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"
