Linear & Direct Relationships

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  • 0:01 Linear Relationships
  • 1:52 Direct Relationships
  • 2:58 Interpreting Slope
  • 4:37 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Learn how to tell when you're dealing with linear and direct relationships and what graphs of these relationships should look like. Then, go further and explore how to interpret the slopes of these graphs.

Linear Relationships

When scientists are working on an experiment, they are often collecting large sets of data. In order to understand what all the numbers mean in these sets of data, it helps to find a way to visualize them. The most common way scientists do this is through graphs. With a graph, we can look at the two sets of numbers forming our data points and try to figure out a relationship between them.

One of the simplest relationships we can see with a graph is a linear relationship. In a linear relationship the data points form a straight, best-fit line. We call the line created from the data points of a graph a curve, even if it is straight. For a linear relationship, we can represent the curve mathematically using the slope-intercept form of the equation of a line:

Equation of a line

We call this the slope-intercept form because here m is the slope, and b is the y-intercept. The slope gives us a measure of the steepness of the curve, and the y-intercept tells us the point where the curve passes through the y-axis of the graph. The last two variables, x and y, are the coordinates for any point on the curve.

Let's look at an example of a linear relationship you might run into in a science course. Have you ever had to convert a temperature from Celsius to Fahrenheit? If you have, then you used the following formula to do so:

temperature conversion

If we look at our equation of a line from before, we can see that this conversion equation is in the exact same form, where 9/5 is the slope and 32 is the y-intercept. When we graph a range of Fahrenheit temperatures vs. Celsius temperatures, we can see that it does indeed form a linear relationship.

Fahrenheit vs. Celsius
graph of fahrenheit vs celsius temps

Direct Relationships

In a direct relationship, as one variable increases, the other increases, or as one decreases, the other decreases. Often, you'll run across a special form of direct relationship called a directly proportional relationship where the variables are increasing or decreasing at the same rate and the curve passes through the origin, (0,0) point, of the graph. This is a special form of linear relationship that gives us a y-intercept of zero, changing our equation of a line to the following:

directly proportional

In a directly proportional relationship, our slope is a constant, meaning it is a number that never changes. For a directly proportional relationship, we call this the constant of proportionality.

In an introductory physics course, you'll find that directly proportional relationships show up quite a lot. A famous example of this is Newton's 2nd law:


Here, force (F) and acceleration (a) are directly proportional to each other as long as the object's mass (m) isn't changing.

Interpreting the Slope

In both linear and direct relationships, slope has been an important concept, but what does it mean? We already said in general that slope is a measure of the steepness of a graph's curve, but let's get a little more specific. Slope is defined as rise over run. Rise is the vertical change between two points on a line, and run is the horizontal change between two points on a line. Mathematically, we can write this as follows:

Slope equation

Let's visualize what a slope of 1/2, for example, would look like on our graph. This would mean that for every 1 unit we move vertically on the graph, we move horizontally 2 units.

Slope of 1/2
graph showing slope of 1/2

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