To analyze the transformations of the function [tex]\( y = \sqrt{-4x - 36} \)[/tex] compared to the parent square root function [tex]\( y = \sqrt{x} \)[/tex], follow these steps:
1. Reflected over the y-axis:
The expression inside the square root, [tex]\( -4x \)[/tex], indicates that the graph is reflected over the y-axis. This is because multiplying [tex]\( x \)[/tex] by -1 reflects the graph over the y-axis.
2. Stretch by a factor:
The coefficient -4 inside the square root affects the horizontal stretching. Specifically, the term [tex]\( -4 \)[/tex] can be factored out as [tex]\( -4 = -(2^2) \)[/tex]. This tells us that there is a horizontal stretch by a factor of 2.
3. Translate horizontally:
To determine the horizontal translation, rewrite the expression inside the square root:
[tex]\[
y = \sqrt{-4(x + 9)}
\][/tex]
Here, you can see that the term [tex]\( (x + 9) \)[/tex] inside the square root indicates a horizontal shift. A value of +9 means the shift is 9 units to the left.
Combining these observations:
- The graph is stretched by a factor of 2.
- It is reflected over the y-axis.
- It is translated 9 units to the left.
Therefore, the correct description is:
- Stretched by a factor of 2
- Reflected over the y-axis
- Translated 9 units left
So, the correct answer is:
stretched by a factor of 2, reflected over the [tex]$y$[/tex]-axis, and translated 9 units left.
