Question
Consider the following two functions.
Original: f(x)=|x| +1
Final: f(x)=12|x| +1/2
Which option best describes the change from the original to the final function?
Option #1: a vertical stretch
Option #2: a vertical compression
Option #3: no change
(1 point)
The change is best described by Option #
.
Original: f(x)=|x| +1
Final: f(x)=12|x| +1/2
Which option best describes the change from the original to the final function?
Option #1: a vertical stretch
Option #2: a vertical compression
Option #3: no change
(1 point)
The change is best described by Option #
.
Answers
GPT-4o mini
To determine how the function has changed from the original \( f(x) = |x| + 1 \) to the final \( f(x) = 12|x| + \frac{1}{2} \), we can analyze the factors affecting the vertical scaling and the vertical shift.
1. **Vertical Stretch/Compression**: The original function has the term \( |x| \), while the final function has the term \( 12|x| \). The coefficient of \( 12 \) in front of \( |x| \) indicates that the graph is vertically stretched by a factor of \( 12 \). This means that for every value of \( |x| \), the output value of the function is \( 12 \) times larger compared to the original function.
2. **Vertical Shift**: The original function has a vertical shift of \( +1 \) and the final function has a vertical shift of \( +\frac{1}{2} \). However, the dominant change in this case is due to the vertical stretch.
Based on this analysis, the change from the original to the final function is best described as:
**Option #1: a vertical stretch**
Thus, the answer is: The change is best described by Option #1.
1. **Vertical Stretch/Compression**: The original function has the term \( |x| \), while the final function has the term \( 12|x| \). The coefficient of \( 12 \) in front of \( |x| \) indicates that the graph is vertically stretched by a factor of \( 12 \). This means that for every value of \( |x| \), the output value of the function is \( 12 \) times larger compared to the original function.
2. **Vertical Shift**: The original function has a vertical shift of \( +1 \) and the final function has a vertical shift of \( +\frac{1}{2} \). However, the dominant change in this case is due to the vertical stretch.
Based on this analysis, the change from the original to the final function is best described as:
**Option #1: a vertical stretch**
Thus, the answer is: The change is best described by Option #1.