Trigonometry Examples

$y = \sqrt{x^{4} - 25 x^{2}}$

Step 1

Find the $y$ value at $x = - 5$ .

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Step 1.1

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

$f (- 5) = | - 5 | \sqrt{((- 5) + 5) ((- 5) - 5)}$

Step 1.2

Simplify the result.

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Step 1.2.1

The absolute value is the distance between a number and zero. The distance between $- 5$ and $0$ is $5$ .

$f (- 5) = 5 \sqrt{((- 5) + 5) ((- 5) - 5)}$

Step 1.2.2

Add $- 5$ and $5$ .

$f (- 5) = 5 \sqrt{0 (- 5 - 5)}$

Step 1.2.3

Subtract $5$ from $- 5$ .

$f (- 5) = 5 \sqrt{0 \cdot - 10}$

Step 1.2.4

Multiply $0$ by $- 10$ .

$f (- 5) = 5 \sqrt{0}$

Step 1.2.5

Rewrite $0$ as $0^{2}$ .

$f (- 5) = 5 \sqrt{0^{2}}$

Step 1.2.6

Multiply by zero.

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Step 1.2.6.1

Pull terms out from under the radical, assuming positive real numbers.

$f (- 5) = 5 \cdot 0$

Step 1.2.6.2

Multiply $5$ by $0$ .

$f (- 5) = 0$

Step 1.2.7

The final answer is $0$ .

$0$

Step 1.3

The $y$ value at $x = - 5$ is $0$ .

$y = 0$

Step 2

Find the $y$ value at $x = - 6$ .

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Step 2.1

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

$f (- 6) = | - 6 | \sqrt{((- 6) + 5) ((- 6) - 5)}$

Step 2.2

Simplify the result.

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Step 2.2.1

The absolute value is the distance between a number and zero. The distance between $- 6$ and $0$ is $6$ .

$f (- 6) = 6 \sqrt{((- 6) + 5) ((- 6) - 5)}$

Step 2.2.2

Add $- 6$ and $5$ .

$f (- 6) = 6 \sqrt{- 1 (- 6 - 5)}$

Step 2.2.3

Subtract $5$ from $- 6$ .

$f (- 6) = 6 \sqrt{- 1 \cdot - 11}$

Step 2.2.4

Multiply $- 1$ by $- 11$ .

$f (- 6) = 6 \sqrt{11}$

Step 2.2.5

The final answer is $6 \sqrt{11}$ .

$6 \sqrt{11}$

Step 2.3

The $y$ value at $x = - 6$ is $6 \sqrt{11}$ .

$y = 6 \sqrt{11}$

Step 3

Find the $y$ value at $x = 5$ .

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Step 3.1

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

$f (5) = | 5 | \sqrt{((5) + 5) ((5) - 5)}$

Step 3.2

Simplify the result.

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Step 3.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$f (5) = 5 \sqrt{((5) + 5) ((5) - 5)}$

Step 3.2.2

Add $5$ and $5$ .

$f (5) = 5 \sqrt{10 (5 - 5)}$

Step 3.2.3

Subtract $5$ from $5$ .

$f (5) = 5 \sqrt{10 \cdot 0}$

Step 3.2.4

Multiply $10$ by $0$ .

$f (5) = 5 \sqrt{0}$

Step 3.2.5

Rewrite $0$ as $0^{2}$ .

$f (5) = 5 \sqrt{0^{2}}$

Step 3.2.6

Multiply by zero.

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Step 3.2.6.1

Pull terms out from under the radical, assuming positive real numbers.

$f (5) = 5 \cdot 0$

Step 3.2.6.2

Multiply $5$ by $0$ .

$f (5) = 0$

Step 3.2.7

The final answer is $0$ .

$0$

Step 3.3

The $y$ value at $x = 5$ is $0$ .

$y = 0$

Step 4

Find the $y$ value at $x = 6$ .

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Step 4.1

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

$f (6) = | 6 | \sqrt{((6) + 5) ((6) - 5)}$

Step 4.2

Simplify the result.

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Step 4.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$f (6) = 6 \sqrt{((6) + 5) ((6) - 5)}$

Step 4.2.2

Add $6$ and $5$ .

$f (6) = 6 \sqrt{11 (6 - 5)}$

Step 4.2.3

Subtract $5$ from $6$ .

$f (6) = 6 \sqrt{11 \cdot 1}$

Step 4.2.4

Multiply $11$ by $1$ .

$f (6) = 6 \sqrt{11}$

Step 4.2.5

The final answer is $6 \sqrt{11}$ .

$6 \sqrt{11}$

Step 4.3

The $y$ value at $x = 6$ is $6 \sqrt{11}$ .

$y = 6 \sqrt{11}$

Step 5

List the points to graph.

$(- 5, 0), (- 6, 19.89974874), (5, 0), (6, 19.89974874)$

Step 6

Select a few points to graph.

$\begin{array}{c} x & y \\ - 6 & 19.9 \\ - 5 & 0 \\ 5 & 0 \\ 6 & 19.9 \end{array}$

Step 7