Trigonometry Examples

$y^{2} \cdot (6 y) - 12 x - 15 = 0$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = \frac{\sqrt[3]{4 (4 (- 2) + 5)}}{2}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Multiply $4$ by $- 2$ .

$f (- 2) = \frac{\sqrt[3]{4 (- 8 + 5)}}{2}$

Step 1.2.1.2

Add $- 8$ and $5$ .

$f (- 2) = \frac{\sqrt[3]{4 \cdot - 3}}{2}$

Step 1.2.1.3

Multiply $4$ by $- 3$ .

$f (- 2) = \frac{\sqrt[3]{- 12}}{2}$

Step 1.2.1.4

Rewrite $- 12$ as ${(- 1)}^{3} \cdot 12$ .

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Step 1.2.1.4.1

Rewrite $- 12$ as $- 1 (12)$ .

$f (- 2) = \frac{\sqrt[3]{- 1 \cdot 12}}{2}$

Step 1.2.1.4.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f (- 2) = \frac{\sqrt[3]{{(- 1)}^{3} \cdot 12}}{2}$

Step 1.2.1.5

Pull terms out from under the radical.

$f (- 2) = \frac{- 1 \sqrt[3]{12}}{2}$

Step 1.2.1.6

Rewrite $- 1 \sqrt[3]{12}$ as $- \sqrt[3]{12}$ .

$f (- 2) = \frac{- \sqrt[3]{12}}{2}$

Step 1.2.2

Move the negative in front of the fraction.

$f (- 2) = - \frac{\sqrt[3]{12}}{2}$

Step 1.2.3

The final answer is $- \frac{\sqrt[3]{12}}{2}$ .

$- \frac{\sqrt[3]{12}}{2}$

Step 1.3

Convert $- \frac{\sqrt[3]{12}}{2}$ to decimal.

$y = - 1.14471424$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = \frac{\sqrt[3]{4 (4 (- 1) + 5)}}{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $4$ by $- 1$ .

$f (- 1) = \frac{\sqrt[3]{4 (- 4 + 5)}}{2}$

Step 2.2.1.2

Add $- 4$ and $5$ .

$f (- 1) = \frac{\sqrt[3]{4 \cdot 1}}{2}$

Step 2.2.1.3

Multiply $4$ by $1$ .

$f (- 1) = \frac{\sqrt[3]{4}}{2}$

Step 2.2.2

The final answer is $\frac{\sqrt[3]{4}}{2}$ .

$\frac{\sqrt[3]{4}}{2}$

Step 2.3

Convert $\frac{\sqrt[3]{4}}{2}$ to decimal.

$y = 0.79370052$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = \frac{\sqrt[3]{4 (4 (0) + 5)}}{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Multiply $4$ by $0$ .

$f (0) = \frac{\sqrt[3]{4 (0 + 5)}}{2}$

Step 3.2.1.2

Add $0$ and $5$ .

$f (0) = \frac{\sqrt[3]{4 \cdot 5}}{2}$

Step 3.2.1.3

Multiply $4$ by $5$ .

$f (0) = \frac{\sqrt[3]{20}}{2}$

Step 3.2.2

The final answer is $\frac{\sqrt[3]{20}}{2}$ .

$\frac{\sqrt[3]{20}}{2}$

Step 3.3

Convert $\frac{\sqrt[3]{20}}{2}$ to decimal.

$y = 1.3572088$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = \frac{\sqrt[3]{4 (4 (1) + 5)}}{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Multiply $4$ by $1$ .

$f (1) = \frac{\sqrt[3]{4 (4 + 5)}}{2}$

Step 4.2.1.2

Add $4$ and $5$ .

$f (1) = \frac{\sqrt[3]{4 \cdot 9}}{2}$

Step 4.2.1.3

Multiply $4$ by $9$ .

$f (1) = \frac{\sqrt[3]{36}}{2}$

Step 4.2.2

The final answer is $\frac{\sqrt[3]{36}}{2}$ .

$\frac{\sqrt[3]{36}}{2}$

Step 4.3

Convert $\frac{\sqrt[3]{36}}{2}$ to decimal.

$y = 1.65096362$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = \frac{\sqrt[3]{4 (4 (2) + 5)}}{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $4$ by $2$ .

$f (2) = \frac{\sqrt[3]{4 (8 + 5)}}{2}$

Step 5.2.1.2

Add $8$ and $5$ .

$f (2) = \frac{\sqrt[3]{4 \cdot 13}}{2}$

Step 5.2.1.3

Multiply $4$ by $13$ .

$f (2) = \frac{\sqrt[3]{52}}{2}$

Step 5.2.2

The final answer is $\frac{\sqrt[3]{52}}{2}$ .

$\frac{\sqrt[3]{52}}{2}$

Step 5.3

Convert $\frac{\sqrt[3]{52}}{2}$ to decimal.

$y = 1.86625557$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & - 1.145 \\ - 1 & 0.794 \\ 0 & 1.357 \\ 1 & 1.651 \\ 2 & 1.866 \end{array}$

Step 7

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

Not a polynomial

$\begin{array}{c} x & y \\ - 2 & - 1.145 \\ - 1 & 0.794 \\ 0 & 1.357 \\ 1 & 1.651 \\ 2 & 1.866 \end{array}$

Step 8