Algebra Examples

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

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

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

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

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

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

Step 1.2.1.2

Divide $0$ by $3$ .

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

Step 1.2.1.3

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

$f (0) = 0 - 2 \cdot 0$

Step 1.2.1.4

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0$

Step 1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

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

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

Cancel the common factor of ${(3)}^{3}$ and $3$ .

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Step 2.2.1.1.1

Factor $3$ out of ${(3)}^{3}$ .

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

Step 2.2.1.1.2

Cancel the common factors.

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Step 2.2.1.1.2.1

Factor $3$ out of $3$ .

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

Step 2.2.1.1.2.2

Cancel the common factor.

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

Step 2.2.1.1.2.3

Rewrite the expression.

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

Step 2.2.1.1.2.4

Divide $3^{2}$ by $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

$f (3) = 9 - 2 \cdot 9$

Step 2.2.1.4

Multiply $- 2$ by $9$ .

$f (3) = 9 - 18$

Step 2.2.2

Subtract $18$ from $9$ .

$f (3) = - 9$

Step 2.2.3

The final answer is $- 9$ .

$- 9$

Step 2.3

Convert $- 9$ to decimal.

$y = - 9$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

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

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

One to any power is one.

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

Step 3.2.1.2

One to any power is one.

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

Step 3.2.1.3

Multiply $- 2$ by $1$ .

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

Step 3.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.2.3

Combine $- 2$ and $\frac{3}{3}$ .

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 2$ by $3$ .

$f (1) = \frac{1 - 6}{3}$

Step 3.2.5.2

Subtract $6$ from $1$ .

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

Step 3.2.6

Move the negative in front of the fraction.

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

Step 3.2.7

The final answer is $- \frac{5}{3}$ .

$- \frac{5}{3}$

Step 3.3

Convert $- \frac{5}{3}$ to decimal.

$y = - 1.\overline{6}$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

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

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

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $- 2$ by $4$ .

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

Step 4.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.2.3

Combine $- 8$ and $\frac{3}{3}$ .

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 8$ by $3$ .

$f (2) = \frac{8 - 24}{3}$

Step 4.2.5.2

Subtract $24$ from $8$ .

$f (2) = \frac{- 16}{3}$

Step 4.2.6

Move the negative in front of the fraction.

$f (2) = - \frac{16}{3}$

Step 4.2.7

The final answer is $- \frac{16}{3}$ .

$- \frac{16}{3}$

Step 4.3

Convert $- \frac{16}{3}$ to decimal.

$y = - 5.\overline{3}$

Step 5

Find the point at $x = 4$ .

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Step 5.1

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

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

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

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

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

Step 5.2.1.2

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

$f (4) = \frac{64}{3} - 2 \cdot 16$

Step 5.2.1.3

Multiply $- 2$ by $16$ .

$f (4) = \frac{64}{3} - 32$

Step 5.2.2

To write $- 32$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (4) = \frac{64}{3} - 32 \cdot \frac{3}{3}$

Step 5.2.3

Combine $- 32$ and $\frac{3}{3}$ .

$f (4) = \frac{64}{3} + \frac{- 32 \cdot 3}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (4) = \frac{64 - 32 \cdot 3}{3}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 32$ by $3$ .

$f (4) = \frac{64 - 96}{3}$

Step 5.2.5.2

Subtract $96$ from $64$ .

$f (4) = \frac{- 32}{3}$

Step 5.2.6

Move the negative in front of the fraction.

$f (4) = - \frac{32}{3}$

Step 5.2.7

The final answer is $- \frac{32}{3}$ .

$- \frac{32}{3}$

Step 5.3

Convert $- \frac{32}{3}$ to decimal.

$y = - 10.\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & - 1.667 \\ 2 & - 5.333 \\ 3 & - 9 \\ 4 & - 10.667 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & - 1.667 \\ 2 & - 5.333 \\ 3 & - 9 \\ 4 & - 10.667 \end{array}$

Step 8