To understand how the graph of the function [tex]\( y = \sqrt{-4x - 36} \)[/tex] is transformed compared to the parent function [tex]\( y = \sqrt{x} \)[/tex], we need to examine each component of the function in detail.
First, let's rewrite the function in a more recognizable form:
[tex]\[
y = \sqrt{-4(x + 9)}
\][/tex]
Now, we can break this down:
1. Inside the square root function:
- The term [tex]\(-4\)[/tex] inside the square root can be analyzed in pieces. The term [tex]\( -4x \)[/tex] indicates two transformations:
- The negative sign reflects the graph over the [tex]\( y \)[/tex]-axis.
- The coefficient [tex]\(-4\)[/tex] can be interpreted as affecting the horizontal scaling. Since the [tex]\( x \)[/tex] axis is scaled by a factor of 4 inside the square root, it results in horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
2. Horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex]:
- The function [tex]\( y = \sqrt{-4x} \)[/tex] compresses horizontally, but this appears as if the graph is stretched in the opposite direction compared to the parent function.
3. Reflection over the [tex]\( y \)[/tex]-axis:
- The negative sign in front of the [tex]\( 4x \)[/tex] means that instead of opening to the right, the graph opens to the left, reflecting over the [tex]\( y \)[/tex]-axis.
4. Horizontal translation:
- The [tex]\( (x + 9) \)[/tex] inside the function translates the graph horizontally by [tex]\( -9 \)[/tex] units. This means the graph is shifted 9 units to the left.
Putting these transformations together, we determine that the transformations are as follows:
- The graph is stretched horizontally by a factor of 2 (compression in the [tex]\( x \)[/tex]-direction).
- The graph is reflected over the [tex]\( y \)[/tex]-axis.
- The graph is translated 9 units to the left.
Thus, the correct option that describes the transformation is:
- stretched by a factor of 2, reflected over the [tex]\( y \)[/tex]-axis, and translated 9 units left
So, the answer is:
4. stretched by a factor of 2 , reflected over the [tex]\( y \)[/tex]-axis, and translated 9 units left
