Question
Interpret the key features of the graph to determine the transformation that occurred from its original function y=x−−√3.
(1 point)
Responses
A. This graph represents a shift to the right due to the change in the x-intercept from the parent function.
B. This graph represents a shift to the left due to the change in the x-intercept from the parent function.
C. This graph represents a shift up due to the change in the y -intercept from the parent function.
D. This graph represents a shift down due to the change in the y-intercept from the parent function.
(1 point)
Responses
A. This graph represents a shift to the right due to the change in the x-intercept from the parent function.
B. This graph represents a shift to the left due to the change in the x-intercept from the parent function.
C. This graph represents a shift up due to the change in the y -intercept from the parent function.
D. This graph represents a shift down due to the change in the y-intercept from the parent function.
Answers
Answered by
GPT-4o mini
To determine the transformation that occurred from the original function \( y = \sqrt{x} \) (the parent function) to the function \( y = \sqrt{x} - 3 \), consider the following:
1. **Transformation Description**: The function \( y = \sqrt{x} - 3 \) represents a vertical transformation of the parent function \( y = \sqrt{x} \). Specifically, it indicates a downward shift of the graph by 3 units.
2. **X-Intercept**: The x-intercept occurs where \( y = 0 \). For \( y = \sqrt{x} \), the x-intercept is at \( (0, 0) \). For \( y = \sqrt{x} - 3 \), setting \( y = 0 \) gives \( \sqrt{x} - 3 = 0 \) or \( \sqrt{x} = 3 \), which yields \( x = 9 \). Thus, the x-intercept has changed from \( (0, 0) \) to \( (9, 0) \).
3. **Y-Intercept**: The y-intercept occurs where \( x = 0 \). For \( y = \sqrt{x} \), the y-intercept is at \( (0, 0) \). For \( y = \sqrt{x} - 3 \), we find the value at \( x = 0 \) is \( y = \sqrt{0} - 3 = -3 \), giving the y-intercept at \( (0, -3) \).
With this information:
- There is a downward shift by 3 units in the function \( y = \sqrt{x} - 3 \) compared to the original \( y = \sqrt{x} \).
- The change in x-intercept and y-intercept indicates a vertical transformation.
Thus, the correct interpretation of the transformation is:
**D. This graph represents a shift down due to the change in the y-intercept from the parent function.**
1. **Transformation Description**: The function \( y = \sqrt{x} - 3 \) represents a vertical transformation of the parent function \( y = \sqrt{x} \). Specifically, it indicates a downward shift of the graph by 3 units.
2. **X-Intercept**: The x-intercept occurs where \( y = 0 \). For \( y = \sqrt{x} \), the x-intercept is at \( (0, 0) \). For \( y = \sqrt{x} - 3 \), setting \( y = 0 \) gives \( \sqrt{x} - 3 = 0 \) or \( \sqrt{x} = 3 \), which yields \( x = 9 \). Thus, the x-intercept has changed from \( (0, 0) \) to \( (9, 0) \).
3. **Y-Intercept**: The y-intercept occurs where \( x = 0 \). For \( y = \sqrt{x} \), the y-intercept is at \( (0, 0) \). For \( y = \sqrt{x} - 3 \), we find the value at \( x = 0 \) is \( y = \sqrt{0} - 3 = -3 \), giving the y-intercept at \( (0, -3) \).
With this information:
- There is a downward shift by 3 units in the function \( y = \sqrt{x} - 3 \) compared to the original \( y = \sqrt{x} \).
- The change in x-intercept and y-intercept indicates a vertical transformation.
Thus, the correct interpretation of the transformation is:
**D. This graph represents a shift down due to the change in the y-intercept from the parent function.**