Graph 2^3 square root of x+2+4

$2^{3} \sqrt{x + 2} + 4$

Step 1

Find the domain for $y = 2^{3} \sqrt{x + 2} + 4$ 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 $\sqrt{x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$x + 2 \geq 0$

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

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

Interval Notation:

$[- 2, \infty)$

Set-Builder Notation:

${x | x \geq - 2}$

Interval Notation:

$[- 2, \infty)$

Set-Builder Notation:

${x | x \geq - 2}$

Step 2

To find the radical expression end point, substitute the $x$ value $- 2$ , which is the least value in the domain, into $f (x) = 8 \sqrt{x + 2} + 4$ .

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

$f (- 2) = 8 \sqrt{(- 2) + 2} + 4$

Simplify the result.

Simplify each term.

Add $- 2$ and $2$ .

$f (- 2) = 8 \sqrt{0} + 4$

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

$f (- 2) = 8 \sqrt{0^{2}} + 4$

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

$f (- 2) = 8 \cdot 0 + 4$

Multiply $8$ by $0$ .

$f (- 2) = 0 + 4$

Add $0$ and $4$ .

$f (- 2) = 4$

The final answer is $4$ .

$4$

Step 3

The radical expression end point is $(- 2, 4)$ .

$(- 2, 4)$

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 $f (x) = 8 \sqrt{x + 2} + 4$ . In this case, the point is $(- 1, 12)$ .

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

$f (- 1) = 8 \sqrt{(- 1) + 2} + 4$

Simplify the result.

Simplify each term.

Add $- 1$ and $2$ .

$f (- 1) = 8 \sqrt{1} + 4$

Any root of $1$ is $1$ .

$f (- 1) = 8 \cdot 1 + 4$

Multiply $8$ by $1$ .

$f (- 1) = 8 + 4$

Add $8$ and $4$ .

$f (- 1) = 12$

The final answer is $12$ .

$y = 12$

Substitute the $x$ value $0$ into $f (x) = 8 \sqrt{x + 2} + 4$ . In this case, the point is $(0, 8 \sqrt{2} + 4)$ .

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

$f (0) = 8 \sqrt{(0) + 2} + 4$

Simplify the result.

Add $0$ and $2$ .

$f (0) = 8 \sqrt{2} + 4$

The final answer is $8 \sqrt{2} + 4$ .

$y = 8 \sqrt{2} + 4$

The square root can be graphed using the points around the vertex $(- 2, 4), (- 1, 12), (0, 15.31)$

$\begin{array}{c} x & y \\ - 2 & 4 \\ - 1 & 12 \\ 0 & 15.31 \end{array}$

Step 5

Graph 2^3 square root of x+2+4

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