Trigonometry Examples

$y = x^{3} + 6 x^{2} + 19$

Step 1

Find the point at $x = - 4$ .

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Step 1.1

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

$f (- 4) = {(- 4)}^{3} + 6 {(- 4)}^{2} + 19$

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

$f (- 4) = - 64 + 6 {(- 4)}^{2} + 19$

Step 1.2.1.2

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

$f (- 4) = - 64 + 6 \cdot 16 + 19$

Step 1.2.1.3

Multiply $6$ by $16$ .

$f (- 4) = - 64 + 96 + 19$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $- 64$ and $96$ .

$f (- 4) = 32 + 19$

Step 1.2.2.2

Add $32$ and $19$ .

$f (- 4) = 51$

Step 1.2.3

The final answer is $51$ .

$51$

Step 1.3

Convert $51$ to decimal.

$y = 51$

Step 2

Find the point at $x = - 3$ .

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Step 2.1

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

$f (- 3) = {(- 3)}^{3} + 6 {(- 3)}^{2} + 19$

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 $3$ .

$f (- 3) = - 27 + 6 {(- 3)}^{2} + 19$

Step 2.2.1.2

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

$f (- 3) = - 27 + 6 \cdot 9 + 19$

Step 2.2.1.3

Multiply $6$ by $9$ .

$f (- 3) = - 27 + 54 + 19$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $- 27$ and $54$ .

$f (- 3) = 27 + 19$

Step 2.2.2.2

Add $27$ and $19$ .

$f (- 3) = 46$

Step 2.2.3

The final answer is $46$ .

$46$

Step 2.3

Convert $46$ to decimal.

$y = 46$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

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

$f (- 2) = {(- 2)}^{3} + 6 {(- 2)}^{2} + 19$

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

$f (- 2) = - 8 + 6 {(- 2)}^{2} + 19$

Step 3.2.1.2

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

$f (- 2) = - 8 + 6 \cdot 4 + 19$

Step 3.2.1.3

Multiply $6$ by $4$ .

$f (- 2) = - 8 + 24 + 19$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $- 8$ and $24$ .

$f (- 2) = 16 + 19$

Step 3.2.2.2

Add $16$ and $19$ .

$f (- 2) = 35$

Step 3.2.3

The final answer is $35$ .

$35$

Step 3.3

Convert $35$ to decimal.

$y = 35$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

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

$f (- 1) = {(- 1)}^{3} + 6 {(- 1)}^{2} + 19$

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

$f (- 1) = - 1 + 6 {(- 1)}^{2} + 19$

Step 4.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 1 + 6 \cdot 1 + 19$

Step 4.2.1.3

Multiply $6$ by $1$ .

$f (- 1) = - 1 + 6 + 19$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $- 1$ and $6$ .

$f (- 1) = 5 + 19$

Step 4.2.2.2

Add $5$ and $19$ .

$f (- 1) = 24$

Step 4.2.3

The final answer is $24$ .

$24$

Step 4.3

Convert $24$ to decimal.

$y = 24$

Step 5

Find the point at $x = 0$ .

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Step 5.1

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

$f (0) = {(0)}^{3} + 6 {(0)}^{2} + 19$

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

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

$f (0) = 0 + 6 {(0)}^{2} + 19$

Step 5.2.1.2

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

$f (0) = 0 + 6 \cdot 0 + 19$

Step 5.2.1.3

Multiply $6$ by $0$ .

$f (0) = 0 + 0 + 19$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 19$

Step 5.2.2.2

Add $0$ and $19$ .

$f (0) = 19$

Step 5.2.3

The final answer is $19$ .

$19$

Step 5.3

Convert $19$ to decimal.

$y = 19$

Step 6

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

$\begin{array}{c} x & y \\ - 4 & 51 \\ - 3 & 46 \\ - 2 & 35 \\ - 1 & 24 \\ 0 & 19 \end{array}$

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

$\begin{array}{c} x & y \\ - 4 & 51 \\ - 3 & 46 \\ - 2 & 35 \\ - 1 & 24 \\ 0 & 19 \end{array}$

Step 8