Pre-Algebra Examples

$y = \frac{1}{3} x^{3} - 7 x^{2}$

Step 1

Find the point at $x = 6$ .

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Step 1.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{{(6)}^{3}}{3} - 7 {(6)}^{2}$

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

Raise $6$ to the power of $3$ .

$f (6) = \frac{216}{3} - 7 {(6)}^{2}$

Step 1.2.1.2

Divide $216$ by $3$ .

$f (6) = 72 - 7 {(6)}^{2}$

Step 1.2.1.3

Raise $6$ to the power of $2$ .

$f (6) = 72 - 7 \cdot 36$

Step 1.2.1.4

Multiply $- 7$ by $36$ .

$f (6) = 72 - 252$

Step 1.2.2

Subtract $252$ from $72$ .

$f (6) = - 180$

Step 1.2.3

The final answer is $- 180$ .

$- 180$

Step 1.3

Convert $- 180$ to decimal.

$y = - 180$

Step 2

Find the point at $x = 9$ .

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Step 2.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \frac{{(9)}^{3}}{3} - 7 {(9)}^{2}$

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 $9$ to the power of $3$ .

$f (9) = \frac{729}{3} - 7 {(9)}^{2}$

Step 2.2.1.2

Divide $729$ by $3$ .

$f (9) = 243 - 7 {(9)}^{2}$

Step 2.2.1.3

Raise $9$ to the power of $2$ .

$f (9) = 243 - 7 \cdot 81$

Step 2.2.1.4

Multiply $- 7$ by $81$ .

$f (9) = 243 - 567$

Step 2.2.2

Subtract $567$ from $243$ .

$f (9) = - 324$

Step 2.2.3

The final answer is $- 324$ .

$- 324$

Step 2.3

Convert $- 324$ to decimal.

$y = - 324$

Step 3

Find the point at $x = 5$ .

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Step 3.1

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

$f (5) = \frac{{(5)}^{3}}{3} - 7 {(5)}^{2}$

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

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

$f (5) = \frac{125}{3} - 7 {(5)}^{2}$

Step 3.2.1.2

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

$f (5) = \frac{125}{3} - 7 \cdot 25$

Step 3.2.1.3

Multiply $- 7$ by $25$ .

$f (5) = \frac{125}{3} - 175$

Step 3.2.2

To write $- 175$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (5) = \frac{125}{3} - 175 \cdot \frac{3}{3}$

Step 3.2.3

Combine $- 175$ and $\frac{3}{3}$ .

$f (5) = \frac{125}{3} + \frac{- 175 \cdot 3}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (5) = \frac{125 - 175 \cdot 3}{3}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 175$ by $3$ .

$f (5) = \frac{125 - 525}{3}$

Step 3.2.5.2

Subtract $525$ from $125$ .

$f (5) = \frac{- 400}{3}$

Step 3.2.6

Move the negative in front of the fraction.

$f (5) = - \frac{400}{3}$

Step 3.2.7

The final answer is $- \frac{400}{3}$ .

$- \frac{400}{3}$

Step 3.3

Convert $- \frac{400}{3}$ to decimal.

$y = - 133.\overline{3}$

Step 4

Find the point at $x = 7$ .

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Step 4.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{{(7)}^{3}}{3} - 7 {(7)}^{2}$

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 $7$ to the power of $3$ .

$f (7) = \frac{343}{3} - 7 {(7)}^{2}$

Step 4.2.1.2

Raise $7$ to the power of $2$ .

$f (7) = \frac{343}{3} - 7 \cdot 49$

Step 4.2.1.3

Multiply $- 7$ by $49$ .

$f (7) = \frac{343}{3} - 343$

Step 4.2.2

To write $- 343$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (7) = \frac{343}{3} - 343 \cdot \frac{3}{3}$

Step 4.2.3

Combine $- 343$ and $\frac{3}{3}$ .

$f (7) = \frac{343}{3} + \frac{- 343 \cdot 3}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (7) = \frac{343 - 343 \cdot 3}{3}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 343$ by $3$ .

$f (7) = \frac{343 - 1029}{3}$

Step 4.2.5.2

Subtract $1029$ from $343$ .

$f (7) = \frac{- 686}{3}$

Step 4.2.6

Move the negative in front of the fraction.

$f (7) = - \frac{686}{3}$

Step 4.2.7

The final answer is $- \frac{686}{3}$ .

$- \frac{686}{3}$

Step 4.3

Convert $- \frac{686}{3}$ to decimal.

$y = - 228.\overline{6}$

Step 5

Find the point at $x = 8$ .

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Step 5.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \frac{{(8)}^{3}}{3} - 7 {(8)}^{2}$

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 $8$ to the power of $3$ .

$f (8) = \frac{512}{3} - 7 {(8)}^{2}$

Step 5.2.1.2

Raise $8$ to the power of $2$ .

$f (8) = \frac{512}{3} - 7 \cdot 64$

Step 5.2.1.3

Multiply $- 7$ by $64$ .

$f (8) = \frac{512}{3} - 448$

Step 5.2.2

To write $- 448$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (8) = \frac{512}{3} - 448 \cdot \frac{3}{3}$

Step 5.2.3

Combine $- 448$ and $\frac{3}{3}$ .

$f (8) = \frac{512}{3} + \frac{- 448 \cdot 3}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (8) = \frac{512 - 448 \cdot 3}{3}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 448$ by $3$ .

$f (8) = \frac{512 - 1344}{3}$

Step 5.2.5.2

Subtract $1344$ from $512$ .

$f (8) = \frac{- 832}{3}$

Step 5.2.6

Move the negative in front of the fraction.

$f (8) = - \frac{832}{3}$

Step 5.2.7

The final answer is $- \frac{832}{3}$ .

$- \frac{832}{3}$

Step 5.3

Convert $- \frac{832}{3}$ to decimal.

$y = - 277.\overline{3}$

Step 6

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

$\begin{array}{c} x & y \\ 5 & - 133.333 \\ 6 & - 180 \\ 7 & - 228.667 \\ 8 & - 277.333 \\ 9 & - 324 \end{array}$

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

$\begin{array}{c} x & y \\ 5 & - 133.333 \\ 6 & - 180 \\ 7 & - 228.667 \\ 8 & - 277.333 \\ 9 & - 324 \end{array}$

Step 8