A student was asked to decide whether the function below is linear or nonlinear using both a table and a graph. She said the function is nonlinear because the rate of change is constant and the graph of the function is not a line. Decide whether the function is linear or nonlinear using both a table and a graph. What mistake might the student have made? Complete the table of values for the function. X -2 -1 y 1 2
To determine whether a function is linear or nonlinear, we need to analyze the rate of change. If the rate of change is constant, the function is linear. If the rate of change is not constant, the function is nonlinear.
In this case, the student correctly observed that the rate of change is constant. However, they mistakenly concluded that the function is nonlinear because the graph of the function is not a line. The graph of a linear function will always be a straight line, but a nonlinear function can have a curved graph.
To confirm the linearity of the function, we can check if the difference in y-values is proportional to the difference in x-values.
From the given values, we have:
x: -2 --> -1: y: 1 --> 2
The difference in x-values is: -1 - (-2) = 1
The difference in y-values is: 2 - 1 = 1
Since the difference in y-values is equal to the difference in x-values, the function is indeed linear.
The student's mistake was concluding that the function is nonlinear based solely on the graph's appearance, without considering the constant rate of change.
Completing the table using the linear relationship between x and y-values:
X -2 -1 0 1 2
y 1 2 3 4 5
Based on the given values, let's complete the table of values for the function:
X | Y
-------------
-2 | 1
-1 | 2
From the table, we can calculate the differences in the Y-values:
ΔY = 2 - 1 = 1
Now, let's plug in the values into the slope formula (rate of change):
m = ΔY / ΔX = (2 - 1) / (-1 - (-2)) = 1 / (-1 + 2) = 1 / 1 = 1
From this calculation, we can see that the rate of change is constant (equal to 1) for both the table and the graph. Therefore, the student's claim that the rate of change is constant is correct. However, the student's claim that the function is nonlinear because the graph is not a line is incorrect.
From the provided values in the table, it is not possible to determine the exact nature of the function. The given values are consistent with both linear and nonlinear functions. To make a definitive conclusion, we would need more values and information about the function. Without that information, we cannot determine whether the function is linear or nonlinear.
To determine whether a function is linear or nonlinear, we need to consider two things: the rate of change and the graph of the function.
If the rate of change (also known as the slope) is constant, then the function is linear. This means that for every unit increase in the independent variable (x), there will be a consistent change in the dependent variable (y).
In the given example, the student claimed that the function is nonlinear because the rate of change is constant. However, it's important to note that constant rate of change is a characteristic of linear functions, not nonlinear functions. So, the student made a mistake in saying that the function is nonlinear based on a constant rate of change.
To further analyze the function, let's complete the table of values:
X | Y
-2 | 1
-1 | 2
Now let's plot these points on a graph:
(-2, 1) and (-1, 2)
Upon plotting the points, we can observe that they fall in a straight line. This straight line suggests that the graph of the function is linear.
Therefore, the student made a mistake by concluding that the function is nonlinear based on a constant rate of change. In fact, the function is linear, as both the rate of change is constant and the graph is a straight line.