Which of the following describes the graph of [tex][tex]$y=\sqrt{-4 x-36}$[/tex][/tex] compared to the parent square root function?

A. Stretched by a factor of 2, reflected over the x-axis, and translated 9 units right

B. Stretched by a factor of 2, reflected over the x-axis, and translated 9 units left

C. Stretched by a factor of 2, reflected over the y-axis, and translated 9 units right

D. Stretched by a factor of 2, reflected over the y-axis, and translated 9 units left



Answer :

To understand how the graph of the function [tex]\( y = \sqrt{-4x - 36} \)[/tex] transforms compared to the parent function [tex]\( y = \sqrt{x} \)[/tex], let's break down the expression inside the square root.

1. Start by considering the expression inside the square root: [tex]\(-4x - 36\)[/tex].

First, factor out the [tex]\(-4\)[/tex] from the terms inside the square root:
[tex]\[ -4x - 36 = -4(x + 9) \][/tex]

2. Now, the function can be rewritten as:
[tex]\[ y = \sqrt{-4(x + 9)} \][/tex]

Next, analyze each part of this expression for transformations:

- The [tex]\(-4\)[/tex] inside the square root includes two transformations:
- The negative sign ([tex]\(-\)[/tex]) indicates a reflection over the [tex]\(y\)[/tex]-axis.
- The factor of [tex]\(4\)[/tex] outside can be rewritten as [tex]\(2^2\)[/tex] under the square root, meaning it represents a vertical stretch by a factor of 2.

- The [tex]\((x + 9)\)[/tex] term indicates a horizontal translation. The plus sign (+9) means the function is translated 9 units to the left.

Given these observations, the transformations compared to the parent function [tex]\( y = \sqrt{x} \)[/tex] are:
- Stretched by a factor of 2
- Reflected over the [tex]\(y\)[/tex]-axis
- Translated 9 units left

Among the given options, this describes:
- Stretched by a factor of 2, reflected over the [tex]\(y\)[/tex]-axis, and translated 9 units left.

Thus, the correct description is the fourth option:
4. "stretched by a factor of 2, reflected over the [tex]\(y\)[/tex]-axis, and translated 9 units left"