Certainly! Let's break down the transformations for the function [tex]\(y = -3\sqrt{x-6}\)[/tex] from the parent function [tex]\(y = \sqrt{x}\)[/tex].
1. Translation:
The function [tex]\(y = \sqrt{x}\)[/tex] is modified to [tex]\(y = \sqrt{x-6}\)[/tex]. The expression [tex]\(x-6\)[/tex] inside the square root indicates a horizontal translation. Specifically, we replace [tex]\(x\)[/tex] with [tex]\(x-6\)[/tex], which shifts the graph to the right by 6 units.
2. Reflection:
The next transformation involves a reflection. The function [tex]\(y = \sqrt{x-6}\)[/tex] becomes [tex]\(y = -\sqrt{x-6}\)[/tex]. The negative sign in front of the square root indicates that the graph is reflected across the x-axis. This means that if a point [tex]\( (a, b) \)[/tex] on the original graph becomes [tex]\( (a, -b) \)[/tex].
3. Vertical Scaling:
Finally, the function [tex]\(y = -\sqrt{x-6}\)[/tex] is modified to [tex]\(y = -3\sqrt{x-6}\)[/tex]. The coefficient [tex]\(-3\)[/tex] outside the square root indicates a vertical scaling by a factor of 3, combined with a reflection (due to the negative sign). Vertical scaling by a factor of 3 means that the graph is stretched by a factor of 3 in the vertical direction.
Combining all these individual transformations step-by-step:
- The parent function [tex]\(y = \sqrt{x}\)[/tex] is translated 6 units to the right.
- The translated function [tex]\(y = \sqrt{x-6}\)[/tex] is then reflected across the x-axis resulting in [tex]\(y = -\sqrt{x-6}\)[/tex].
- Lastly, the reflected function [tex]\(y = -\sqrt{x-6}\)[/tex] is vertically scaled by a factor of 3 to become [tex]\(y = -3\sqrt{x-6}\)[/tex].
Therefore, the graph of [tex]\(y = \sqrt{x}\)[/tex] is transformed to [tex]\(y = -3\sqrt{x-6}\)[/tex] by:
1. Translating 6 units to the right.
2. Reflecting across the x-axis.
3. Vertically scaling by a factor of [tex]\(-3\)[/tex].
These combined transformations provide the complete graphing of the function [tex]\(y = -3\sqrt{x-6}\)[/tex].
