2.2: The slope of a straight line is a rate of change

Learning Objectives

1. Define rate of change for a linear relationship.
2. Compute the rate of change for a linear relationship.

In the examples discussed so far, we have plotted data and used words to describe trends. Our goal now is to formalize the idea of change and rate of change. Let us consider the simplest case where a variable of interest, \(y\) depends linearly on time, \(t\). This was approximately true in some examples seen previously (Figure 2.2a, parts of Figure 2.3). We can describe this kind of relationship by the idealized equation

\[y(t)=m t+b \]

A graph of \(y\) versus \(t\) is then a straight line with slope \(m\) and intercept \(b\).

The equation of a straight line (\(\PageIndex{1}\)) specifies the slope m and the y intercept b of the line, as shown by manipulating the sliders on this interactive graph.

Definition: 2.1 (Rate of change for a linear relationship)

For a straight line, we define the rate of change of \(y\) with respect to time \(t\) as the ratio:

\[\frac{\text { Change in } y}{\text { Change in } t} \text {. } \nonumber \]

We now make an important observation.

Theorem

The slope \(m\) of the straight line in Equation (\(\PageIndex{1}\)) corresponds to definition \(2.1\) of the rate of change of a linear relationship.

Proof: Taking any two points \(\left(t_{1}, y_{1}\right)\) and \(\left(t_{2}, y_{2}\right)\) on that line, and using the notation \(\Delta y, \Delta t\) to represent the change in \(y\) and \(t\) we compute the ratio \(\Delta y / \Delta t\) and simplify algebraically to find:

\[\begin{aligned} \frac{\text { Change in } y}{\text { Change in } t}=\frac{\Delta y}{\Delta t} & =\frac{y_{2}-y_{1}}{t_{2}-t_{1}} \\ & =\frac{\left(m t_{2}+b\right)-\left(m t_{1}+b\right)}{t_{2}-t_{1}}=\frac{m t_{2}-m t_{1}}{t_{2}-t_{1}}=m . \end{aligned} \nonumber \]

Thus, the slope m corresponds exactly to the notion of change of y per unit time which we call henceforth the rate of change of y with respect to time. It is important to notice that this calculation leads to the same result no matter which two points we pick on the graph of the straight line.

Mastered Material Check

4. What does it mean for two variables to have a linear relationship?
5. Why are we ’idealizing’?
6. Compute the rate of change for a linear relationship which goes through points \((1,4)\) and \((2,2)\).

Hint: A “Theorem” is just a mathematical statement that can be established rigorously by an argument called a “proof”. While we will not use such terminology often here, it is a staple of mathematics.

Featured Problem 2.1 (Velocity of growing microtubule tips) Shown in Figure 2.5 is a part of the kymograph from Figure 2.1, but with a more "conventional" view of position \(y\) (in \(\mu \mathrm{m}\) ) on the vertical axis versus time t (in seconds) on the horizontal axis. The bright streaks are the tips of microtubules (MT) at various positions as they grow inside a cell.

1. Use the image and the superimposed grid to estimate the average velocities of microtubule tips (up to one significant digit). This is best done on the lower panel, where the position versus time graphs can be most easily viewed.
2. Based on the lines in the graph, explain whether the three microtubule tips shown are moving at similar or at quite different speeds.
3. Compare the normal and treated cells shown in two panels of Figure 2.1, carefully noting the fact that the coordinate system differs from that of Figure 2.5. How does the treatment affect the speed of microtubule tips?
4. Explain what could account for the apparent "zigzags" and "curves" (not straight lines) seen in both panels of Figure 2.1.

Figure 2.5: Part of the image from Figure 2.5 is shown (top panel) on a \(yt\) (distance in µm vs time in seconds) coordinate system, with a grid superimposed. The red lines represent positions of proteins that track the tips of growing microtubules over time. In the lower panel, the same lines are shown on the grid alone; we can use this to estimate average velocity