Pre-Algebra Examples

$y = 2 + \sqrt{x^{2} - 3}$

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.

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

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

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

Step 1.2.1.2

Subtract $3$ from $4$ .

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

Step 1.2.1.3

Any root of $1$ is $1$ .

$f (- 2) = 2 + 1$

Step 1.2.2

Add $2$ and $1$ .

$f (- 2) = 3$

Step 1.2.3

The final answer is $3$ .

$3$

Step 1.3

The $y$ value at $x = - 2$ is $3$ .

$y = 3$

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.

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

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 $2$ .

$f (- 3) = 2 + \sqrt{9 - 3}$

Step 2.2.1.2

Subtract $3$ from $9$ .

$f (- 3) = 2 + \sqrt{6}$

Step 2.2.2

The final answer is $2 + \sqrt{6}$ .

$2 + \sqrt{6}$

Step 2.3

The $y$ value at $x = - 3$ is $2 + \sqrt{6}$ .

$y = 2 + \sqrt{6}$

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.

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

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

$f (- 4) = 2 + \sqrt{16 - 3}$

Step 3.2.1.2

Subtract $3$ from $16$ .

$f (- 4) = 2 + \sqrt{13}$

Step 3.2.2

The final answer is $2 + \sqrt{13}$ .

$2 + \sqrt{13}$

Step 3.3

The $y$ value at $x = - 4$ is $2 + \sqrt{13}$ .

$y = 2 + \sqrt{13}$

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.

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

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 $2$ to the power of $2$ .

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

Step 4.2.1.2

Subtract $3$ from $4$ .

$f (2) = 2 + \sqrt{1}$

Step 4.2.1.3

Any root of $1$ is $1$ .

$f (2) = 2 + 1$

Step 4.2.2

Add $2$ and $1$ .

$f (2) = 3$

Step 4.2.3

The final answer is $3$ .

$3$

Step 4.3

The $y$ value at $x = 2$ is $3$ .

$y = 3$

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.

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

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

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

$f (3) = 2 + \sqrt{9 - 3}$

Step 5.2.1.2

Subtract $3$ from $9$ .

$f (3) = 2 + \sqrt{6}$

Step 5.2.2

The final answer is $2 + \sqrt{6}$ .

$2 + \sqrt{6}$

Step 5.3

The $y$ value at $x = 3$ is $2 + \sqrt{6}$ .

$y = 2 + \sqrt{6}$

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.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$f (4) = 2 + \sqrt{16 - 3}$

Step 6.2.1.2

Subtract $3$ from $16$ .

$f (4) = 2 + \sqrt{13}$

Step 6.2.2

The final answer is $2 + \sqrt{13}$ .

$2 + \sqrt{13}$

Step 6.3

The $y$ value at $x = 4$ is $2 + \sqrt{13}$ .

$y = 2 + \sqrt{13}$

Step 7

List the points to graph.

$(- 2, 3), (- 3, 4.44948974), (- 4, 5.60555127), (2, 3), (3, 4.44948974), (4, 5.60555127)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 4 & 5.606 \\ - 3 & 4.449 \\ - 2 & 3 \\ 2 & 3 \\ 3 & 4.449 \\ 4 & 5.606 \end{array}$

Step 9