Graph y=-5/2 square root of x+6

$y = - \frac{5}{2} \sqrt{x} + 6$

Step 1

Find the domain for $y = - \frac{5}{2} \cdot \sqrt{x} + 6$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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

$x \geq 0$

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = - \frac{5 \sqrt{x}}{2} + 6$ .

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

$f (0) = - \frac{5 \sqrt{0}}{2} + 6$

Simplify the result.

Simplify each term.

Simplify the numerator.

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

$f (0) = - \frac{5 \sqrt{0^{2}}}{2} + 6$

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

$f (0) = - \frac{5 \cdot 0}{2} + 6$

Multiply $5$ by $0$ .

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

Divide $0$ by $2$ .

$f (0) = - 0 + 6$

Multiply $- 1$ by $0$ .

$f (0) = 0 + 6$

Add $0$ and $6$ .

$f (0) = 6$

The final answer is $6$ .

$6$

Step 3

The radical expression end point is $(0, 6)$ .

$(0, 6)$

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.

Substitute the $x$ value $1$ into $f (x) = - \frac{5 \sqrt{x}}{2} + 6$ . In this case, the point is $(1, \frac{7}{2})$ .

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

$f (1) = - \frac{5 \sqrt{1}}{2} + 6$

Simplify the result.

Simplify each term.

Any root of $1$ is $1$ .

$f (1) = - \frac{5 \cdot 1}{2} + 6$

Multiply $5$ by $1$ .

$f (1) = - \frac{5}{2} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (1) = - \frac{5}{2} + 6 \cdot \frac{2}{2}$

Combine $6$ and $\frac{2}{2}$ .

$f (1) = - \frac{5}{2} + \frac{6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (1) = \frac{- 5 + 6 \cdot 2}{2}$

Simplify the numerator.

Multiply $6$ by $2$ .

$f (1) = \frac{- 5 + 12}{2}$

Add $- 5$ and $12$ .

$f (1) = \frac{7}{2}$

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

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

Substitute the $x$ value $2$ into $f (x) = - \frac{5 \sqrt{x}}{2} + 6$ . In this case, the point is $(2, - \frac{5 \sqrt{2}}{2} + 6)$ .

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

$f (2) = - \frac{5 \sqrt{2}}{2} + 6$

The final answer is $- \frac{5 \sqrt{2}}{2} + 6$ .

$y = - \frac{5 \sqrt{2}}{2} + 6$

The square root can be graphed using the points around the vertex $(0, 6), (1, 3.5), (2, 2.46)$

$\begin{array}{c} x & y \\ 0 & 6 \\ 1 & 3.5 \\ 2 & 2.46 \end{array}$

Step 5

Graph y=-5/2 square root of x+6

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