Is the function represented by this table linear or nonlinear and why?

Question

Responses

The function is linear because it decreases at a constant rate.
The function is linear because it decreases at a constant rate.

The function is not linear because the x-values and y-values are increasing in opposite directions.
The function is not linear because the x- values and y- values are increasing in opposite directions.

The function is not linear because there is no x-value of 0.
The function is not linear because there is no x -value of 0.

The function is linear because all of the values on the table are positive.
The function is linear because all of the values on the table are positive.
x y
2 10
4 9
6 8
8 7

Answer 1

To determine if the function represented by the table is linear or nonlinear, we need to examine the relationship between the x-values and y-values.

Let's look at the pairs of values you've provided:

x | y
-----
2 | 10
4 | 9
6 | 8
8 | 7

To check if the function is linear, we need to see if there is a constant rate of change (slope) between the y-values as the x-values increase.

1. Calculate the change in y (Δy) and the change in x (Δx):

   • From (2, 10) to (4, 9): Δy = 9 - 10 = -1, Δx = 4 - 2 = 2
   • From (4, 9) to (6, 8): Δy = 8 - 9 = -1, Δx = 6 - 4 = 2
   • From (6, 8) to (8, 7): Δy = 7 - 8 = -1, Δx = 8 - 6 = 2

2. The slope (rate of change) can be calculated as:

   • Slope (m) = Δy / Δx
   • Between each of our intervals, this is:
     • (−1) / (2) = −0.5

Since the slope remains consistent (−0.5) across all intervals, this indicates a constant rate of change.

Therefore, we can conclude that the function is linear because it decreases at a constant rate. The statement "The function is linear because it decreases at a constant rate," is correct.

The other statements provided either present incorrect reasoning or focus on irrelevant factors, such as the sign of the values or the absence of x-value 0, which do not determine linearity.