Final answer:
The function is linear if the rate of change is constant and the graph is a straight line. The student incorrectly associated a constant rate of change with a nonlinear function, but it is actually indicative of a linear function. Therefore, the correct answer is that the function is linear.
Step-by-step explanation:
To determine whether the function represented by the table is linear or nonlinear, we need to consider the definition of a linear function and how we can recognize one both in tabular form and graphically.
A linear function is one whose graph in a coordinate plane is a straight line. This means that for any two points on the line, the rate of change, or the slope, is constant. The equation of a linear function is typically given by y = mx + b, where m is the slope and b is the y-intercept.
To decide if the function is linear using a table, we would look for a constant change in y for each unit change in x. If the change in y is the same for each interval of x, then the function is linear. However, in this problem, we are not given the actual y values. Nonetheless, we can analyze the concept that if the rate of change is constant, then the function should indeed be linear and not nonlinear as the student claims.
For the graphical approach, plotting the given points and analyzing the graph can confirm if it represents a straight line. If it does, then the function is linear. If it curves in any way, the function is nonlinear. Given that a straight line has a constant slope, if a student observes that the graph of the function is not a straight line, then she should conclude the function is nonlinear. But, the student has made a mistake by stating that a constant rate of change indicates a nonlinear function; it is the opposite.
Possible Error Made by the Student
The mistake the student may have made lies in the understanding of what a constant rate of change means. A constant rate of change is a feature of a linear function and results in a straight-line graph. Therefore, if the graph is not a line, the rate of change is not constant, and the function must be nonlinear.
Consequently, the correct statement would be: The function is linear, and the student mistakenly thought it was nonlinear. Thus, the correct answer would be '(a) The function is linear, and the student mistakenly thought it was nonlinear.'