Question
Interpret the key features of the graph to determine the transformation that occurred from it’s original function y = ^3 square root x
Points on graph (-11,-2), (-4,-1),(-3,0),(-2,1), and (5,2)
Shift to left due to change in x-intercept
Shift to the right due to change in x-intercept
Shift up due to change in y-intercept
Shift down due to change in y -intercept
Points on graph (-11,-2), (-4,-1),(-3,0),(-2,1), and (5,2)
Shift to left due to change in x-intercept
Shift to the right due to change in x-intercept
Shift up due to change in y-intercept
Shift down due to change in y -intercept
Answers
Answered by
GPT-4o mini
To analyze the key features of the graph of the function \( y = \sqrt[3]{x} \) and determine the transformation given the new points you provided, let's first examine the original function and then compare the given points.
1. **Original Function**: The function \( y = \sqrt[3]{x} \) is defined for all real numbers and has the following characteristics:
- The graph passes through the origin (0, 0).
- It is an odd function, symmetric about the origin.
- As \( x \) increases, \( y \) also increases; as \( x \) decreases, \( y \) decreases.
2. **Points on Graph**: The points you provided are:
- (-11, -2)
- (-4, -1)
- (-3, 0)
- (-2, 1)
- (5, 2)
3. **Analyzing the Transformation**:
- From the original function \( y = \sqrt[3]{x} \), the crucial point to note is that it originally passes through the point (0, 0).
- In the new set of points, (5, 2) corresponds to a higher value than what \( \sqrt[3]{5} \) would yield (which is less than 2), indicating that the entire graph has been shifted upwards.
- The point (−3, 0) was originally the y-intercept of the cubic root function at (0, 0), suggesting a downshift in the x-values.
### Key Transformation:
Given these observations, the key transformation that occurred is a **shift upwards**, as indicated by the fact that the new points, particularly the point (5, 2), fall higher than where they would under the original function \( y = \sqrt[3]{x} \).
Thus, the correct interpretation based on the provided points is that the graph of \( y = \sqrt[3]{x} \) has undergone a:
- **Shift up due to change in y-intercept.**
The shifts to the left or right do not occur in this context since the x-coordinates of the critical intercept points did not move horizontally in a way that would require a left/right transformation. Based on the points given, the overall transformation is a vertical shift upward.
1. **Original Function**: The function \( y = \sqrt[3]{x} \) is defined for all real numbers and has the following characteristics:
- The graph passes through the origin (0, 0).
- It is an odd function, symmetric about the origin.
- As \( x \) increases, \( y \) also increases; as \( x \) decreases, \( y \) decreases.
2. **Points on Graph**: The points you provided are:
- (-11, -2)
- (-4, -1)
- (-3, 0)
- (-2, 1)
- (5, 2)
3. **Analyzing the Transformation**:
- From the original function \( y = \sqrt[3]{x} \), the crucial point to note is that it originally passes through the point (0, 0).
- In the new set of points, (5, 2) corresponds to a higher value than what \( \sqrt[3]{5} \) would yield (which is less than 2), indicating that the entire graph has been shifted upwards.
- The point (−3, 0) was originally the y-intercept of the cubic root function at (0, 0), suggesting a downshift in the x-values.
### Key Transformation:
Given these observations, the key transformation that occurred is a **shift upwards**, as indicated by the fact that the new points, particularly the point (5, 2), fall higher than where they would under the original function \( y = \sqrt[3]{x} \).
Thus, the correct interpretation based on the provided points is that the graph of \( y = \sqrt[3]{x} \) has undergone a:
- **Shift up due to change in y-intercept.**
The shifts to the left or right do not occur in this context since the x-coordinates of the critical intercept points did not move horizontally in a way that would require a left/right transformation. Based on the points given, the overall transformation is a vertical shift upward.