Precalculus Examples

$\sqrt{x^{2} - 4}$

Step 1

Find the $y$ value at $x = - 2$ .

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Step 1.1

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

$f (- 2) = \sqrt{((- 2) + 2) ((- 2) - 2)}$

Step 1.2

Simplify the result.

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Step 1.2.1

Add $- 2$ and $2$ .

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

Step 1.2.2

Subtract $2$ from $- 2$ .

$f (- 2) = \sqrt{0 \cdot - 4}$

Step 1.2.3

Multiply $0$ by $- 4$ .

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

Step 1.2.4

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

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

Step 1.2.5

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

$f (- 2) = 0$

Step 1.2.6

The final answer is $0$ .

$0$

Step 1.3

The $y$ value at $x = - 2$ is $0$ .

$y = 0$

Step 2

Find the $y$ value at $x = - 3$ .

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Step 2.1

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

$f (- 3) = \sqrt{((- 3) + 2) ((- 3) - 2)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 3$ and $2$ .

$f (- 3) = \sqrt{- 1 (- 3 - 2)}$

Step 2.2.2

Subtract $2$ from $- 3$ .

$f (- 3) = \sqrt{- 1 \cdot - 5}$

Step 2.2.3

Multiply $- 1$ by $- 5$ .

$f (- 3) = \sqrt{5}$

Step 2.2.4

The final answer is $\sqrt{5}$ .

$\sqrt{5}$

Step 2.3

The $y$ value at $x = - 3$ is $\sqrt{5}$ .

$y = \sqrt{5}$

Step 3

Find the $y$ value at $x = - 4$ .

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Step 3.1

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

$f (- 4) = \sqrt{((- 4) + 2) ((- 4) - 2)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $- 4$ and $2$ .

$f (- 4) = \sqrt{- 2 (- 4 - 2)}$

Step 3.2.2

Subtract $2$ from $- 4$ .

$f (- 4) = \sqrt{- 2 \cdot - 6}$

Step 3.2.3

Multiply $- 2$ by $- 6$ .

$f (- 4) = \sqrt{12}$

Step 3.2.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.2.4.1

Factor $4$ out of $12$ .

$f (- 4) = \sqrt{4 (3)}$

Step 3.2.4.2

Rewrite $4$ as $2^{2}$ .

$f (- 4) = \sqrt{2^{2} \cdot 3}$

Step 3.2.5

Pull terms out from under the radical.

$f (- 4) = 2 \sqrt{3}$

Step 3.2.6

The final answer is $2 \sqrt{3}$ .

$2 \sqrt{3}$

Step 3.3

The $y$ value at $x = - 4$ is $2 \sqrt{3}$ .

$y = 2 \sqrt{3}$

Step 4

Find the $y$ value at $x = 2$ .

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Step 4.1

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

$f (2) = \sqrt{((2) + 2) ((2) - 2)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $2$ and $2$ .

$f (2) = \sqrt{4 (2 - 2)}$

Step 4.2.2

Subtract $2$ from $2$ .

$f (2) = \sqrt{4 \cdot 0}$

Step 4.2.3

Multiply $4$ by $0$ .

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

Step 4.2.4

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

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

Step 4.2.5

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

$f (2) = 0$

Step 4.2.6

The final answer is $0$ .

$0$

Step 4.3

The $y$ value at $x = 2$ is $0$ .

$y = 0$

Step 5

Find the $y$ value at $x = 3$ .

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Step 5.1

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

$f (3) = \sqrt{((3) + 2) ((3) - 2)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Add $3$ and $2$ .

$f (3) = \sqrt{5 (3 - 2)}$

Step 5.2.2

Subtract $2$ from $3$ .

$f (3) = \sqrt{5 \cdot 1}$

Step 5.2.3

Multiply $5$ by $1$ .

$f (3) = \sqrt{5}$

Step 5.2.4

The final answer is $\sqrt{5}$ .

$\sqrt{5}$

Step 5.3

The $y$ value at $x = 3$ is $\sqrt{5}$ .

$y = \sqrt{5}$

Step 6

Find the $y$ value at $x = 4$ .

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Step 6.1

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

$f (4) = \sqrt{((4) + 2) ((4) - 2)}$

Step 6.2

Simplify the result.

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Step 6.2.1

Add $4$ and $2$ .

$f (4) = \sqrt{6 (4 - 2)}$

Step 6.2.2

Subtract $2$ from $4$ .

$f (4) = \sqrt{6 \cdot 2}$

Step 6.2.3

Multiply $6$ by $2$ .

$f (4) = \sqrt{12}$

Step 6.2.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.2.4.1

Factor $4$ out of $12$ .

$f (4) = \sqrt{4 (3)}$

Step 6.2.4.2

Rewrite $4$ as $2^{2}$ .

$f (4) = \sqrt{2^{2} \cdot 3}$

Step 6.2.5

Pull terms out from under the radical.

$f (4) = 2 \sqrt{3}$

Step 6.2.6

The final answer is $2 \sqrt{3}$ .

$2 \sqrt{3}$

Step 6.3

The $y$ value at $x = 4$ is $2 \sqrt{3}$ .

$y = 2 \sqrt{3}$

Step 7

List the points to graph.

$(- 2, 0), (- 3, 2.23606797), (- 4, 3.46410161), (2, 0), (3, 2.23606797), (4, 3.46410161)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 4 & 3.464 \\ - 3 & 2.236 \\ - 2 & 0 \\ 2 & 0 \\ 3 & 2.236 \\ 4 & 3.464 \end{array}$

Step 9