A student was asked to decide whether the function below is linear or nonlinear using both a table and a graph. He

Question

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?
yequals
minus6xsquared
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Part 1
Complete the table of values for the function.
x

y
minus
2


enter your response here
minus
1


enter your response here
0


enter your response here
1


enter your response here
2


enter your response here

Answer 1

The student has made a mistake in their reasoning. A linear function is one in which the rate of change is constant, but the graph of the function is a straight line. If the function is not a straight line when graphed, it is nonlinear.

The function given is:
y = -6x^2

This function is a quadratic function, not linear, because of the x^2 term. The graph of this function is a parabola, which is a curve, not a line.

Now let's complete the table of values for the function to see why this is the case:

For x = -2:
y = -6(-2)^2 = -6(4) = -24

For x = -1:
y = -6(-1)^2 = -6(1) = -6

For x = 0:
y = -6(0)^2 = -6(0) = 0

For x = 1:
y = -6(1)^2 = -6(1) = -6

For x = 2:
y = -6(2)^2 = -6(4) = -24

The completed table is:

x | y
-----
-2 | -24
-1 | -6
0 | 0
1 | -6
2 | -24

As shown, when x increases or decreases by 1, the change in y is not consistent, which already indicates that the function is not linear. The graph would show a parabola with the vertex at the origin (0,0) opening downwards, further indicating that it is a nonlinear function. Thus, the student's assertion that the function is nonlinear is correct, but their reasoning that the rate of change is constant is incorrect - the rate of change of a linear function is what's constant, not for a quadratic function. In this quadratic function, the rate of change itself changes as x changes.