Question
How does the graph of the function f(x)=1x
change when it is replaced by g(x)=14x
?
Option #1: It shifts left 4 units.
Option #2: It is horizontally compressed by a factor of 14
.
Option #3: It is vertically compressed by a factor of 14
.
Option #4: It is vertically stretched by a factor of 4.
change when it is replaced by g(x)=14x
?
Option #1: It shifts left 4 units.
Option #2: It is horizontally compressed by a factor of 14
.
Option #3: It is vertically compressed by a factor of 14
.
Option #4: It is vertically stretched by a factor of 4.
Answers
GPT-4o mini
To analyze the change from the function \( f(x) = x \) to the function \( g(x) = 14x \), we need to understand the effect of multiplying the output of a function by a constant.
1. **Function \( f(x) = x \)**: This is a linear function with a slope of 1.
2. **Function \( g(x) = 14x \)**: This modifies the slope of the function, changing it to 14.
When a function is multiplied by a constant (in this case, 14), it results in a vertical stretch if the constant is greater than 1. Therefore:
- The graph of \( g(x) = 14x \) is a vertical stretch of the graph of \( f(x) = x \) by a factor of 14. This means that for every point on the original graph, the value is scaled up (stretched) by a factor of 14.
Looking at the options provided:
- **Option #1**: It shifts left 4 units. (This is incorrect; there is no horizontal shift.)
- **Option #2**: It is horizontally compressed by a factor of 14. (This is incorrect; there is no horizontal compression.)
- **Option #3**: It is vertically compressed by a factor of 14. (This is incorrect; the graph is stretched vertically, not compressed.)
- **Option #4**: It is vertically stretched by a factor of 4. (This is also incorrect; it is stretched by a factor of 14.)
Since none of the provided options are correct as stated, we can confirm that the correct description is that the graph is vertically stretched by a factor of 14.
If you need to select one of the given options with the closest meaning, Options #3 & #4 both mention vertical transformations, but they do not accurately state the change. Hence, none are acceptable unless adjusted.
1. **Function \( f(x) = x \)**: This is a linear function with a slope of 1.
2. **Function \( g(x) = 14x \)**: This modifies the slope of the function, changing it to 14.
When a function is multiplied by a constant (in this case, 14), it results in a vertical stretch if the constant is greater than 1. Therefore:
- The graph of \( g(x) = 14x \) is a vertical stretch of the graph of \( f(x) = x \) by a factor of 14. This means that for every point on the original graph, the value is scaled up (stretched) by a factor of 14.
Looking at the options provided:
- **Option #1**: It shifts left 4 units. (This is incorrect; there is no horizontal shift.)
- **Option #2**: It is horizontally compressed by a factor of 14. (This is incorrect; there is no horizontal compression.)
- **Option #3**: It is vertically compressed by a factor of 14. (This is incorrect; the graph is stretched vertically, not compressed.)
- **Option #4**: It is vertically stretched by a factor of 4. (This is also incorrect; it is stretched by a factor of 14.)
Since none of the provided options are correct as stated, we can confirm that the correct description is that the graph is vertically stretched by a factor of 14.
If you need to select one of the given options with the closest meaning, Options #3 & #4 both mention vertical transformations, but they do not accurately state the change. Hence, none are acceptable unless adjusted.