Trigonometry Examples

$f (x) = - 6 (x - 3) {(x + 4)}^{2}$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

$f (- 3) = - 6 \cdot - 3 {((- 3) + 4)}^{2} + 18 {((- 3) + 4)}^{2}$

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

Multiply $- 6$ by $- 3$ .

$f (- 3) = 18 {((- 3) + 4)}^{2} + 18 {((- 3) + 4)}^{2}$

Step 1.2.1.2

Add $- 3$ and $4$ .

$f (- 3) = 18 \cdot 1^{2} + 18 {((- 3) + 4)}^{2}$

Step 1.2.1.3

One to any power is one.

$f (- 3) = 18 \cdot 1 + 18 {((- 3) + 4)}^{2}$

Step 1.2.1.4

Multiply $18$ by $1$ .

$f (- 3) = 18 + 18 {((- 3) + 4)}^{2}$

Step 1.2.1.5

Add $- 3$ and $4$ .

$f (- 3) = 18 + 18 \cdot 1^{2}$

Step 1.2.1.6

One to any power is one.

$f (- 3) = 18 + 18 \cdot 1$

Step 1.2.1.7

Multiply $18$ by $1$ .

$f (- 3) = 18 + 18$

Step 1.2.2

Add $18$ and $18$ .

$f (- 3) = 36$

Step 1.2.3

The final answer is $36$ .

$36$

Step 1.3

Convert $36$ to decimal.

$y = 36$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

$f (- 2) = - 6 \cdot - 2 {((- 2) + 4)}^{2} + 18 {((- 2) + 4)}^{2}$

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

Multiply $- 6$ by $- 2$ .

$f (- 2) = 12 {((- 2) + 4)}^{2} + 18 {((- 2) + 4)}^{2}$

Step 2.2.1.2

Add $- 2$ and $4$ .

$f (- 2) = 12 \cdot 2^{2} + 18 {((- 2) + 4)}^{2}$

Step 2.2.1.3

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

$f (- 2) = 12 \cdot 4 + 18 {((- 2) + 4)}^{2}$

Step 2.2.1.4

Multiply $12$ by $4$ .

$f (- 2) = 48 + 18 {((- 2) + 4)}^{2}$

Step 2.2.1.5

Add $- 2$ and $4$ .

$f (- 2) = 48 + 18 \cdot 2^{2}$

Step 2.2.1.6

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

$f (- 2) = 48 + 18 \cdot 4$

Step 2.2.1.7

Multiply $18$ by $4$ .

$f (- 2) = 48 + 72$

Step 2.2.2

Add $48$ and $72$ .

$f (- 2) = 120$

Step 2.2.3

The final answer is $120$ .

$120$

Step 2.3

Convert $120$ to decimal.

$y = 120$

Step 3

Find the point at $x = - 1$ .

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Step 3.1

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

$f (- 1) = - 6 \cdot - 1 {((- 1) + 4)}^{2} + 18 {((- 1) + 4)}^{2}$

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

Multiply $- 6$ by $- 1$ .

$f (- 1) = 6 {((- 1) + 4)}^{2} + 18 {((- 1) + 4)}^{2}$

Step 3.2.1.2

Add $- 1$ and $4$ .

$f (- 1) = 6 \cdot 3^{2} + 18 {((- 1) + 4)}^{2}$

Step 3.2.1.3

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

$f (- 1) = 6 \cdot 9 + 18 {((- 1) + 4)}^{2}$

Step 3.2.1.4

Multiply $6$ by $9$ .

$f (- 1) = 54 + 18 {((- 1) + 4)}^{2}$

Step 3.2.1.5

Add $- 1$ and $4$ .

$f (- 1) = 54 + 18 \cdot 3^{2}$

Step 3.2.1.6

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

$f (- 1) = 54 + 18 \cdot 9$

Step 3.2.1.7

Multiply $18$ by $9$ .

$f (- 1) = 54 + 162$

Step 3.2.2

Add $54$ and $162$ .

$f (- 1) = 216$

Step 3.2.3

The final answer is $216$ .

$216$

Step 3.3

Convert $216$ to decimal.

$y = 216$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = - 6 \cdot 0 {((0) + 4)}^{2} + 18 {((0) + 4)}^{2}$

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

Multiply $- 6$ by $0$ .

$f (0) = 0 {((0) + 4)}^{2} + 18 {((0) + 4)}^{2}$

Step 4.2.1.2

Add $0$ and $4$ .

$f (0) = 0 \cdot 4^{2} + 18 {((0) + 4)}^{2}$

Step 4.2.1.3

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

$f (0) = 0 \cdot 16 + 18 {((0) + 4)}^{2}$

Step 4.2.1.4

Multiply $0$ by $16$ .

$f (0) = 0 + 18 {((0) + 4)}^{2}$

Step 4.2.1.5

Add $0$ and $4$ .

$f (0) = 0 + 18 \cdot 4^{2}$

Step 4.2.1.6

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

$f (0) = 0 + 18 \cdot 16$

Step 4.2.1.7

Multiply $18$ by $16$ .

$f (0) = 0 + 288$

Step 4.2.2

Add $0$ and $288$ .

$f (0) = 288$

Step 4.2.3

The final answer is $288$ .

$288$

Step 4.3

Convert $288$ to decimal.

$y = 288$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

$f (1) = - 6 \cdot 1 {((1) + 4)}^{2} + 18 {((1) + 4)}^{2}$

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 $- 6$ by $1$ .

$f (1) = - 6 {((1) + 4)}^{2} + 18 {((1) + 4)}^{2}$

Step 5.2.1.2

Add $1$ and $4$ .

$f (1) = - 6 \cdot 5^{2} + 18 {((1) + 4)}^{2}$

Step 5.2.1.3

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

$f (1) = - 6 \cdot 25 + 18 {((1) + 4)}^{2}$

Step 5.2.1.4

Multiply $- 6$ by $25$ .

$f (1) = - 150 + 18 {((1) + 4)}^{2}$

Step 5.2.1.5

Add $1$ and $4$ .

$f (1) = - 150 + 18 \cdot 5^{2}$

Step 5.2.1.6

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

$f (1) = - 150 + 18 \cdot 25$

Step 5.2.1.7

Multiply $18$ by $25$ .

$f (1) = - 150 + 450$

Step 5.2.2

Add $- 150$ and $450$ .

$f (1) = 300$

Step 5.2.3

The final answer is $300$ .

$300$

Step 5.3

Convert $300$ to decimal.

$y = 300$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 36 \\ - 2 & 120 \\ - 1 & 216 \\ 0 & 288 \\ 1 & 300 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 36 \\ - 2 & 120 \\ - 1 & 216 \\ 0 & 288 \\ 1 & 300 \end{array}$

Step 8