Build and use linear models

Here's an amount, in liters, measured every $3$ minutes. How does it change?

Each input step is $3$ minutes, and each output step is

liters. The same $18$ liters gets added in every $3$-minute stretch. Per minute, that's

That steady change is what makes a relationship linear. A linear model is one whose output changes at a constant rate, so equal steps in the input always add the same amount to the output.

Word problems usually give it away:

• "At a constant rate" or "the same amount per hour" (or per item, or per mile) tells you a fixed amount comes with each unit of input.
• A fixed amount plus an amount for each unit, like a sign-up fee plus a charge for each month, is linear too.
• A table like the one above, with equal jumps over equal input steps, is the same signal in numbers.

Here's a trap, though. Going up or down doesn't tell you which kind of model you have, because linear and exponential models can both do either. What matters is how the output changes. A constant difference over equal steps means linear: you add the same amount each time. A constant ratio over equal steps means exponential: you multiply by the same number each time.

Compare $40,52,64,76$, which adds $12$ each time, so it's linear, with $40,52,67.6,87.88$, which multiplies by $1.3$ each time, so it's exponential. Both jump from $40$ to $52$ first, so you can't tell them apart until the third value. Exponential models get their own lesson later, in Build, identify, and interpret exponential models.

All of this assumes the problem gives you an exact constant rate. If it gives several measured points that only roughly follow a line, use a best-fit model instead.

Sometimes the story never tells you the value at the start. Say a freezer is at $-6^\circ\text{C}$ four hours after a power outage, and its temperature is rising at a constant rate of $1.5^\circ\text{C}$ per hour. Let $T(h)$ be the temperature $h$ hours after the outage.

Before you write an equation, sort out four things. Here they are for the freezer:

1. The input is the time since the outage, in hours.
2. The output is the temperature, in degrees Celsius.
3. The rate is the change in output for each unit of input, here $1.5^\circ\text{C}$ per hour. It's positive because the freezer is warming up. If the temperature were falling, the rate would get a minus sign.
4. You know one input and the output that goes with it: $-6^\circ\text{C}$ at $4$ hours. That's the point $(4,-6)$, and it's your anchor.

Now build the model from the anchor:

Here's how to read it. The part $h-4$ is how far the time has moved from the known point, in hours. Multiplying by $1.5$ turns those hours into degrees of change. Adding $-6$ starts you from the temperature you already know.

The same pattern works for any rate $m$ and known point $(x_0,y_0)$:

Every model like this says the same thing: start from the point you know, then add the rate times how far you've moved.

A quick check tells you the point went in correctly. At $h=4$, the input change $h-4$ is zero, so the model should give back the known temperature, $-6^\circ\text{C}$. It does.

You can also expand the model:

Both forms describe the same line. The first keeps the known point in view. The second shows the value at $h=0$: the model puts the freezer at $-12^\circ\text{C}$ when the power went out.

When the problem tells you the output at input zero directly, use the familiar form

where $b$ is that starting output, the real value when the input is $0$.

The tank problem shows the big idea here: the equation doesn't know the story, but you do. The line goes on forever, straight past $800$ liters, while the real tank can't hold a drop more than $800$. An equation will just as happily hand you a negative number of tablets or half a person. So once you've calculated a prediction, hold it up against the story with three questions:

1. Is it the right quantity, in the right units? A question about a total charge wants dollars, not dollars per hour.
2. Can that value really happen? Negative time may not make sense, a count has to be a whole number, and some amounts can't drop below zero or go past a capacity.
3. Does the constant rate really last that far? A prediction inside the range of values you were given is usually safer than one far outside it. Going past them doesn't make the arithmetic wrong, but it makes the real-world answer less certain.