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Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function [tex] y = \sqrt{x} [/tex] and show all stages. Be sure to show at least three key points. Find the domain and the range of the function.

[tex] h(x) = \sqrt{-x - 4} [/tex]

Which transformations are needed to graph the function [tex] h(x) = \sqrt{-x - 4} [/tex]? Choose the correct answer below:

A. The graph of [tex] y = \sqrt{x} [/tex] should be vertically shifted down by 4 units and reflected about the y-axis.

B. The graph of [tex] y = \sqrt{x} [/tex] should be horizontally shifted to the right by 4 units and reflected about the x-axis.

C. The graph of [tex] y = \sqrt{x} [/tex] should be vertically shifted up by 4 units and reflected about the x-axis.

D. The graph of [tex] y = \sqrt{x} [/tex] should be horizontally shifted to the left by 4 units and reflected about the y-axis.

Sagot :

GinnyAnsweridn
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]
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