Pre-Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 1.2.2.1.1

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

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Step 1.2.2.1.1.1

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

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

Step 1.2.2.1.1.2

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

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

Step 1.2.2.1.2

Add $1$ and $3$ .

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

Step 1.2.2.2

Raise $- 1$ to the power of $4$ .

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

Step 1.2.3

Subtract $4$ from $1$ .

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

Step 1.2.4

Simplify each term.

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Step 1.2.4.1

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

$f (- 1) = 1 + \frac{- 3}{3}$

Step 1.2.4.2

Divide $- 3$ by $3$ .

$f (- 1) = 1 - 1$

Step 1.2.5

Subtract $1$ from $1$ .

$f (- 1) = 0$

Step 1.2.6

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = - \frac{{(0)}^{3}}{3} + {(0)}^{2} - \frac{4}{3}$

Step 2.2

Simplify the result.

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Step 2.2.1

Combine the numerators over the common denominator.

$f (0) = {(0)}^{2} + \frac{- {(0)}^{3} - 4}{3}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

$f (0) = {(0)}^{2} + \frac{- 0 - 4}{3}$

Step 2.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = {(0)}^{2} + \frac{0 - 4}{3}$

Step 2.2.3

Subtract $4$ from $0$ .

$f (0) = {(0)}^{2} + \frac{- 4}{3}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$f (0) = 0 + \frac{- 4}{3}$

Step 2.2.4.2

Move the negative in front of the fraction.

$f (0) = 0 - \frac{4}{3}$

Step 2.2.5

Subtract $\frac{4}{3}$ from $0$ .

$f (0) = - \frac{4}{3}$

Step 2.2.6

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

Step 2.3

Convert $- \frac{4}{3}$ to decimal.

$y = - 1.\overline{3}$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

One to any power is one.

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

Step 3.2.2.2

Multiply $- 1$ by $1$ .

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

Step 3.2.3

Subtract $4$ from $- 1$ .

$f (1) = {(1)}^{2} + \frac{- 5}{3}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

One to any power is one.

$f (1) = 1 + \frac{- 5}{3}$

Step 3.2.4.2

Move the negative in front of the fraction.

$f (1) = 1 - \frac{5}{3}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Write $1$ as a fraction with a common denominator.

$f (1) = \frac{3}{3} - \frac{5}{3}$

Step 3.2.5.2

Combine the numerators over the common denominator.

$f (1) = \frac{3 - 5}{3}$

Step 3.2.5.3

Subtract $5$ from $3$ .

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

Step 3.2.5.4

Move the negative in front of the fraction.

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

Step 3.2.6

The final answer is $- \frac{2}{3}$ .

$- \frac{2}{3}$

Step 3.3

Convert $- \frac{2}{3}$ to decimal.

$y = - 0.\overline{6}$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

$f (3) = {(3)}^{2} + \frac{- 1 \cdot 27 - 4}{3}$

Step 4.2.2.2

Multiply $- 1$ by $27$ .

$f (3) = {(3)}^{2} + \frac{- 27 - 4}{3}$

Step 4.2.3

Subtract $4$ from $- 27$ .

$f (3) = {(3)}^{2} + \frac{- 31}{3}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

$f (3) = 9 + \frac{- 31}{3}$

Step 4.2.4.2

Move the negative in front of the fraction.

$f (3) = 9 - \frac{31}{3}$

Step 4.2.5

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (3) = 9 \cdot \frac{3}{3} - \frac{31}{3}$

Step 4.2.6

Combine $9$ and $\frac{3}{3}$ .

$f (3) = \frac{9 \cdot 3}{3} - \frac{31}{3}$

Step 4.2.7

Combine the numerators over the common denominator.

$f (3) = \frac{9 \cdot 3 - 31}{3}$

Step 4.2.8

Simplify the numerator.

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Step 4.2.8.1

Multiply $9$ by $3$ .

$f (3) = \frac{27 - 31}{3}$

Step 4.2.8.2

Subtract $31$ from $27$ .

$f (3) = \frac{- 4}{3}$

Step 4.2.9

Move the negative in front of the fraction.

$f (3) = - \frac{4}{3}$

Step 4.2.10

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

Step 4.3

Convert $- \frac{4}{3}$ to decimal.

$y = - 1.\overline{3}$

Step 5

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

$\begin{array}{c} x & y \\ - 1 & 0 \\ 0 & - 1.333 \\ 1 & - 0.667 \\ 2 & 0 \\ 3 & - 1.333 \end{array}$

Step 6

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 1 & 0 \\ 0 & - 1.333 \\ 1 & - 0.667 \\ 2 & 0 \\ 3 & - 1.333 \end{array}$

Step 7