Question

Choose all of the sets of points that define non-linear functions.
Responses
A (−4, 0), (−3, 3), (−2, 6), (−1, 0)(−4, 0), (−3, 3), (−2, 6), (−1, 0)
B (−4, 16), (−2, 4), (2, 4), (4, 16)(−4, 16), (−2, 4), (2, 4), (4, 16)
C (−4, −10), (−2, −8), (2, −4), (4, −2)(−4, −10), (−2, −8), (2, −4), (4, −2)
D (−4, −1), (−2, 1), (2, 5), (4, 7)

Answer 1

To determine which sets of points define non-linear functions, we need to check if the relationship between the x and y values can be expressed as a linear function, which has the form \(y = mx + b\). In a linear function, a change in x results in a constant change in y, meaning the y-values would form a linear pattern (a straight line).

Let's analyze each set of points:

**Set A:** \((-4, 0), (-3, 3), (-2, 6), (-1, 0)\)
- The differences in y-values are: \(3 - 0 = 3\), \(6 - 3 = 3\), and \(0 - 6 = -6\).
- Since the changes are not consistent, this set of points forms a non-linear function.

**Set B:** \((-4, 16), (-2, 4), (2, 4), (4, 16)\)
- The differences in y-values are: \(4 - 16 = -12\), \(4 - 4 = 0\), and \(16 - 4 = 12\).
- This set forms a parabolic pattern (specifically resembling a quadratic function), thus it is non-linear.

**Set C:** \((-4, -10), (-2, -8), (2, -4), (4, -2)\)
- The differences in y-values are: \(-8 - (-10) = 2\), \(-4 - (-8) = 4\), and \(-2 - (-4) = 2\).
- The changes are not consistent (especially between the second and last points), which indicates this set forms a non-linear function.

**Set D:** \((-4, -1), (-2, 1), (2, 5), (4, 7)\)
- The changes in y-values are: \(1 - (-1) = 2\), \(5 - 1 = 4\), and \(7 - 5 = 2\).
- The changes here do not create a linear relationship (the incremented values are not consistent), indicating this is also a non-linear function.

Thus, all sets A, B, C, and D define non-linear functions.

**All sets A, B, C, and D** define non-linear functions.