Pre-Algebra Examples

$h (x) = \sqrt{- 4 x + 8.5}$

Step 1

Find the domain for $y = \sqrt{- 4 x + 8.5}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{- 8 x + 17}$ greater than or equal to $0$ to find where the expression is defined.

$- 8 x + 17 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $17$ from both sides of the inequality.

$- 8 x \geq - 17$

Step 1.2.2

Divide each term in $- 8 x \geq - 17$ by $- 8$ and simplify.

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Step 1.2.2.1

Divide each term in $- 8 x \geq - 17$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{- 17}{- 8}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{- 17}{- 8}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 17}{- 8}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{17}{8}$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, \frac{17}{8}]$

Set-Builder Notation:

${x | x \leq \frac{17}{8}}$

Interval Notation:

$(- \infty, \frac{17}{8}]$

Set-Builder Notation:

${x | x \leq \frac{17}{8}}$

Step 2

To find the radical expression end point, substitute the $x$ value $\frac{17}{8}$ , which is the least value in the domain, into $f (x) = 0.70710678 \sqrt{- 8 x + 17}$ .

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Step 2.1

Replace the variable $x$ with $\frac{17}{8}$ in the expression.

$f (\frac{17}{8}) = 0.70710678 \sqrt{- 8 (\frac{17}{8}) + 17}$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Factor $8$ out of $- 8$ .

$f (\frac{17}{8}) = 0.70710678 \sqrt{8 (- 1) (\frac{17}{8}) + 17}$

Step 2.2.1.2

Cancel the common factor.

$f (\frac{17}{8}) = 0.70710678 \sqrt{8 \cdot (- 1 (\frac{17}{8})) + 17}$

Step 2.2.1.3

Rewrite the expression.

$f (\frac{17}{8}) = 0.70710678 \sqrt{- 1 \cdot 17 + 17}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Multiply $- 1$ by $17$ .

$f (\frac{17}{8}) = 0.70710678 \sqrt{- 17 + 17}$

Step 2.2.2.2

Add $- 17$ and $17$ .

$f (\frac{17}{8}) = 0.70710678 \sqrt{0}$

Step 2.2.2.3

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

$f (\frac{17}{8}) = 0.70710678 \sqrt{0^{2}}$

Step 2.2.2.4

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

$f (\frac{17}{8}) = 0.70710678 \cdot 0$

Step 2.2.2.5

Multiply $0.70710678$ by $0$ .

$f (\frac{17}{8}) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(\frac{17}{8}, 0)$ .

$(\frac{17}{8}, 0)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $2$ into $f (x) = 0.70710678 \sqrt{- 8 x + 17}$ . In this case, the point is $(2, 0.70710678)$ .

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Step 4.1.1

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

$f (2) = 0.70710678 \sqrt{- 8 \cdot 2 + 17}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 8$ by $2$ .

$f (2) = 0.70710678 \sqrt{- 16 + 17}$

Step 4.1.2.2

Add $- 16$ and $17$ .

$f (2) = 0.70710678 \sqrt{1}$

Step 4.1.2.3

Any root of $1$ is $1$ .

$f (2) = 0.70710678 \cdot 1$

Step 4.1.2.4

Multiply $0.70710678$ by $1$ .

$f (2) = 0.70710678$

Step 4.1.2.5

The final answer is $0.70710678$ .

$y = 0.70710678$

Step 4.2

The square root can be graphed using the points around the vertex $(2.125, 0), (2, 0.71)$

$\begin{array}{c} x & y \\ 2 & 0.71 \\ 2.125 & 0 \end{array}$

Step 5