8.F.A.3 Tutor Intake -- Linear vs. Nonlinear Functions

Because for any input $x$, the rule "multiply by $m$, then add $b$" produces exactly one output $y$.

Because $m$ and $b$ are always positive numbers.

Because the equation can be graphed on a coordinate plane.

Because $x$ and $y$ are both variables.

Because the output is always positive.

Because equal horizontal steps always produce equal vertical steps, so the points line up in a consistent, uniform pattern.

Because the rate of change equals the y-intercept.

It doesn't have to be a straight line -- a constant rate of change just means it is a function, not that it is linear.

Linear -- the rate of change is constant at 3 for every step.

Nonlinear -- the output values keep increasing.

Linear -- because all the outputs are positive.

Nonlinear -- because the table has four rows.

Nonlinear -- the y-differences (2, 4, 6) are not constant.

Nonlinear -- the x-values do not increase by the same amount each row, so the function cannot be linear.

Linear -- the x-steps are 1, 2, 3, and computing (change in y)/(change in x) for each gives 2/1 = 2, 4/2 = 2, 6/3 = 2, which is constant.

It cannot be determined without a graph.