Algebra Examples

$y = \frac{1}{3} x^{3} - x^{2} - \frac{4}{3}$

Step 1

Find the point at $x = 1$ .

Tap for more steps...

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.

Tap for more steps...

Step 1.2.1

Combine fractions.

Tap for more steps...

Step 1.2.1.1

Combine the numerators over the common denominator.

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

Step 1.2.1.2

Simplify the expression.

Tap for more steps...

Step 1.2.1.2.1

One to any power is one.

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

Step 1.2.1.2.2

Subtract $4$ from $1$ .

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

Step 1.2.2

Simplify each term.

Tap for more steps...

Step 1.2.2.1

One to any power is one.

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

Step 1.2.2.2

Multiply $- 1$ by $1$ .

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

Step 1.2.2.3

Divide $- 3$ by $3$ .

$f (1) = - 1 - 1$

Step 1.2.3

Subtract $1$ from $- 1$ .

$f (1) = - 2$

Step 1.2.4

The final answer is $- 2$ .

$- 2$

Step 1.3

Convert $- 2$ to decimal.

$y = - 2$

Step 2

Find the point at $x = 4$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Combine fractions.

Tap for more steps...

Step 2.2.1.1

Combine the numerators over the common denominator.

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

Step 2.2.1.2

Simplify the expression.

Tap for more steps...

Step 2.2.1.2.1

Raise $4$ to the power of $3$ .

$f (4) = - {(4)}^{2} + \frac{64 - 4}{3}$

Step 2.2.1.2.2

Subtract $4$ from $64$ .

$f (4) = - {(4)}^{2} + \frac{60}{3}$

Step 2.2.2

Simplify each term.

Tap for more steps...

Step 2.2.2.1

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

$f (4) = - 1 \cdot 16 + \frac{60}{3}$

Step 2.2.2.2

Multiply $- 1$ by $16$ .

$f (4) = - 16 + \frac{60}{3}$

Step 2.2.2.3

Divide $60$ by $3$ .

$f (4) = - 16 + 20$

Step 2.2.3

Add $- 16$ and $20$ .

$f (4) = 4$

Step 2.2.4

The final answer is $4$ .

$4$

Step 2.3

Convert $4$ to decimal.

$y = 4$

Step 3

Find the point at $x = - 1$ .

Tap for more steps...

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.

Tap for more steps...

Step 3.2.1

Combine fractions.

Tap for more steps...

Step 3.2.1.1

Combine the numerators over the common denominator.

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

Step 3.2.1.2

Simplify the expression.

Tap for more steps...

Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

Subtract $4$ from $- 1$ .

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

Step 3.2.2

Simplify each term.

Tap for more steps...

Step 3.2.2.1

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

Tap for more steps...

Step 3.2.2.1.1

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

Tap for more steps...

Step 3.2.2.1.1.1

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

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

Step 3.2.2.1.1.2

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

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

Step 3.2.2.1.2

Add $1$ and $2$ .

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Move the negative in front of the fraction.

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

Step 3.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.2.4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

Tap for more steps...

Step 3.2.6.1

Multiply $- 1$ by $3$ .

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

Step 3.2.6.2

Subtract $5$ from $- 3$ .

$f (- 1) = \frac{- 8}{3}$

Step 3.2.7

Move the negative in front of the fraction.

$f (- 1) = - \frac{8}{3}$

Step 3.2.8

The final answer is $- \frac{8}{3}$ .

$- \frac{8}{3}$

Step 3.3

Convert $- \frac{8}{3}$ to decimal.

$y = - 2.\overline{6}$

Step 4

Find the point at $x = 0$ .

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Combine fractions.

Tap for more steps...

Step 4.2.1.1

Combine the numerators over the common denominator.

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

Step 4.2.1.2

Simplify the expression.

Tap for more steps...

Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Subtract $4$ from $0$ .

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

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

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

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

Step 4.2.2.2

Multiply $- 1$ by $0$ .

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

Step 4.2.2.3

Move the negative in front of the fraction.

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

Step 4.2.3

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

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

Step 4.2.4

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

Find the point at $x = 2$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Combine fractions.

Tap for more steps...

Step 5.2.1.1

Combine the numerators over the common denominator.

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

Step 5.2.1.2

Simplify the expression.

Tap for more steps...

Step 5.2.1.2.1

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

$f (2) = - {(2)}^{2} + \frac{8 - 4}{3}$

Step 5.2.1.2.2

Subtract $4$ from $8$ .

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

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $- 1$ by $4$ .

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

Step 5.2.3

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2.4

Combine $- 4$ and $\frac{3}{3}$ .

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

Step 5.2.5

Combine the numerators over the common denominator.

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

Step 5.2.6

Simplify the numerator.

Tap for more steps...

Step 5.2.6.1

Multiply $- 4$ by $3$ .

$f (2) = \frac{- 12 + 4}{3}$

Step 5.2.6.2

Add $- 12$ and $4$ .

$f (2) = \frac{- 8}{3}$

Step 5.2.7

Move the negative in front of the fraction.

$f (2) = - \frac{8}{3}$

Step 5.2.8

The final answer is $- \frac{8}{3}$ .

$- \frac{8}{3}$

Step 5.3

Convert $- \frac{8}{3}$ to decimal.

$y = - 2.\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & - 2.667 \\ 0 & - 1.333 \\ 1 & - 2 \\ 2 & - 2.667 \\ 4 & 4 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & - 2.667 \\ 0 & - 1.333 \\ 1 & - 2 \\ 2 & - 2.667 \\ 4 & 4 \end{array}$

Step 8