Trigonometry Examples

$f (x) = - \frac{1}{3} x^{3} - 4 x^{2} + 5 x + 36$

Step 1

Find the point at $x = - 6$ .

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Step 1.1

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

$f (- 6) = - \frac{{(- 6)}^{3}}{3} - 4 {(- 6)}^{2} + 5 (- 6) + 36$

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 $3$ .

$f (- 6) = - \frac{- 216}{3} - 4 {(- 6)}^{2} + 5 (- 6) + 36$

Step 1.2.1.2

Divide $- 216$ by $3$ .

$f (- 6) = 72 - 4 {(- 6)}^{2} + 5 (- 6) + 36$

Step 1.2.1.3

Multiply $- 1$ by $- 72$ .

$f (- 6) = 72 - 4 {(- 6)}^{2} + 5 (- 6) + 36$

Step 1.2.1.4

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

$f (- 6) = 72 - 4 \cdot 36 + 5 (- 6) + 36$

Step 1.2.1.5

Multiply $- 4$ by $36$ .

$f (- 6) = 72 - 144 + 5 (- 6) + 36$

Step 1.2.1.6

Multiply $5$ by $- 6$ .

$f (- 6) = 72 - 144 - 30 + 36$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $144$ from $72$ .

$f (- 6) = - 72 - 30 + 36$

Step 1.2.2.2

Subtract $30$ from $- 72$ .

$f (- 6) = - 102 + 36$

Step 1.2.2.3

Add $- 102$ and $36$ .

$f (- 6) = - 66$

Step 1.2.3

The final answer is $- 66$ .

$- 66$

Step 1.3

Convert $- 66$ to decimal.

$y = - 66$

Step 2

Find the point at $x = - 3$ .

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Step 2.1

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

$f (- 3) = - \frac{{(- 3)}^{3}}{3} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

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

Rewrite $- 3$ as $- 1 (3)$ .

$f (- 3) = - \frac{{(- 1 \cdot 3)}^{3}}{3} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.2

Apply the product rule to $- 1 (3)$ .

$f (- 3) = - \frac{{(- 1)}^{3} \cdot 3^{3}}{3} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.3

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

$f (- 3) = - \frac{- 1 \cdot 3^{3}}{3} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.4

Factor $3$ out of $- 1 \cdot 3^{3}$ .

$f (- 3) = - \frac{3 (- 1 \cdot 3^{2})}{3} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.5

Cancel the common factors.

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Step 2.2.1.1.5.1

Factor $3$ out of $3$ .

$f (- 3) = - \frac{3 (- 1 \cdot 3^{2})}{3 (1)} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.5.2

Cancel the common factor.

$f (- 3) = - \frac{3 (- 1 \cdot 3^{2})}{3 \cdot 1} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.5.3

Rewrite the expression.

$f (- 3) = - \frac{- 1 \cdot 3^{2}}{1} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.1.5.4

Divide $- 1 \cdot 3^{2}$ by $1$ .

$f (- 3) = - (- 1 \cdot 3^{2}) - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.2

Rewrite $- 1 \cdot 3^{2}$ as $- 3^{2}$ .

$f (- 3) = 3^{2} - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.3

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

$f (- 3) = - (- 1 \cdot 9) - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.4

Multiply $- (- 1 \cdot 9)$ .

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Step 2.2.1.4.1

Multiply $- 1$ by $9$ .

$f (- 3) = 9 - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.4.2

Multiply $- 1$ by $- 9$ .

$f (- 3) = 9 - 4 {(- 3)}^{2} + 5 (- 3) + 36$

Step 2.2.1.5

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

$f (- 3) = 9 - 4 \cdot 9 + 5 (- 3) + 36$

Step 2.2.1.6

Multiply $- 4$ by $9$ .

$f (- 3) = 9 - 36 + 5 (- 3) + 36$

Step 2.2.1.7

Multiply $5$ by $- 3$ .

$f (- 3) = 9 - 36 - 15 + 36$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $36$ from $9$ .

$f (- 3) = - 27 - 15 + 36$

Step 2.2.2.2

Subtract $15$ from $- 27$ .

$f (- 3) = - 42 + 36$

Step 2.2.2.3

Add $- 42$ and $36$ .

$f (- 3) = - 6$

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 $x = - 5$ .

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Step 3.1

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

$f (- 5) = - \frac{{(- 5)}^{3}}{3} - 4 {(- 5)}^{2} + 5 (- 5) + 36$

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 $3$ .

$f (- 5) = - \frac{- 125}{3} - 4 {(- 5)}^{2} + 5 (- 5) + 36$

Step 3.2.1.2

Move the negative in front of the fraction.

$f (- 5) = \frac{125}{3} - 4 {(- 5)}^{2} + 5 (- 5) + 36$

Step 3.2.1.3

Multiply $- - \frac{125}{3}$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 5) = 1 (\frac{125}{3}) - 4 {(- 5)}^{2} + 5 (- 5) + 36$

Step 3.2.1.3.2

Multiply $\frac{125}{3}$ by $1$ .

$f (- 5) = \frac{125}{3} - 4 {(- 5)}^{2} + 5 (- 5) + 36$

Step 3.2.1.4

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

$f (- 5) = \frac{125}{3} - 4 \cdot 25 + 5 (- 5) + 36$

Step 3.2.1.5

Multiply $- 4$ by $25$ .

$f (- 5) = \frac{125}{3} - 100 + 5 (- 5) + 36$

Step 3.2.1.6

Multiply $5$ by $- 5$ .

$f (- 5) = \frac{125}{3} - 100 - 25 + 36$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Write $- 100$ as a fraction with denominator $1$ .

$f (- 5) = \frac{125}{3} + \frac{- 100}{1} - 25 + 36$

Step 3.2.2.2

Multiply $\frac{- 100}{1}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{125}{3} + \frac{- 100}{1} \cdot \frac{3}{3} - 25 + 36$

Step 3.2.2.3

Multiply $\frac{- 100}{1}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} - 25 + 36$

Step 3.2.2.4

Write $- 25$ as a fraction with denominator $1$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} + \frac{- 25}{1} + 36$

Step 3.2.2.5

Multiply $\frac{- 25}{1}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} + \frac{- 25}{1} \cdot \frac{3}{3} + 36$

Step 3.2.2.6

Multiply $\frac{- 25}{1}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} + \frac{- 25 \cdot 3}{3} + 36$

Step 3.2.2.7

Write $36$ as a fraction with denominator $1$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} + \frac{- 25 \cdot 3}{3} + \frac{36}{1}$

Step 3.2.2.8

Multiply $\frac{36}{1}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} + \frac{- 25 \cdot 3}{3} + \frac{36}{1} \cdot \frac{3}{3}$

Step 3.2.2.9

Multiply $\frac{36}{1}$ by $\frac{3}{3}$ .

$f (- 5) = \frac{125}{3} + \frac{- 100 \cdot 3}{3} + \frac{- 25 \cdot 3}{3} + \frac{36 \cdot 3}{3}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- 5) = \frac{125 - 100 \cdot 3 - 25 \cdot 3 + 36 \cdot 3}{3}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $- 100$ by $3$ .

$f (- 5) = \frac{125 - 300 - 25 \cdot 3 + 36 \cdot 3}{3}$

Step 3.2.4.2

Multiply $- 25$ by $3$ .

$f (- 5) = \frac{125 - 300 - 75 + 36 \cdot 3}{3}$

Step 3.2.4.3

Multiply $36$ by $3$ .

$f (- 5) = \frac{125 - 300 - 75 + 108}{3}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract $300$ from $125$ .

$f (- 5) = \frac{- 175 - 75 + 108}{3}$

Step 3.2.5.2

Subtract $75$ from $- 175$ .

$f (- 5) = \frac{- 250 + 108}{3}$

Step 3.2.5.3

Add $- 250$ and $108$ .

$f (- 5) = \frac{- 142}{3}$

Step 3.2.5.4

Move the negative in front of the fraction.

$f (- 5) = - \frac{142}{3}$

Step 3.2.6

The final answer is $- \frac{142}{3}$ .

$- \frac{142}{3}$

Step 3.3

Convert $- \frac{142}{3}$ to decimal.

$y = - 47.\overline{3}$

Step 4

Find the point at $x = - 4$ .

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Step 4.1

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

$f (- 4) = - \frac{{(- 4)}^{3}}{3} - 4 {(- 4)}^{2} + 5 (- 4) + 36$

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

$f (- 4) = - \frac{- 64}{3} - 4 {(- 4)}^{2} + 5 (- 4) + 36$

Step 4.2.1.2

Move the negative in front of the fraction.

$f (- 4) = \frac{64}{3} - 4 {(- 4)}^{2} + 5 (- 4) + 36$

Step 4.2.1.3

Multiply $- - \frac{64}{3}$ .

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Step 4.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 4) = 1 (\frac{64}{3}) - 4 {(- 4)}^{2} + 5 (- 4) + 36$

Step 4.2.1.3.2

Multiply $\frac{64}{3}$ by $1$ .

$f (- 4) = \frac{64}{3} - 4 {(- 4)}^{2} + 5 (- 4) + 36$

Step 4.2.1.4

Multiply $- 4$ by ${(- 4)}^{2}$ by adding the exponents.

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Step 4.2.1.4.1

Multiply $- 4$ by ${(- 4)}^{2}$ .

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Step 4.2.1.4.1.1

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

$f (- 4) = \frac{64}{3} + (- 4) {(- 4)}^{2} + 5 (- 4) + 36$

Step 4.2.1.4.1.2

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

$f (- 4) = \frac{64}{3} + {(- 4)}^{1 + 2} + 5 (- 4) + 36$

Step 4.2.1.4.2

Add $1$ and $2$ .

$f (- 4) = \frac{64}{3} + {(- 4)}^{3} + 5 (- 4) + 36$

Step 4.2.1.5

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

$f (- 4) = \frac{64}{3} - 64 + 5 (- 4) + 36$

Step 4.2.1.6

Multiply $5$ by $- 4$ .

$f (- 4) = \frac{64}{3} - 64 - 20 + 36$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $- 64$ as a fraction with denominator $1$ .

$f (- 4) = \frac{64}{3} + \frac{- 64}{1} - 20 + 36$

Step 4.2.2.2

Multiply $\frac{- 64}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{64}{3} + \frac{- 64}{1} \cdot \frac{3}{3} - 20 + 36$

Step 4.2.2.3

Multiply $\frac{- 64}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} - 20 + 36$

Step 4.2.2.4

Write $- 20$ as a fraction with denominator $1$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} + \frac{- 20}{1} + 36$

Step 4.2.2.5

Multiply $\frac{- 20}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} + \frac{- 20}{1} \cdot \frac{3}{3} + 36$

Step 4.2.2.6

Multiply $\frac{- 20}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} + \frac{- 20 \cdot 3}{3} + 36$

Step 4.2.2.7

Write $36$ as a fraction with denominator $1$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} + \frac{- 20 \cdot 3}{3} + \frac{36}{1}$

Step 4.2.2.8

Multiply $\frac{36}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} + \frac{- 20 \cdot 3}{3} + \frac{36}{1} \cdot \frac{3}{3}$

Step 4.2.2.9

Multiply $\frac{36}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{64}{3} + \frac{- 64 \cdot 3}{3} + \frac{- 20 \cdot 3}{3} + \frac{36 \cdot 3}{3}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (- 4) = \frac{64 - 64 \cdot 3 - 20 \cdot 3 + 36 \cdot 3}{3}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 64$ by $3$ .

$f (- 4) = \frac{64 - 192 - 20 \cdot 3 + 36 \cdot 3}{3}$

Step 4.2.4.2

Multiply $- 20$ by $3$ .

$f (- 4) = \frac{64 - 192 - 60 + 36 \cdot 3}{3}$

Step 4.2.4.3

Multiply $36$ by $3$ .

$f (- 4) = \frac{64 - 192 - 60 + 108}{3}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Subtract $192$ from $64$ .

$f (- 4) = \frac{- 128 - 60 + 108}{3}$

Step 4.2.5.2

Subtract $60$ from $- 128$ .

$f (- 4) = \frac{- 188 + 108}{3}$

Step 4.2.5.3

Add $- 188$ and $108$ .

$f (- 4) = \frac{- 80}{3}$

Step 4.2.5.4

Move the negative in front of the fraction.

$f (- 4) = - \frac{80}{3}$

Step 4.2.6

The final answer is $- \frac{80}{3}$ .

$- \frac{80}{3}$

Step 4.3

Convert $- \frac{80}{3}$ to decimal.

$y = - 26.\overline{6}$

Step 5

Find the point at $x = - 2$ .

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Step 5.1

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

$f (- 2) = - \frac{{(- 2)}^{3}}{3} - 4 {(- 2)}^{2} + 5 (- 2) + 36$

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

$f (- 2) = - \frac{- 8}{3} - 4 {(- 2)}^{2} + 5 (- 2) + 36$

Step 5.2.1.2

Move the negative in front of the fraction.

$f (- 2) = \frac{8}{3} - 4 {(- 2)}^{2} + 5 (- 2) + 36$

Step 5.2.1.3

Multiply $- - \frac{8}{3}$ .

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Step 5.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 2) = 1 (\frac{8}{3}) - 4 {(- 2)}^{2} + 5 (- 2) + 36$

Step 5.2.1.3.2

Multiply $\frac{8}{3}$ by $1$ .

$f (- 2) = \frac{8}{3} - 4 {(- 2)}^{2} + 5 (- 2) + 36$

Step 5.2.1.4

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

$f (- 2) = \frac{8}{3} - 4 \cdot 4 + 5 (- 2) + 36$

Step 5.2.1.5

Multiply $- 4$ by $4$ .

$f (- 2) = \frac{8}{3} - 16 + 5 (- 2) + 36$

Step 5.2.1.6

Multiply $5$ by $- 2$ .

$f (- 2) = \frac{8}{3} - 16 - 10 + 36$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Write $- 16$ as a fraction with denominator $1$ .

$f (- 2) = \frac{8}{3} + \frac{- 16}{1} - 10 + 36$

Step 5.2.2.2

Multiply $\frac{- 16}{1}$ by $\frac{3}{3}$ .

$f (- 2) = \frac{8}{3} + \frac{- 16}{1} \cdot \frac{3}{3} - 10 + 36$

Step 5.2.2.3

Multiply $\frac{- 16}{1}$ by $\frac{3}{3}$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} - 10 + 36$

Step 5.2.2.4

Write $- 10$ as a fraction with denominator $1$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} + \frac{- 10}{1} + 36$

Step 5.2.2.5

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} + \frac{- 10}{1} \cdot \frac{3}{3} + 36$

Step 5.2.2.6

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} + \frac{- 10 \cdot 3}{3} + 36$

Step 5.2.2.7

Write $36$ as a fraction with denominator $1$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} + \frac{- 10 \cdot 3}{3} + \frac{36}{1}$

Step 5.2.2.8

Multiply $\frac{36}{1}$ by $\frac{3}{3}$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} + \frac{- 10 \cdot 3}{3} + \frac{36}{1} \cdot \frac{3}{3}$

Step 5.2.2.9

Multiply $\frac{36}{1}$ by $\frac{3}{3}$ .

$f (- 2) = \frac{8}{3} + \frac{- 16 \cdot 3}{3} + \frac{- 10 \cdot 3}{3} + \frac{36 \cdot 3}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$f (- 2) = \frac{8 - 16 \cdot 3 - 10 \cdot 3 + 36 \cdot 3}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Multiply $- 16$ by $3$ .

$f (- 2) = \frac{8 - 48 - 10 \cdot 3 + 36 \cdot 3}{3}$

Step 5.2.4.2

Multiply $- 10$ by $3$ .

$f (- 2) = \frac{8 - 48 - 30 + 36 \cdot 3}{3}$

Step 5.2.4.3

Multiply $36$ by $3$ .

$f (- 2) = \frac{8 - 48 - 30 + 108}{3}$

Step 5.2.5

Simplify by adding and subtracting.

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Step 5.2.5.1

Subtract $48$ from $8$ .

$f (- 2) = \frac{- 40 - 30 + 108}{3}$

Step 5.2.5.2

Subtract $30$ from $- 40$ .

$f (- 2) = \frac{- 70 + 108}{3}$

Step 5.2.5.3

Add $- 70$ and $108$ .

$f (- 2) = \frac{38}{3}$

Step 5.2.6

The final answer is $\frac{38}{3}$ .

$\frac{38}{3}$

Step 5.3

Convert $\frac{38}{3}$ to decimal.

$y = 12.\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ - 6 & - 66 \\ - 5 & - 47.333 \\ - 4 & - 26.667 \\ - 3 & - 6 \\ - 2 & 12.667 \end{array}$

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

$\begin{array}{c} x & y \\ - 6 & - 66 \\ - 5 & - 47.333 \\ - 4 & - 26.667 \\ - 3 & - 6 \\ - 2 & 12.667 \end{array}$

Step 8