Sure! Let's carefully analyze the transformations applied to the parent function [tex]\( y = |x| \)[/tex] to obtain the given function [tex]\( g(x) = \frac{1}{2}|x-1| + 5 \)[/tex].
### Step-by-Step Solution:
1. Parent Function: The parent function is [tex]\( y = |x| \)[/tex].
2. Transformations Applied:
- [tex]\( \frac{1}{2} \)[/tex] outside the absolute value. This indicates a vertical compression by a factor of [tex]\( \frac{1}{2} \)[/tex]. So, the graph is "squished" vertically to half its original height.
- [tex]\( |x-1| \)[/tex] represents a horizontal shift to the right by 1 unit. In general, [tex]\( |x - h| \)[/tex] shifts the graph horizontally to the right by [tex]\( h \)[/tex] units.
- [tex]\( +5 \)[/tex] outside the absolute value records a vertical shift up by 5 units. So, the entire graph is moved up by 5 units.
### Conclusion:
Let's summarize these transformations:
- Reflect across the x-axis: No (indicated by 0).
- Vertical stretch: No (indicated by 0).
- Vertical compression: Yes, by a factor of [tex]\( \frac{1}{2} \)[/tex] (indicated by 1).
- Horizontal shift of 1 right: Yes, by 1 unit (indicated by 1).
- Horizontal shift of 1 left: No (indicated by 0).
- Vertical shift of 5 up: Yes, by 5 units (indicated by 5).
- Vertical shift of 5 down: No (indicated by 0).
So, the correct transformations for [tex]\( g(x) = \frac{1}{2}|x-1|+5 \)[/tex] are:
- Reflect across the x-axis: 0
- Vertical stretch: 0
- Vertical compression: 1
- Horizontal shift of 1 right: 1
- Horizontal shift of 1 left: 0
- Vertical shift of 5 up: 5
- Vertical shift of 5 down: 0.
These are the transformations that have occurred.
