Final answer:
The given data set could represent a quadratic function because the second differences in the Y values are constant, suggesting a parabolic shape which implies a quadratic relationship.
Step-by-step explanation:
To determine if the data set could represent a quadratic function, we look for a pattern in which the change in Y values is not constant but the changes in those changes (second differences) are constant. For the given data set X: -8 -4 0 4 8, and Y: 5 11 14 14 11, we can observe the first differences (changes in Y) and the second differences (changes between the first differences).
- From X = -8 to -4, Y changes from 5 to 11, which is an increase of 6.
- From X = -4 to 0, Y changes from 11 to 14, which is an increase of 3.
- From X = 0 to 4, Y changes from 14 to 14, which is an increase of 0.
- From X = 4 to 8, Y changes from 14 to 11, which is a decrease of 3.
The second differences are as follows:
- The change between 6 and 3 is -3.
- The change between 3 and 0 is -3.
- The change between 0 and -3 is -3.
Since the second differences are constant (-3), this is indicative of a quadratic relationship. Therefore, the data set could indeed represent a quadratic function, as opposed to a linear one where first differences would be constant. In terms of a scatter plot, if we were to graph these points, we would see a curved, parabolic shape typical of a quadratic function, with an axis of symmetry likely at X=0 where the maximum Y value occurs.