Graph f(x)=4 square root of 2x-1+3

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

Step 1

Find the domain for $y = 4 \sqrt{2 x - 1} + 3$ 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 $\sqrt{2 x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$2 x - 1 \geq 0$

Solve for $x$ .

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Divide each term in $2 x \geq 1$ by $2$ and simplify.

Divide each term in $2 x \geq 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{1}{2}$

Divide $x$ by $1$ .

$x \geq \frac{1}{2}$

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

Interval Notation:

$[\frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{1}{2}}$

Interval Notation:

$[\frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{1}{2}}$

Step 2

To find the radical expression end point, substitute the $x$ value $\frac{1}{2}$ , which is the least value in the domain, into $f (x) = 4 \sqrt{2 x - 1} + 3$ .

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Simplify the result.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $1$ from $1$ .

$f (\frac{1}{2}) = 4 \sqrt{0} + 3$

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

$f (\frac{1}{2}) = 4 \sqrt{0^{2}} + 3$

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

$f (\frac{1}{2}) = 4 \cdot 0 + 3$

Multiply $4$ by $0$ .

$f (\frac{1}{2}) = 0 + 3$

Add $0$ and $3$ .

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

The final answer is $3$ .

$3$

Step 3

The radical expression end point is $(\frac{1}{2}, 3)$ .

$(\frac{1}{2}, 3)$

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 $f (x) = 4 \sqrt{2 x - 1} + 3$ . In this case, the point is $(1, 7)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $1$ .

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

Subtract $1$ from $2$ .

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

Any root of $1$ is $1$ .

$f (1) = 4 \cdot 1 + 3$

Multiply $4$ by $1$ .

$f (1) = 4 + 3$

Add $4$ and $3$ .

$f (1) = 7$

The final answer is $7$ .

$y = 7$

Substitute the $x$ value $2$ into $f (x) = 4 \sqrt{2 x - 1} + 3$ . In this case, the point is $(2, 4 \sqrt{3} + 3)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $2$ .

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

Subtract $1$ from $4$ .

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

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

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

Substitute the $x$ value $3$ into $f (x) = 4 \sqrt{2 x - 1} + 3$ . In this case, the point is $(3, 4 \sqrt{5} + 3)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $3$ .

$f (3) = 4 \sqrt{6 - 1} + 3$

Subtract $1$ from $6$ .

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

The final answer is $4 \sqrt{5} + 3$ .

$y = 4 \sqrt{5} + 3$

The square root can be graphed using the points around the vertex $(0.5, 3), (1, 7), (2, 9.93), (3, 11.94)$

$\begin{array}{c} x & y \\ 0.5 & 3 \\ 1 & 7 \\ 2 & 9.93 \\ 3 & 11.94 \end{array}$

Step 5

Graph f(x)=4 square root of 2x-1+3

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