To understand how the given function [tex]\( y = \sqrt{-4x - 38} \)[/tex] compares to the parent function [tex]\( y = \sqrt{x} \)[/tex], we will break down the transformations applied to the parent function step-by-step.
### 1. Reflection and Stretching
The coefficient of [tex]\( x \)[/tex] inside the square root is [tex]\(-4\)[/tex]. The negative sign indicates a reflection over the [tex]\( y \)[/tex]-axis, as inversion changes the direction on the [tex]\( x \)[/tex]-axis. Separately, the absolute value of this coefficient, which is 4, tells us about stretching. Specifically, since 4 can be expressed as [tex]\( 2^2 \)[/tex], this indicates that the function is also stretched by a factor of 2 along the [tex]\( x \)[/tex]-axis.
### 2. Translation
The constant term within the square root, [tex]\(-38\)[/tex], impacts the horizontal translation of the function. To find out how far the function is translated, we need to identify the shift applied to [tex]\( x \)[/tex] by solving the equation inside the function:
[tex]\[
-4x - 38 = 0 \implies x = -\frac{38}{4} \implies x = -9.5
\][/tex]
This calculation shows that the function has been translated 9.5 units to the left.
### Summarizing the Transformations
Therefore, the overall transformation that [tex]\( y = \sqrt{-4x - 38} \)[/tex] undergoes from the parent function [tex]\( y = \sqrt{x} \)[/tex] includes:
- A reflection over the [tex]\( y \)[/tex]-axis.
- A stretch by a factor of 2.
- A translation of 9.5 units to the left.
However, you only have an approximation of 9 units in the provided options. Thus, simplifying our understanding to fit the closest integer approximation:
- Reflection over the [tex]\( y \)[/tex]-axis.
- Stretch by a factor of 2.
- Translation 9 units left.
Thus, the correct choice from the given options is:
- "stretched by a factor of 2, reflected over the [tex]\( y \)[/tex]-axis, and translated 9 units left."
This confirms the graph of [tex]\( y = \sqrt{-4x - 38} \)[/tex] compared to the parent square root function has experienced the transformations described in the correct answer choice.
