Graph y=1- square root of 2x+3

$y = 1 - \sqrt{2 x + 3}$

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

$2 x + 3 \geq 0$

Solve for $x$ .

Subtract $3$ from both sides of the inequality.

$2 x \geq - 3$

Divide each term in $2 x \geq - 3$ by $2$ and simplify.

Divide each term in $2 x \geq - 3$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 3}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{3}{2}$

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

Interval Notation:

$[- \frac{3}{2}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{3}{2}}$

Interval Notation:

$[- \frac{3}{2}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{3}{2}}$

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

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

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

Simplify the result.

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $- 3$ and $3$ .

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

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

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

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

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

Multiply $- 1$ by $0$ .

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

Add $1$ and $0$ .

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

The final answer is $1$ .

$1$

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

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

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 $f (x) = 1 - \sqrt{2 x + 3}$ . In this case, the point is $(- 1, 0)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $- 1$ .

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

Add $- 2$ and $3$ .

$f (- 1) = 1 - \sqrt{1}$

Any root of $1$ is $1$ .

$f (- 1) = 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (- 1) = 1 - 1$

Subtract $1$ from $1$ .

$f (- 1) = 0$

The final answer is $0$ .

$y = 0$

Substitute the $x$ value $0$ into $f (x) = 1 - \sqrt{2 x + 3}$ . In this case, the point is $(0, 1 - \sqrt{3})$ .

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $0$ .

$f (0) = 1 - \sqrt{0 + 3}$

Add $0$ and $3$ .

$f (0) = 1 - \sqrt{3}$

The final answer is $1 - \sqrt{3}$ .

$y = 1 - \sqrt{3}$

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

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $1$ .

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

Add $2$ and $3$ .

$f (1) = 1 - \sqrt{5}$

The final answer is $1 - \sqrt{5}$ .

$y = 1 - \sqrt{5}$

The square root can be graphed using the points around the vertex $(- 1.5, 1), (- 1, 0), (0, - 0.73), (1, - 1.24)$

$\begin{array}{c} x & y \\ - 1.5 & 1 \\ - 1 & 0 \\ 0 & - 0.73 \\ 1 & - 1.24 \end{array}$

Graph y=1- square root of 2x+3

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