Certainly! Let's analyze the function [tex]\( y = a \sqrt{x-h} + k \)[/tex] and the effect of various transformations on its graph.
In the context of transformations of functions, we have four possible scenarios:
1. Vertical stretch by a factor of 8.
2. Vertical stretch by a factor of 2.
3. Vertical compression by a factor of 8.
4. Vertical compression by a factor of 2.
A vertical stretch or compression modifies the coefficient [tex]\( a \)[/tex] in the function [tex]\( y = a \sqrt{x-h} + k \)[/tex]. Specifically:
- A vertical stretch by a factor increases the value of [tex]\( a \)[/tex].
- A vertical compression by a factor decreases the value of [tex]\( a \)[/tex].
Given the problem statement, we know there's a vertical stretch by a factor of 8.
To determine the value of [tex]\( a \)[/tex] in this scenario:
- A vertical stretch by a factor of 8 means that [tex]\( a \)[/tex] should be multiplied by 8.
Given this information, the value of [tex]\( a \)[/tex] is confirmed to be [tex]\( 8 \)[/tex].
Thus, the final choice is [tex]\( \boxed{\text{stretch by a factor of 8}} \)[/tex].
