Pre-Algebra Examples

$f (x) = 4 x^{3} - 11 x^{2} + 10 x - 3$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = 4 {(- 1)}^{3} - 11 {(- 1)}^{2} + 10 (- 1) - 3$

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

$f (- 1) = 4 \cdot - 1 - 11 {(- 1)}^{2} + 10 (- 1) - 3$

Step 1.2.1.2

Multiply $4$ by $- 1$ .

$f (- 1) = - 4 - 11 {(- 1)}^{2} + 10 (- 1) - 3$

Step 1.2.1.3

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

$f (- 1) = - 4 - 11 \cdot 1 + 10 (- 1) - 3$

Step 1.2.1.4

Multiply $- 11$ by $1$ .

$f (- 1) = - 4 - 11 + 10 (- 1) - 3$

Step 1.2.1.5

Multiply $10$ by $- 1$ .

$f (- 1) = - 4 - 11 - 10 - 3$

Step 1.2.2

Simplify by subtracting numbers.

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Step 1.2.2.1

Subtract $11$ from $- 4$ .

$f (- 1) = - 15 - 10 - 3$

Step 1.2.2.2

Subtract $10$ from $- 15$ .

$f (- 1) = - 25 - 3$

Step 1.2.2.3

Subtract $3$ from $- 25$ .

$f (- 1) = - 28$

Step 1.2.3

The final answer is $- 28$ .

$- 28$

Step 1.3

Convert $- 28$ to decimal.

$y = - 28$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = 4 {(0)}^{3} - 11 {(0)}^{2} + 10 (0) - 3$

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

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

$f (0) = 4 \cdot 0 - 11 {(0)}^{2} + 10 (0) - 3$

Step 2.2.1.2

Multiply $4$ by $0$ .

$f (0) = 0 - 11 {(0)}^{2} + 10 (0) - 3$

Step 2.2.1.3

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

$f (0) = 0 - 11 \cdot 0 + 10 (0) - 3$

Step 2.2.1.4

Multiply $- 11$ by $0$ .

$f (0) = 0 + 0 + 10 (0) - 3$

Step 2.2.1.5

Multiply $10$ by $0$ .

$f (0) = 0 + 0 + 0 - 3$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 - 3$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 3$

Step 2.2.2.3

Subtract $3$ from $0$ .

$f (0) = - 3$

Step 2.2.3

The final answer is $- 3$ .

$- 3$

Step 2.3

Convert $- 3$ to decimal.

$y = - 3$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = 4 {(1)}^{3} - 11 {(1)}^{2} + 10 (1) - 3$

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

One to any power is one.

$f (1) = 4 \cdot 1 - 11 {(1)}^{2} + 10 (1) - 3$

Step 3.2.1.2

Multiply $4$ by $1$ .

$f (1) = 4 - 11 {(1)}^{2} + 10 (1) - 3$

Step 3.2.1.3

One to any power is one.

$f (1) = 4 - 11 \cdot 1 + 10 (1) - 3$

Step 3.2.1.4

Multiply $- 11$ by $1$ .

$f (1) = 4 - 11 + 10 (1) - 3$

Step 3.2.1.5

Multiply $10$ by $1$ .

$f (1) = 4 - 11 + 10 - 3$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $11$ from $4$ .

$f (1) = - 7 + 10 - 3$

Step 3.2.2.2

Add $- 7$ and $10$ .

$f (1) = 3 - 3$

Step 3.2.2.3

Subtract $3$ from $3$ .

$f (1) = 0$

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 $x = 2$ .

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Step 4.1

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

$f (2) = 4 {(2)}^{3} - 11 {(2)}^{2} + 10 (2) - 3$

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 $2$ to the power of $3$ .

$f (2) = 4 \cdot 8 - 11 {(2)}^{2} + 10 (2) - 3$

Step 4.2.1.2

Multiply $4$ by $8$ .

$f (2) = 32 - 11 {(2)}^{2} + 10 (2) - 3$

Step 4.2.1.3

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

$f (2) = 32 - 11 \cdot 4 + 10 (2) - 3$

Step 4.2.1.4

Multiply $- 11$ by $4$ .

$f (2) = 32 - 44 + 10 (2) - 3$

Step 4.2.1.5

Multiply $10$ by $2$ .

$f (2) = 32 - 44 + 20 - 3$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $44$ from $32$ .

$f (2) = - 12 + 20 - 3$

Step 4.2.2.2

Add $- 12$ and $20$ .

$f (2) = 8 - 3$

Step 4.2.2.3

Subtract $3$ from $8$ .

$f (2) = 5$

Step 4.2.3

The final answer is $5$ .

$5$

Step 4.3

Convert $5$ to decimal.

$y = 5$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

$f (3) = 4 {(3)}^{3} - 11 {(3)}^{2} + 10 (3) - 3$

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 $3$ to the power of $3$ .

$f (3) = 4 \cdot 27 - 11 {(3)}^{2} + 10 (3) - 3$

Step 5.2.1.2

Multiply $4$ by $27$ .

$f (3) = 108 - 11 {(3)}^{2} + 10 (3) - 3$

Step 5.2.1.3

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

$f (3) = 108 - 11 \cdot 9 + 10 (3) - 3$

Step 5.2.1.4

Multiply $- 11$ by $9$ .

$f (3) = 108 - 99 + 10 (3) - 3$

Step 5.2.1.5

Multiply $10$ by $3$ .

$f (3) = 108 - 99 + 30 - 3$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $99$ from $108$ .

$f (3) = 9 + 30 - 3$

Step 5.2.2.2

Add $9$ and $30$ .

$f (3) = 39 - 3$

Step 5.2.2.3

Subtract $3$ from $39$ .

$f (3) = 36$

Step 5.2.3

The final answer is $36$ .

$36$

Step 5.3

Convert $36$ to decimal.

$y = 36$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & - 28 \\ 0 & - 3 \\ 1 & 0 \\ 2 & 5 \\ 3 & 36 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & - 28 \\ 0 & - 3 \\ 1 & 0 \\ 2 & 5 \\ 3 & 36 \end{array}$

Step 8