Pre-Algebra Examples

$f (x) = \sqrt{x^{2} + x - 12}$

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Subtract $3$ from $- 4$ .

$f (- 4) = \sqrt{- 7 (- 4 + 4)}$

Step 1.2.2

Add $- 4$ and $4$ .

$f (- 4) = \sqrt{- 7 \cdot 0}$

Step 1.2.3

Multiply $- 7$ by $0$ .

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

Step 1.2.4

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

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

Step 1.2.5

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

$f (- 4) = 0$

Step 1.2.6

The final answer is $0$ .

$0$

Step 1.3

The $y$ value at $x = - 4$ is $0$ .

$y = 0$

Step 2

Find the $y$ value at $x = - 13$ .

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Step 2.1

Replace the variable $x$ with $- 13$ in the expression.

$f (- 13) = \sqrt{((- 13) - 3) ((- 13) + 4)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $3$ from $- 13$ .

$f (- 13) = \sqrt{- 16 (- 13 + 4)}$

Step 2.2.2

Add $- 13$ and $4$ .

$f (- 13) = \sqrt{- 16 \cdot - 9}$

Step 2.2.3

Multiply $- 16$ by $- 9$ .

$f (- 13) = \sqrt{144}$

Step 2.2.4

Rewrite $144$ as $12^{2}$ .

$f (- 13) = \sqrt{12^{2}}$

Step 2.2.5

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

$f (- 13) = 12$

Step 2.2.6

The final answer is $12$ .

$12$

Step 2.3

The $y$ value at $x = - 13$ is $12$ .

$y = 12$

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.

$f (- 5) = \sqrt{((- 5) - 3) ((- 5) + 4)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $3$ from $- 5$ .

$f (- 5) = \sqrt{- 8 (- 5 + 4)}$

Step 3.2.2

Add $- 5$ and $4$ .

$f (- 5) = \sqrt{- 8 \cdot - 1}$

Step 3.2.3

Multiply $- 8$ by $- 1$ .

$f (- 5) = \sqrt{8}$

Step 3.2.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.2.4.1

Factor $4$ out of $8$ .

$f (- 5) = \sqrt{4 (2)}$

Step 3.2.4.2

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

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

Step 3.2.5

Pull terms out from under the radical.

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

Step 3.2.6

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

$2 \sqrt{2}$

Step 3.3

The $y$ value at $x = - 5$ is $2 \sqrt{2}$ .

$y = 2 \sqrt{2}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $3$ from $3$ .

$f (3) = \sqrt{0 (3 + 4)}$

Step 4.2.2

Add $3$ and $4$ .

$f (3) = \sqrt{0 \cdot 7}$

Step 4.2.3

Multiply $0$ by $7$ .

$f (3) = \sqrt{0}$

Step 4.2.4

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

$f (3) = \sqrt{0^{2}}$

Step 4.2.5

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

$f (3) = 0$

Step 4.2.6

The final answer is $0$ .

$0$

Step 4.3

The $y$ value at $x = 3$ is $0$ .

$y = 0$

Step 5

Find the $y$ value at $x = 12$ .

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Step 5.1

Replace the variable $x$ with $12$ in the expression.

$f (12) = \sqrt{((12) - 3) ((12) + 4)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Subtract $3$ from $12$ .

$f (12) = \sqrt{9 (12 + 4)}$

Step 5.2.2

Add $12$ and $4$ .

$f (12) = \sqrt{9 \cdot 16}$

Step 5.2.3

Multiply $9$ by $16$ .

$f (12) = \sqrt{144}$

Step 5.2.4

Rewrite $144$ as $12^{2}$ .

$f (12) = \sqrt{12^{2}}$

Step 5.2.5

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

$f (12) = 12$

Step 5.2.6

The final answer is $12$ .

$12$

Step 5.3

The $y$ value at $x = 12$ is $12$ .

$y = 12$

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.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Subtract $3$ from $4$ .

$f (4) = \sqrt{1 (4 + 4)}$

Step 6.2.2

Multiply $4 + 4$ by $1$ .

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

Step 6.2.3

Add $4$ and $4$ .

$f (4) = \sqrt{8}$

Step 6.2.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.2.4.1

Factor $4$ out of $8$ .

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

Step 6.2.4.2

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

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

Step 6.2.5

Pull terms out from under the radical.

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

Step 6.2.6

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

$2 \sqrt{2}$

Step 6.3

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

$y = 2 \sqrt{2}$

Step 7

List the points to graph.

$(- 4, 0), (- 13, 12), (- 5, 2.82842712), (3, 0), (12, 12), (4, 2.82842712)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 13 & 12 \\ - 5 & 2.828 \\ - 4 & 0 \\ 3 & 0 \\ 4 & 2.828 \\ 12 & 12 \end{array}$

Step 9