$Linear Change and Rate$

modelling Level 1

Linear Change and Rate

The simplest non-trivial model is a straight line. When temperature drops with elevation, when population grows with time, when erosion deepens with distance from a ridge — these are all linear relationships. Understanding slope means understanding rate, and rate is the language of change.

Prerequisites: linear functions, slope, rate of change, units of rate

26 Feb 2026 Updated 12 Mar 2026 9 min read

1. The Question

How do we describe change that happens at a constant rate?

Temperature drops as you climb a mountain. Population grows year by year. A river’s gradient flattens as it approaches the sea. Some of these relationships are approximately linear over the range we care about. Others are exactly linear by design (a road built at 6% grade, a tax that increases by $50 per bracket).

The question is: How do we write down a linear relationship mathematically? What does the equation tell us? What does the slope mean, and why do the units matter?

2. The Conceptual Model

A linear relationship is one where equal changes in input produce equal changes in output.

Walk up 100 meters in elevation → temperature drops 0.6°C
Walk up another 100 meters → temperature drops another 0.6°C

This is linearity: constant rate of change.

The general form is:

\[y = mx + b\]

Where:

$y$ is the output (the thing that responds)
$x$ is the input (the thing we control or observe changing)
$m$ is the slope (the rate of change)
$b$ is the intercept (the value of $y$ when $x = 0$)

Slope is the engine of the model. It tells you how much $y$ changes for each unit change in $x$.

3. Building the Mathematical Model

Deriving Slope from Two Points

Suppose we measure temperature at two elevations:

Elevation (m)	Temperature (°C)
1000	15.0
1500	12.0

How do we find the rate of change?

Change in temperature:
$\Delta T = T_2 - T_1 = 12.0 - 15.0 = -3.0 \text{ °C}$

Change in elevation:
$\Delta z = z_2 - z_1 = 1500 - 1000 = 500 \text{ m}$

Rate of change (slope):
$m = \frac{\Delta T}{\Delta z} = \frac{-3.0 \text{ °C}}{500 \text{ m}} = -0.006 \text{ °C/m}$

The negative sign tells us temperature decreases with elevation. The magnitude tells us by how much per meter.

Writing the Full Equation

We know $m = -0.006$ °C/m. We need the intercept $b$.

Using the point $(z_1, T_1) = (1000, 15.0)$:

$T = mz + b$ $$15.0 = (-0.006)(1000) + b$$$15.0 = -6.0 + b$ $b = 21.0 \text{ °C}$

Full model:
$T(z) = -0.006z + 21.0$

Interpretation:

At sea level ($z = 0$), temperature would be 21.0°C
For every meter of elevation gained, temperature drops by 0.006°C
At 2000 m: $T = -0.006(2000) + 21.0 = 9.0$ °C

4. Worked Example by Hand

Problem: A glacier is retreating. In 1990, its terminus was at 2400 m elevation. In 2020, it had retreated to 2700 m elevation. Assuming constant retreat rate:

(a) What is the rate of retreat in meters per year?
(b) Write the equation for terminus elevation as a function of year.
(c) When will the terminus reach 3000 m if the trend continues?

Solution

(a) Rate of retreat

$\Delta z = 2700 - 2400 = 300 \text{ m}$ $\Delta t = 2020 - 1990 = 30 \text{ years}$ $\text{rate} = \frac{300 \text{ m}}{30 \text{ yr}} = 10 \text{ m/yr}$

The glacier terminus is moving upslope at 10 m/yr.

(b) Equation

Let $z(t)$ be elevation, $t$ be year.

\[z(t) = mt + b\]

We know $m = 10$ m/yr. Using the point $(1990, 2400)$:

$$2400 = 10(1990) + bb = 2400 - 19900 = -17500$$

Model:
$z(t) = 10t - 17500$

Check: $z(2020) = 10(2020) - 17500 = 20200 - 17500 = 2700$ ✓

(c) When will $z = 3000$?

$$3000 = 10t - 17500$$$10t = 20500$ $t = 2050$

If the linear trend holds, the terminus will reach 3000 m in 2050.

5. Computational Implementation

Below is a simple interactive model. Adjust the parameters and watch the line respond.

Slope (m): 0.5 Intercept (b): 2

Try this:

Set $m = 0$. What happens? (A horizontal line — no change with $x$)
Set $m = 1$ and $b = 0$. The line passes through the origin at 45°.
Set $m = -1$. The line tilts the other way.
Set $m = 2$ and $b = -5$. Where does the line cross the x-axis? (Solve $0 = 2x - 5 \Rightarrow x = 2.5$)

6. Interpretation

What Slope Means Physically

The slope $m$ is not just a number. It is a ratio of two physical quantities, and its units tell you what it measures.

Examples:

Context	$y$	$x$	Slope $m$	Units	Meaning
Elevation profile	elevation	horizontal distance	$\frac{\Delta z}{\Delta x}$	m/m or %	gradient
Temperature lapse	temperature	elevation	$\frac{\Delta T}{\Delta z}$	°C/m	lapse rate
Population growth	population	time	$\frac{\Delta P}{\Delta t}$	people/year	growth rate
Cost function	total cost	quantity	$\frac{\Delta C}{\Delta q}$	$/unit	marginal cost

Units are not decorative. They encode what the model means.

Positive vs. Negative Slope

$m > 0$: $y$ increases as $x$ increases (uphill, warming, growth)
$m < 0$: $y$ decreases as $x$ increases (downhill, cooling, retreat)
$m = 0$: no change (flat, constant, equilibrium)

Steepness

The absolute value $	m	$ measures how fast the change happens:
- $	m	$ small: gradual change (prairie, slow warming, stable population)
- $	m	$ large: rapid change (cliff, thermal inversion, population boom)

7. What Could Go Wrong?

Assuming Linearity Beyond the Data Range

Our glacier model predicted the terminus would reach 3000 m in 2050. But what if:

The glacier disappears entirely before then?
Climate policy reduces warming, slowing retreat?
A cold decade causes temporary advance?

Linear models are local approximations. They work well over the range where you measured them. Extrapolation is prediction, and prediction requires caution.

Ignoring Units

A common student error:

“The slope is 0.006.”

Wrong. The slope is $-0.006$ °C/m. Without units, you don’t know if you’re talking about temperature change, population change, or the cost of bananas.

Confusing Slope with Intercept

The intercept $b$ tells you where the line crosses the $y$-axis. It is often physically meaningful (sea-level temperature, initial population, fixed cost), but it is not the rate of change. That’s the slope’s job.

Mixing Up Dependent and Independent Variables

In $T = -0.006z + 21$, temperature depends on elevation. You don’t choose the temperature and look up the elevation. The convention is:

Independent variable (the one you control or measure): on the x-axis
Dependent variable (the one that responds): on the y-axis

Flipping these gives you a different slope. $\frac{\Delta z}{\Delta T}$ is the inverse of $\frac{\Delta T}{\Delta z}$.

8. Extension: From Lines to Gradients

In the next model, we’ll ask: what happens when the rate of change itself changes? When growth accelerates or slows? When temperature doesn’t drop at a constant rate, but follows an exponential curve?

But before we get there, notice this: the slope of a line is the simplest possible derivative.

If $y = mx + b$, then:

\[\frac{dy}{dx} = m\]

The derivative is the rate of change. For a line, the rate is constant. For curves, it varies. That’s where calculus begins.

But you don’t need calculus to understand that slope is rate, and rate is how we measure change.

Summary

A linear model has the form $y = mx + b$
Slope $m = \frac{\Delta y}{\Delta x}$ measures the rate of change
Units of slope are (units of $y$) / (units of $x$)
Positive slope: increasing; negative slope: decreasing; zero slope: constant
Linear models are local approximations — valid over the range you measured
The derivative of a linear function is its slope

Math Refresher: Solving for x

If you’re rusty on algebra, here’s the pattern:

Given: $y = mx + b$, and you know $y$, $m$, $b$ — solve for $x$.

Steps:

Subtract $b$ from both sides: $y - b = mx$
Divide both sides by $m$: $x = \frac{y - b}{m}$

Example: When does $T = -0.006z + 21$ reach $T = 10$°C?

$$10 = -0.006z + 21-11 = -0.006zz = \frac{-11}{-0.006} = 1833.3 \text{ m}$$

At 1833 meters elevation, temperature would be 10°C.