Precalculus Examples

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

Step 1

Find the $y$ value at $x = - 1$ .

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Step 1.1

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

$f (- 1) = \sqrt{{(- 1)}^{2} - 6 \cdot - 1 + 2}$

Step 1.2

Simplify the result.

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Step 1.2.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = \sqrt{1 - 6 \cdot - 1 + 2}$

Step 1.2.2

Multiply $- 6$ by $- 1$ .

$f (- 1) = \sqrt{1 + 6 + 2}$

Step 1.2.3

Add $1$ and $6$ .

$f (- 1) = \sqrt{7 + 2}$

Step 1.2.4

Add $7$ and $2$ .

$f (- 1) = \sqrt{9}$

Step 1.2.5

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.6

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

$f (- 1) = 3$

Step 1.2.7

The final answer is $3$ .

$3$

Step 1.3

The $y$ value at $x = - 1$ is $3$ .

$y = 3$

Step 2

Find the $y$ value at $x = 0$ .

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Step 2.1

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

$f (0) = \sqrt{{(0)}^{2} - 6 \cdot 0 + 2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \sqrt{0 - 6 \cdot 0 + 2}$

Step 2.2.2

Multiply $- 6$ by $0$ .

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

Step 2.2.3

Add $0$ and $0$ .

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

Step 2.2.4

Add $0$ and $2$ .

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

Step 2.2.5

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

$\sqrt{2}$

Step 2.3

The $y$ value at $x = 0$ is $\sqrt{2}$ .

$y = \sqrt{2}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

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

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

Step 3.2.2

Multiply $- 6$ by $- 2$ .

$f (- 2) = \sqrt{4 + 12 + 2}$

Step 3.2.3

Add $4$ and $12$ .

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

Step 3.2.4

Add $16$ and $2$ .

$f (- 2) = \sqrt{18}$

Step 3.2.5

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.2.5.1

Factor $9$ out of $18$ .

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

Step 3.2.5.2

Rewrite $9$ as $3^{2}$ .

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

Step 3.2.6

Pull terms out from under the radical.

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

Step 3.2.7

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

$3 \sqrt{2}$

Step 3.3

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

$y = 3 \sqrt{2}$

Step 4

Find the $y$ value at $x = 7$ .

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Step 4.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \sqrt{{(7)}^{2} - 6 \cdot 7 + 2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $7$ to the power of $2$ .

$f (7) = \sqrt{49 - 6 \cdot 7 + 2}$

Step 4.2.2

Multiply $- 6$ by $7$ .

$f (7) = \sqrt{49 - 42 + 2}$

Step 4.2.3

Subtract $42$ from $49$ .

$f (7) = \sqrt{7 + 2}$

Step 4.2.4

Add $7$ and $2$ .

$f (7) = \sqrt{9}$

Step 4.2.5

Rewrite $9$ as $3^{2}$ .

$f (7) = \sqrt{3^{2}}$

Step 4.2.6

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

$f (7) = 3$

Step 4.2.7

The final answer is $3$ .

$3$

Step 4.3

The $y$ value at $x = 7$ is $3$ .

$y = 3$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Raise $6$ to the power of $2$ .

$f (6) = \sqrt{36 - 6 \cdot 6 + 2}$

Step 5.2.2

Multiply $- 6$ by $6$ .

$f (6) = \sqrt{36 - 36 + 2}$

Step 5.2.3

Subtract $36$ from $36$ .

$f (6) = \sqrt{0 + 2}$

Step 5.2.4

Add $0$ and $2$ .

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

Step 5.2.5

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

$\sqrt{2}$

Step 5.3

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

$y = \sqrt{2}$

Step 6

Find the $y$ value at $x = 8$ .

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Step 6.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \sqrt{{(8)}^{2} - 6 \cdot 8 + 2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Raise $8$ to the power of $2$ .

$f (8) = \sqrt{64 - 6 \cdot 8 + 2}$

Step 6.2.2

Multiply $- 6$ by $8$ .

$f (8) = \sqrt{64 - 48 + 2}$

Step 6.2.3

Subtract $48$ from $64$ .

$f (8) = \sqrt{16 + 2}$

Step 6.2.4

Add $16$ and $2$ .

$f (8) = \sqrt{18}$

Step 6.2.5

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 6.2.5.1

Factor $9$ out of $18$ .

$f (8) = \sqrt{9 (2)}$

Step 6.2.5.2

Rewrite $9$ as $3^{2}$ .

$f (8) = \sqrt{3^{2} \cdot 2}$

Step 6.2.6

Pull terms out from under the radical.

$f (8) = 3 \sqrt{2}$

Step 6.2.7

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

$3 \sqrt{2}$

Step 6.3

The $y$ value at $x = 8$ is $3 \sqrt{2}$ .

$y = 3 \sqrt{2}$

Step 7

List the points to graph.

$(- 1, 3), (0, 1.41421356), (- 2, 4.24264068), (7, 3), (6, 1.41421356), (8, 4.24264068)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 2 & 4.243 \\ - 1 & 3 \\ 0 & 1.414 \\ 6 & 1.414 \\ 7 & 3 \\ 8 & 4.243 \end{array}$

Step 9