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.
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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.
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Points on Graph: The points you provided are:
- (-11, -2)
- (-4, -1)
- (-3, 0)
- (-2, 1)
- (5, 2)
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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.
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