Linear vs. Nonlinear Functions

Linear and Nonlinear Functions

By the end of this lesson, you will be able to decide whether a function is linear from its table, graph or equation, and explain the one property that decides it: whether the rate of change stays the same.

One Question Decides It

A function is linear when it changes at a constant rate. That is the whole definition, and every test below is that one question asked of a different representation.

$$\text{linear} \iff \text{the same change in } x \text{ always causes the same change in } y$$

A taxi charging $2 a mile is linear: mile ten costs the same as mile one. A savings account paying interest is not: the growth gets bigger as the balance does, because it is a percentage of a number that keeps changing.

That is the real distinction, and "makes a straight line" is a consequence of it rather than the definition. Constant rate of change IS a straight line, drawn.

From a Table: Look at the Differences

Take the differences between consecutive outputs, but only when the inputs step evenly.

$x$	$0$	$1$	$2$	$3$
$y$	$2$	$5$	$8$	$11$

The $x$ values go up by $1$ each time. The $y$ values go $+3, +3, +3$. Constant, so linear.

$x$	$0$	$1$	$2$	$3$
$y$	$1$	$2$	$4$	$8$

Now the differences are $+1, +2, +4$. Not constant, so not linear. (This one is doubling, which is exponential, and the unit on exponentials is where it goes.)

The trap. The differences only mean anything when the inputs are evenly spaced. With $x$ at $0, 1, 5$ the output differences compare unequal gaps, so they tell you nothing. Divide by the change in $x$ instead:

$$\text{rate} = \frac{\text{change in } y}{\text{change in } x}$$

which is the slope, and it is the version that always works.

Example 1: Testing a Table

Here is the whole method on one table, worked all the way through.

$x$	$2$	$4$	$6$	$8$
$y$	$10$	$16$	$22$	$28$

• By how much does x step each time? 2

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