To identify the transformations of the function [tex]\( y = \sqrt{x} \)[/tex] when transformed into [tex]\( y = 2 \sqrt{\frac{1}{3}(x-1)} - 5 \)[/tex], we need to look at how each component of the new function affects [tex]\( y = \sqrt{x} \)[/tex].
Here are the transformations step-by-step:
1. Horizontal Translation:
The function inside the square root is [tex]\( (x - 1) \)[/tex]. This means that the function is translated to the right by 1 unit. In general, for [tex]\( y = f(x - h) \)[/tex], the graph is shifted to the right by [tex]\( h \)[/tex]. So, for our function,
[tex]\[
x \rightarrow x - 1
\][/tex]
corresponds to a horizontal translation of [tex]\( +1 \)[/tex] units.
2. Horizontal Compression:
The term [tex]\(\frac{1}{3}\)[/tex] inside the square root affects the horizontal stretch/compression of the function. Specifically, [tex]\( y = \sqrt{\frac{1}{k}x} \)[/tex] compresses the graph horizontally by a factor of [tex]\( k \)[/tex]. Here, [tex]\( k \)[/tex] is 3. So, [tex]\(\frac{1}{3} \left( x - 1 \right) \)[/tex] indicates a horizontal compression by a factor of 3.
3. Vertical Scaling:
The multiplier 2 outside the square root indicates a vertical scaling. For [tex]\( y = a \cdot f(x) \)[/tex], the graph is vertically stretched (or compressed) by a factor of [tex]\( a \)[/tex]. In this case, [tex]\( y = 2 \cdot \sqrt{\frac{1}{3}(x-1)} \)[/tex], the function is scaled vertically by a factor of 2.
4. Vertical Translation:
Finally, the term [tex]\(-5\)[/tex] at the end of the function shifts the graph vertically downward. For [tex]\( y = f(x) + k \)[/tex], the graph shifts upward by [tex]\( k \)[/tex] if [tex]\( k \)[/tex] is positive, and downward by [tex]\( k \)[/tex] if [tex]\( k \)[/tex] is negative. Here, the -5 translates the function downward by 5 units.
Summarizing these transformations:
- Horizontal translation to the right by [tex]\( +1 \)[/tex] unit.
- Horizontal compression by a factor of 3.
- Vertical scaling by a factor of 2.
- Vertical translation downward by 5 units.
So the transformations of the function [tex]\( y = \sqrt{x} \)[/tex] into [tex]\( y = 2 \sqrt{\frac{1}{3}(x-1)} - 5 \)[/tex] are as follows:
1. Horizontal translation of [tex]\( +1 \)[/tex] unit.
2. Horizontal compression by a factor of 3.
3. Vertical scaling by a factor of 2.
4. Vertical translation downward by 5 units.
