Linear vs Nonlinear Functions: How to Tell Them Apart

Check the power of $x$ and where it sits.

Yes.

$x$ appears to the first power, in the numerator, and is not multiplied by another variable. The graph is a line of slope $7$.

No.

The cube means the rate of change varies with $x$, so the graph curves.

Yes.

The variable is in the numerator; dividing by $3$ is a coefficient of $\tfrac{1}{3}$. The function is linear with slope $\tfrac{1}{3}$.

Yes.

The $x$-steps are all $1$ and the $y$-steps are all $3$, so the rate is a constant $3$. The function is $y = 3x + 2$.

Look at whether the outputs are being added to or multiplied.

No.

The $y$-steps are $3$, $6$ and $12$, which grow rather than repeat. Each output is double the previous one, so this is exponential rather than linear.

Yes.

Both variables are to the first power and neither is in a denominator. Rearranged it is $y = \tfrac{5}{2}x - 4$.

The $x$-steps are uneven, so compute rates rather than comparing $y$-steps.

Yes.

First interval: $\tfrac{17 - 5}{3 - 0} = 4$.

Second interval: $\tfrac{37 - 17}{8 - 3} = \tfrac{20}{5} = 4$.

Both rates are $4$, so the relationship is linear: $y = 4x + 5$. The $y$-steps of $12$ and $20$ looked unequal only because the $x$-steps were.

Build a small table for each and compare the steps.

The second is linear; the doubling one is not.

Doubling: $100, 200, 400, 800$. The steps are $100$, $200$, $400$ — growing, so the rate is not constant. Nonlinear.

Adding: $100, 200, 300, 400$. Every step is $100$, so the rate is a constant $100$ cells per hour, giving $y = 100x + 100$. Linear.

Both start at $100$ and both reach $200$ after one hour, so a single early reading cannot tell them apart. You need at least three points to see whether the steps stay equal.