Let's graph the function [tex]\( h(x) = \sqrt{-x - 4} \)[/tex] by breaking down the transformations step-by-step. We'll start with the basic function [tex]\( y = \sqrt{x} \)[/tex] and apply the necessary transformations one by one.
1. Start with [tex]\( y = \sqrt{x} \)[/tex]: This is the basic square root function. It passes through the points [tex]\((0, 0)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((4, 2)\)[/tex].
2. Reflect about the y-axis: Reflecting [tex]\( y = \sqrt{x} \)[/tex] about the y-axis results in [tex]\( y = \sqrt{-x} \)[/tex]. This changes the points as follows:
- [tex]\((0, 0)\)[/tex] remains [tex]\((0, 0)\)[/tex].
- [tex]\((1, 1)\)[/tex] becomes [tex]\((-1, 1)\)[/tex].
- [tex]\((4, 2)\)[/tex] becomes [tex]\((-4, 2)\)[/tex].
3. Horizontal shift to the left by 4 units: Shifting [tex]\( y = \sqrt{-x} \)[/tex] to the left by 4 units changes the function to [tex]\( y = \sqrt{-(x + 4)} \)[/tex]. The points are now:
- [tex]\((0, 0)\)[/tex] becomes [tex]\((-4, 0)\)[/tex].
- [tex]\((-1, 1)\)[/tex] becomes [tex]\((-5, 1)\)[/tex].
- [tex]\((-4, 2)\)[/tex] becomes [tex]\((-8, 2)\)[/tex].
So, the transformations needed to graph [tex]\( h(x) = \sqrt{-x - 4} \)[/tex] are:
- Reflect [tex]\( y = \sqrt{x} \)[/tex] about the y-axis.
- Horizontally shift the resulting graph to the left by 4 units.
Given these steps, the correct answer is:
D. The graph of [tex]\( y = \sqrt{x} \)[/tex] should be horizontally shifted to the left by 4 units, reflected about the y-axis.
### Domain and Range:
- Domain: For [tex]\( h(x) = \sqrt{-(x + 4)} \)[/tex] to be defined, the expression inside the square root must be non-negative.
[tex]\[
-(x + 4) \geq 0 \\
-x - 4 \geq 0 \\
-x \geq 4 \\
x \leq -4
\][/tex]
So, the domain is [tex]\( (-\infty, -4] \)[/tex].
- Range: Since [tex]\( \sqrt{-(x + 4)} \)[/tex] gives non-negative values (as the square root function only produces non-negative results),
the range of [tex]\( h(x) \)[/tex] is [tex]\([0, \infty)\)[/tex].
### Key Points:
- [tex]\((-4, 0)\)[/tex]
- [tex]\((-5, 1)\)[/tex]
- [tex]\((-8, 2)\)[/tex]
