To determine the type of function (linear,
quadratic, or neither), we look at the finite differences between consecutive y-values (known as the first difference). If these first differences are constant, then the function is linear. If the first differences are not constant but the second differences (differences of the first differences) are constant, then the function is quadratic. If neither the first nor second differences are constant, the function is neither linear nor quadratic.
Given the table:
x | y
----------
-1 | 21
0 | 11
1 | 5
2 | 3
3 | 5
Let's compute the first differences:
From x = -1 to x = 0:
11 - 21 = -10
From x = 0 to x = 1:
5 - 11 = -6
From x = 1 to x = 2:
3 - 5 = -2
From x = 2 to x = 3:
5 - 3 = 2
The first differences are: -10, -6, -2, and 2. Since they're not constant, the function isn't linear.
Now, let's compute the second differences:
-6 - (-10) = 4
-2 - (-6) = 4
2 - (-2) = 4
The second differences are all constant and equal to 4, which suggests that this function is quadratic.
Conclusion:
The function represented by the table is quadratic because while the first differences are not constant, the second differences are constant. This constant second difference is characteristic of quadratic functions.