Question

Willis analyzed the following table to determine if the function it represents is linear or non-linear. First he found the differences in the y-values as 7 – 1 = 6, 17 – 7 = 10, and 31 – 17 = 14. Then he concluded that since the differences of 6, 10, and 14 are increasing by 4 each time, the function has a constant rate of change and is linear. What was Willis’s mistake?

x
y
1
1
2
7
3
17
4
31
He found the differences in the y-values as 7 – 1 = 6, 17 – 7 = 10, and 31 – 17 = 14.
He determined that the differences of 6, 10, and 14 are increasing by 4 each time.
He concluded that the function has a constant rate of change.
He reasoned that a function that has a constant rate of change is linear.

Answer 1

Willis made a mistake in concluding that the function is linear based on the increasing differences in the y-values. The correct approach is to check if the ratios of the differences are constant. If the ratios are constant, then the function is linear. For the given table, the ratios of the differences are not constant, indicating that the function is non-linear.

Step-by-step explanation:

Willis made a mistake in concluding that the function is linear based on the increasing differences in the y-values. While it is true that a linear function will have a constant rate of change, the differences can also increase for a non-linear function. Willis should have checked if the ratios of the differences were constant instead. If the ratios were constant, then the function would be linear.

For the given table, the differences in the y-values are 6, 10, and 14. Taking the ratios of these differences, we get 10/6 = 1.67 and 14/10 = 1.4. Since the ratios are not constant, the function is non-linear.