Algebra Examples

$y = 2 + \frac{3}{2} \sqrt{5 - x}$

Step 1

Find the domain for $y = 2 + \frac{3}{2} \cdot \sqrt{5 - x}$ 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 $\sqrt{5 - x}$ greater than or equal to $0$ to find where the expression is defined.

$5 - x \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

Step 1.2.2

Divide each term in $- x \geq - 5$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 5}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 5}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 5$ by $- 1$ .

$x \leq 5$

Step 1.3

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

Interval Notation:

$(- \infty, 5]$

Set-Builder Notation:

${x | x \leq 5}$

Interval Notation:

$(- \infty, 5]$

Set-Builder Notation:

${x | x \leq 5}$

Step 2

To find the radical expression end point, substitute the $x$ value $5$ , which is the least value in the domain, into $f (x) = \frac{4 + 3 \sqrt{5 - x}}{2}$ .

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Step 2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{4 + 3 \sqrt{5 - (5)}}{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $- 1$ by $5$ .

$f (5) = \frac{4 + 3 \sqrt{5 - 5}}{2}$

Step 2.2.1.2

Subtract $5$ from $5$ .

$f (5) = \frac{4 + 3 \sqrt{0}}{2}$

Step 2.2.1.3

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

$f (5) = \frac{4 + 3 \sqrt{0^{2}}}{2}$

Step 2.2.1.4

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

$f (5) = \frac{4 + 3 \cdot 0}{2}$

Step 2.2.1.5

Multiply $3$ by $0$ .

$f (5) = \frac{4 + 0}{2}$

Step 2.2.1.6

Add $4$ and $0$ .

$f (5) = \frac{4}{2}$

Step 2.2.2

Divide $4$ by $2$ .

$f (5) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 3

The radical expression end point is $(5, 2)$ .

$(5, 2)$

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 $3$ into $f (x) = \frac{4 + 3 \sqrt{5 - x}}{2}$ . In this case, the point is $(3, \frac{4 + 3 \sqrt{2}}{2})$ .

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Step 4.1.1

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

$f (3) = \frac{4 + 3 \sqrt{5 - (3)}}{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Multiply $- 1$ by $3$ .

$f (3) = \frac{4 + 3 \sqrt{5 - 3}}{2}$

Step 4.1.2.1.2

Subtract $3$ from $5$ .

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

Step 4.1.2.2

The final answer is $\frac{4 + 3 \sqrt{2}}{2}$ .

$y = \frac{4 + 3 \sqrt{2}}{2}$

Step 4.2

Substitute the $x$ value $4$ into $f (x) = \frac{4 + 3 \sqrt{5 - x}}{2}$ . In this case, the point is $(4, \frac{7}{2})$ .

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Step 4.2.1

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

$f (4) = \frac{4 + 3 \sqrt{5 - (4)}}{2}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Multiply $- 1$ by $4$ .

$f (4) = \frac{4 + 3 \sqrt{5 - 4}}{2}$

Step 4.2.2.1.2

Subtract $4$ from $5$ .

$f (4) = \frac{4 + 3 \sqrt{1}}{2}$

Step 4.2.2.1.3

Any root of $1$ is $1$ .

$f (4) = \frac{4 + 3 \cdot 1}{2}$

Step 4.2.2.1.4

Multiply $3$ by $1$ .

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

Step 4.2.2.1.5

Add $4$ and $3$ .

$f (4) = \frac{7}{2}$

Step 4.2.2.2

The final answer is $\frac{7}{2}$ .

$y = \frac{7}{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(5, 2), (3, 4.12), (4, 3.5)$

$\begin{array}{c} x & y \\ 3 & 4.12 \\ 4 & 3.5 \\ 5 & 2 \end{array}$

Step 5