6.02 Slope as a rate of change

What's the rate of change?

Recall that a rate is a ratio between two measurements with different units.

When we graph these rates, the rate of change can be understood as the slope, steepness or slope of a line. Further, we look at the equations in slope-intercept form (that is, $y=ax+b$y=ax+b, where $a$a is the slope), the larger the absolute value of $a$a, the steeper the slope of the line. Other versions of the slope intercept form of a line are $y=mx+b$y=mx+b or $y=mx+c$y=mx+c . 

For example, a line with a slope of of $4$4 is steeper than a line with a slope of $\frac{2}{3}$23​. Similarly, a line with a slope of $-2$−2 is steeper than a line with a slope of $1$1, even though one is positive and one is negative.

Increasing or decreasing?

The rate of change in a graph can be positive or negative.

The lines below have positive rates of change. Notice how as the values on the $x$x-axis increase, the values on the $y$y-axis also increase.

These next graphs have decreasing rates of change. Unlike graphs with a positive slope, as the values on the $x$x-axis increase, the values on the $y$y-axis decrease.

 

 

Rate of change

The rate of change of a line is a measure of how steep it is.  In mathematics we also call this the slope.

The rate of change is a single value that describes:

• if a line is increasing (has positive slope)
• if a line is decreasing (has negative slope)
• how far up or down the line moves (how the $y$y-value changes) with every step to the right (for every $1$1 unit increase in the $x$x-value)

Take a look at this line, where the horizontal and vertical steps are highlighted:

We call the horizontal measurement the run and the vertical measurement the rise. For this line, a run of $1$1 means a rise of $2$2, so the line has slope $2$2.

Sometimes it is difficult to measure how far the line goes up or down (how much the $y$y value changes) in $1$1 horizontal unit, especially if the line doesn't line up with the grid points on the $xy$xy-plane. In this case we calculate the slope by using a formula:

$\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run ​

Where you take any two points on the line whose coordinates are known or can be easily found, and look for the rise and run between them.

 

Slope from a graph

You can find the rise and run of a line by drawing a right triangle created by any two points on the line. The line itself forms the hypotenuse. 

This line has a slope of $\frac{\text{rise }}{\text{run }}=\frac{4}{3}$rise run ​=43​

In this case, the slope is positive, because over the $3$3 unit increase in the $x$x-values, the $y$y-value has increased. If the $y$y-value decreased as the $x$x-value increases, the slope would be negative.

Slope in action

This applet allows you to see the rise and run between two points on a line of your choosing.

Guiding questions

1. Identify the vertical and then horizontal distances between the two endpoints.  Is there a difference between the vertical distance (rise) in linear functions that are increasing and functions that are decreasing?
2. Does the order in which you express rise and run matter when you are stating the slope?
3. Discuss with a partner whether there is a connection between equivalent ratios (rates of change) and slope.
4. Does the order in which you express the rise and run matter when you express a rate of change?

 

Slope of horizontal and vertical lines

Horizontal lines have no rise value.  The $\text{rise }=0$rise =0.  

So the slope of a horizontal line is $\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run ​$=$=$\frac{0}{\text{run}}$0run​$=$=$0$0.

Vertical lines have no run value.  The $\text{run }=0$run =0.  

So the slope of a vertical line is $\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run ​$=$=$\frac{\text{rise }}{0}$rise 0​. Division by $0$0 results in the value being undefined.

 

Applications of slope

Consider these scenarios:

• The horizontal speed of a projectile
• The reproduction rate of a colony of bacteria
• The revenue growth of a business over a year
• The flow in and out of a body of water

In each of these cases, we are interested in how one measurement varies as another one does. That is, there is a dependent variable that varies with respect to an independent variable. However, we’re not interested in the measurements themselves but in how they vary. We call this the rate of change.

Exploration

Let’s focus on the first scenario listed above. We can’t directly measure speed, but we can measure distance and time. Notice that speed is the amount that distance changes per unit of time. That is, speed is the rate of change of distance with respect to time, so we can use our measurements of distance and time to figure out the speed.

Suppose that the distance ($x$x, in meters) is related to the time ($t$t, in seconds) by the relationship $x=3t$x=3t. Let’s plot this relationship first:

Notice that this is a linear function. Since speed is the rate of change of distance over time, we want to find out how the distance changes over any amount of time. Let’s start by picking two points on the line. We’ll use $\left(1,3\right)$(1,3) and $\left(3,9\right)$(3,9):

First we find the change in the independent and dependent variables:

Change in time	$=$=	$3-1$3−1 m
	$=$=	$2$2 m
Change in distance	$=$=	$9-3$9−3 s
	$=$=	$6$6 s

And then we divide the change in distance by the change of time to get the rate of change:

Rate of change	$=$=	$\frac{6}{2}$62​ m/s
	$=$=	$3$3 m/s

So the speed is $3$3 m/s. We will get the same result no matter which two points we choose. Notice that this is the same as the slope of a linear function. In fact, this is always the case when the function is linear.

 

Practice questions

Question 1

Question 2