$

Step 3.2.1.2

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

$f (2) = 16 - 9 {(2)}^{2} + 12 (2) - 1$

Step 3.2.1.3

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

$f (2) = 16 - 9 \cdot 4 + 12 (2) - 1$

Step 3.2.1.4

Multiply $- 9$ by $4$ .

$f (2) = 16 - 36 + 12 (2) - 1$

Step 3.2.1.5

Multiply $12$ by $2$ .

$f (2) = 16 - 36 + 24 - 1$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $36$ from $16$ .

$f (2) = - 20 + 24 - 1$

Step 3.2.2.2

Add $- 20$ and $24$ .

$f (2) = 4 - 1$

Step 3.2.2.3

Subtract $1$ from $4$ .

$f (2) = 3$

Step 3.2.3

The final answer is $3$ .

$3$

Step 3.3

Convert $3$ to decimal.

$y = 3$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = 2 {(3)}^{3} - 9 {(3)}^{2} + 12 (3) - 1$

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 $3$ to the power of $3$ .

$f (3) = 2 \cdot 27 - 9 {(3)}^{2} + 12 (3) - 1$

Step 4.2.1.2

Multiply $2$ by $27$ .

$f (3) = 54 - 9 {(3)}^{2} + 12 (3) - 1$

Step 4.2.1.3

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

$f (3) = 54 - 9 \cdot 9 + 12 (3) - 1$

Step 4.2.1.4

Multiply $- 9$ by $9$ .

$f (3) = 54 - 81 + 12 (3) - 1$

Step 4.2.1.5

Multiply $12$ by $3$ .

$f (3) = 54 - 81 + 36 - 1$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $81$ from $54$ .

$f (3) = - 27 + 36 - 1$

Step 4.2.2.2

Add $- 27$ and $36$ .

$f (3) = 9 - 1$

Step 4.2.2.3

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.

$f (4) = 2 {(4)}^{3} - 9 {(4)}^{2} + 12 (4) - 1$

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 $4$ to the power of $3$ .

$f (4) = 2 \cdot 64 - 9 {(4)}^{2} + 12 (4) - 1$

Step 5.2.1.2

Multiply $2$ by $64$ .

$f (4) = 128 - 9 {(4)}^{2} + 12 (4) - 1$

Step 5.2.1.3

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

$f (4) = 128 - 9 \cdot 16 + 12 (4) - 1$

Step 5.2.1.4

Multiply $- 9$ by $16$ .

$f (4) = 128 - 144 + 12 (4) - 1$

Step 5.2.1.5

Multiply $12$ by $4$ .

$f (4) = 128 - 144 + 48 - 1$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $144$ from $128$ .

$f (4) = - 16 + 48 - 1$

Step 5.2.2.2

Add $- 16$ and $48$ .

$f (4) = 32 - 1$

Step 5.2.2.3

Subtract $1$ from $32$ .

$f (4) = 31$

Step 5.2.3

The final answer is $31$ .

$31$

Step 5.3

Convert $31$ to decimal.

$y = 31$

Step 6

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

$\begin{array}{c} x & y \\ 0 & - 1 \\ 1 & 4 \\ 2 & 3 \\ 3 & 8 \\ 4 & 31 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & - 1 \\ 1 & 4 \\ 2 & 3 \\ 3 & 8 \\ 4 & 31 \end{array}$

Step 8