Is the function represented by the table non-linear?
| x | y |
|---|---|
| 6 | 4 |
| 7 | 2 |
| 8 | 0 |
| 9 | -2 |
A. Yes, because it has a constant rate of change.
B. Yes, because it does not have a constant rate of change.
C. No, because it has a constant rate of change.
D. No, because it does not have a constant rate of change.
Answer (1)
The rate of change between consecutive points is calculated. Since the rate of change is constant (-2), the function is linear. Therefore, the function is not non-linear. The answer is: No, because it has a constant rate of change.
Explanation
Understanding the Problem We are given a table of x and y values and asked to determine if the function represented by the table is non-linear. To do this, we need to check if the rate of change between consecutive points is constant. If the rate of change is constant, the function is linear. If the rate of change is not constant, the function is non-linear.
Calculating Rate of Change The rate of change between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: Δ x Δ y = x 2 − x 1 y 2 − y 1 We will calculate the rate of change between consecutive points in the table.
Determining Constant Rate of Change
Between the points (6, 4) and (7, 2): 7 − 6 2 − 4 = 1 − 2 = − 2
Between the points (7, 2) and (8, 0): 8 − 7 0 − 2 = 1 − 2 = − 2
Between the points (8, 0) and (9, -2): 9 − 8 − 2 − 0 = 1 − 2 = − 2 The rate of change between all consecutive points is -2.
Conclusion Since the rate of change is constant (-2), the function is linear. Therefore, the function is not non-linear. The correct answer is: No, because it has a constant rate of change.
Examples
In real life, understanding linear and non-linear relationships is crucial in various fields. For instance, in physics, the relationship between distance and time for an object moving at a constant speed is linear. However, if the object is accelerating, the relationship becomes non-linear. Similarly, in economics, demand and price might have a linear relationship under certain conditions, but factors like market saturation can introduce non-linearity. Recognizing these relationships helps in making accurate predictions and informed decisions.
