Algebra Examples

$P (x) = \frac{4}{9} x^{3} - 2 x^{2}$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = \frac{4 {(0)}^{3}}{9} - 2 {(0)}^{2}$

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 $0$ .

$f (0) = \frac{4 \cdot 0}{9} - 2 {(0)}^{2}$

Step 1.2.1.2

Multiply $4$ by $0$ .

$f (0) = \frac{0}{9} - 2 {(0)}^{2}$

Step 1.2.1.3

Divide $0$ by $9$ .

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

Step 1.2.1.4

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

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

Step 1.2.1.5

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.

$f (3) = \frac{4 {(3)}^{3}}{9} - 2 {(3)}^{2}$

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

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

$f (3) = \frac{4 \cdot 27}{9} - 2 {(3)}^{2}$

Step 2.2.1.2

Multiply $4$ by $27$ .

$f (3) = \frac{108}{9} - 2 {(3)}^{2}$

Step 2.2.1.3

Divide $108$ by $9$ .

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

Step 2.2.1.4

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

$f (3) = 12 - 2 \cdot 9$

Step 2.2.1.5

Multiply $- 2$ by $9$ .

$f (3) = 12 - 18$

Step 2.2.2

Subtract $18$ from $12$ .

$f (3) = - 6$

Step 2.2.3

The final answer is $- 6$ .

$- 6$

Step 2.3

Convert $- 6$ to decimal.

$y = - 6$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = \frac{4 {(1)}^{3}}{9} - 2 {(1)}^{2}$

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.

$f (1) = \frac{4 \cdot 1}{9} - 2 {(1)}^{2}$

Step 3.2.1.2

Multiply $4$ by $1$ .

$f (1) = \frac{4}{9} - 2 {(1)}^{2}$

Step 3.2.1.3

One to any power is one.

$f (1) = \frac{4}{9} - 2 \cdot 1$

Step 3.2.1.4

Multiply $- 2$ by $1$ .

$f (1) = \frac{4}{9} - 2$

Step 3.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (1) = \frac{4}{9} - 2 \cdot \frac{9}{9}$

Step 3.2.3

Combine $- 2$ and $\frac{9}{9}$ .

$f (1) = \frac{4}{9} + \frac{- 2 \cdot 9}{9}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{4 - 2 \cdot 9}{9}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 2$ by $9$ .

$f (1) = \frac{4 - 18}{9}$

Step 3.2.5.2

Subtract $18$ from $4$ .

$f (1) = \frac{- 14}{9}$

Step 3.2.6

Move the negative in front of the fraction.

$f (1) = - \frac{14}{9}$

Step 3.2.7

The final answer is $- \frac{14}{9}$ .

$- \frac{14}{9}$

Step 3.3

Convert $- \frac{14}{9}$ to decimal.

$y = - 1.\overline{5}$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = \frac{4 {(2)}^{3}}{9} - 2 {(2)}^{2}$

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

Simplify the numerator.

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Step 4.2.1.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.1.1.2

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

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

Step 4.2.1.1.3

Add $2$ and $3$ .

$f (2) = \frac{2^{5}}{9} - 2 {(2)}^{2}$

Step 4.2.1.2

Raise $2$ to the power of $5$ .

$f (2) = \frac{32}{9} - 2 {(2)}^{2}$

Step 4.2.1.3

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

$f (2) = \frac{32}{9} - 2 \cdot 4$

Step 4.2.1.4

Multiply $- 2$ by $4$ .

$f (2) = \frac{32}{9} - 8$

Step 4.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (2) = \frac{32}{9} - 8 \cdot \frac{9}{9}$

Step 4.2.3

Combine $- 8$ and $\frac{9}{9}$ .

$f (2) = \frac{32}{9} + \frac{- 8 \cdot 9}{9}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{32 - 8 \cdot 9}{9}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 8$ by $9$ .

$f (2) = \frac{32 - 72}{9}$

Step 4.2.5.2

Subtract $72$ from $32$ .

$f (2) = \frac{- 40}{9}$

Step 4.2.6

Move the negative in front of the fraction.

$f (2) = - \frac{40}{9}$

Step 4.2.7

The final answer is $- \frac{40}{9}$ .

$- \frac{40}{9}$

Step 4.3

Convert $- \frac{40}{9}$ to decimal.

$y = - 4.\overline{4}$

Step 5

Find the point at $x = 4$ .

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Step 5.1

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

$f (4) = \frac{4 {(4)}^{3}}{9} - 2 {(4)}^{2}$

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

Multiply $4$ by ${(4)}^{3}$ by adding the exponents.

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Step 5.2.1.1.1

Multiply $4$ by ${(4)}^{3}$ .

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Step 5.2.1.1.1.1

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

$f (4) = \frac{4 {(4)}^{3}}{9} - 2 {(4)}^{2}$

Step 5.2.1.1.1.2

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

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

Step 5.2.1.1.2

Add $1$ and $3$ .

$f (4) = \frac{4^{4}}{9} - 2 {(4)}^{2}$

Step 5.2.1.2

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

$f (4) = \frac{256}{9} - 2 {(4)}^{2}$

Step 5.2.1.3

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

$f (4) = \frac{256}{9} - 2 \cdot 16$

Step 5.2.1.4

Multiply $- 2$ by $16$ .

$f (4) = \frac{256}{9} - 32$

Step 5.2.2

To write $- 32$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (4) = \frac{256}{9} - 32 \cdot \frac{9}{9}$

Step 5.2.3

Combine $- 32$ and $\frac{9}{9}$ .

$f (4) = \frac{256}{9} + \frac{- 32 \cdot 9}{9}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (4) = \frac{256 - 32 \cdot 9}{9}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 32$ by $9$ .

$f (4) = \frac{256 - 288}{9}$

Step 5.2.5.2

Subtract $288$ from $256$ .

$f (4) = \frac{- 32}{9}$

Step 5.2.6

Move the negative in front of the fraction.

$f (4) = - \frac{32}{9}$

Step 5.2.7

The final answer is $- \frac{32}{9}$ .

$- \frac{32}{9}$

Step 5.3

Convert $- \frac{32}{9}$ to decimal.

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

Step 6

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & - 1.556 \\ 2 & - 4.444 \\ 3 & - 6 \\ 4 & - 3.556 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & - 1.556 \\ 2 & - 4.444 \\ 3 & - 6 \\ 4 & - 3.556 \end{array}$

Step 8