Pre-Algebra Examples

$y = \frac{1}{2} \cdot ((x^{2} - 1) (5 - x))$

Step 1

Find the point at $x = 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

One to any power is one.

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

Step 1.2.2.2

Multiply $5$ by $1$ .

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

Step 1.2.2.3

One to any power is one.

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

Step 1.2.2.4

Multiply $- 1$ by $1$ .

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

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Subtract $1$ from $5$ .

$f (1) = \frac{4 - 5 + 1}{2}$

Step 1.2.3.2

Subtract $5$ from $4$ .

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

Step 1.2.3.3

Add $- 1$ and $1$ .

$f (1) = \frac{0}{2}$

Step 1.2.3.4

Divide $0$ by $2$ .

$f (1) = 0$

Step 1.2.4

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.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Combine the numerators over the common denominator.

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

$f (3) = \frac{5 \cdot 9 - {(3)}^{3} - 5 + 3}{2}$

Step 2.2.2.2

Multiply $5$ by $9$ .

$f (3) = \frac{45 - {(3)}^{3} - 5 + 3}{2}$

Step 2.2.2.3

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

$f (3) = \frac{45 - 1 \cdot 27 - 5 + 3}{2}$

Step 2.2.2.4

Multiply $- 1$ by $27$ .

$f (3) = \frac{45 - 27 - 5 + 3}{2}$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Subtract $27$ from $45$ .

$f (3) = \frac{18 - 5 + 3}{2}$

Step 2.2.3.2

Subtract $5$ from $18$ .

$f (3) = \frac{13 + 3}{2}$

Step 2.2.3.3

Add $13$ and $3$ .

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

Step 2.2.3.4

Divide $16$ by $2$ .

$f (3) = 8$

Step 2.2.4

The final answer is $8$ .

$8$

Step 2.3

Convert $8$ to decimal.

$y = 8$

Step 3

Find the point at $x = 5$ .

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Step 3.1

Replace the variable $x$ with $5$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply $5$ by ${(5)}^{2}$ by adding the exponents.

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Step 3.2.2.1.1

Multiply $5$ by ${(5)}^{2}$ .

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Step 3.2.2.1.1.1

Raise $5$ to the power of $1$ .

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

Step 3.2.2.1.1.2

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

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

Step 3.2.2.1.2

Add $1$ and $2$ .

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

Step 3.2.2.2

Raise $5$ to the power of $3$ .

$f (5) = \frac{125 - {(5)}^{3} - 5 + 5}{2}$

Step 3.2.2.3

Raise $5$ to the power of $3$ .

$f (5) = \frac{125 - 1 \cdot 125 - 5 + 5}{2}$

Step 3.2.2.4

Multiply $- 1$ by $125$ .

$f (5) = \frac{125 - 125 - 5 + 5}{2}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Subtract $125$ from $125$ .

$f (5) = \frac{0 - 5 + 5}{2}$

Step 3.2.3.2

Subtract $5$ from $0$ .

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

Step 3.2.3.3

Add $- 5$ and $5$ .

$f (5) = \frac{0}{2}$

Step 3.2.3.4

Divide $0$ by $2$ .

$f (5) = 0$

Step 3.2.4

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = \frac{5 {(0)}^{2}}{2} - \frac{{(0)}^{3}}{2} - \frac{5}{2} + \frac{0}{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Combine the numerators over the common denominator.

$f (0) = \frac{5 {(0)}^{2} - {(0)}^{3} - 5}{2}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

$f (0) = \frac{5 \cdot 0 - {(0)}^{3} - 5}{2}$

Step 4.2.2.2

Multiply $5$ by $0$ .

$f (0) = \frac{0 - {(0)}^{3} - 5}{2}$

Step 4.2.2.3

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

$f (0) = \frac{0 - 0 - 5}{2}$

Step 4.2.2.4

Multiply $- 1$ by $0$ .

$f (0) = \frac{0 + 0 - 5}{2}$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Add $0$ and $0$ .

$f (0) = \frac{0 - 5}{2}$

Step 4.2.3.2

Subtract $5$ from $0$ .

$f (0) = \frac{- 5}{2}$

Step 4.2.3.3

Move the negative in front of the fraction.

$f (0) = - \frac{5}{2}$

Step 4.2.4

The final answer is $- \frac{5}{2}$ .

$- \frac{5}{2}$

Step 4.3

Convert $- \frac{5}{2}$ to decimal.

$y = - 2.5$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Combine the numerators over the common denominator.

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

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

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

$f (2) = \frac{5 \cdot 4 - {(2)}^{3} - 5 + 2}{2}$

Step 5.2.2.2

Multiply $5$ by $4$ .

$f (2) = \frac{20 - {(2)}^{3} - 5 + 2}{2}$

Step 5.2.2.3

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

$f (2) = \frac{20 - 1 \cdot 8 - 5 + 2}{2}$

Step 5.2.2.4

Multiply $- 1$ by $8$ .

$f (2) = \frac{20 - 8 - 5 + 2}{2}$

Step 5.2.3

Simplify by adding and subtracting.

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Step 5.2.3.1

Subtract $8$ from $20$ .

$f (2) = \frac{12 - 5 + 2}{2}$

Step 5.2.3.2

Subtract $5$ from $12$ .

$f (2) = \frac{7 + 2}{2}$

Step 5.2.3.3

Add $7$ and $2$ .

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

Step 5.2.4

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

Step 5.3

Convert $\frac{9}{2}$ to decimal.

$y = 4.5$

Step 6

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

$\begin{array}{c} x & y \\ 0 & - 2.5 \\ 1 & 0 \\ 2 & 4.5 \\ 3 & 8 \\ 5 & 0 \end{array}$

Step 7

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ 0 & - 2.5 \\ 1 & 0 \\ 2 & 4.5 \\ 3 & 8 \\ 5 & 0 \end{array}$

Step 8