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Physics: High School18 chapters | 212 lessons

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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.

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**:

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:

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.

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:

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.

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:

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.

In physics, the slope we find often represents not only just a number but also a real world property. To understand this, let's look at a graph of distance vs. time for a jogger. In our equation for slope, rise would be the change in distance and run a change in time. Well, a change in distance divided by a change in time is what we call velocity:

Let's look at three examples of a distance vs. time graph:

One has a much steeper slope than the other two. Since our *y*-axis represents distance, the steeper the slope, the greater the distance per unit of time that is being traveled. In other words, the steeper the slope, the faster the jogger is moving. Finally, a completely horizontal slope would represent the jogger standing still, as he or she is moving zero distance over an increasing amount of time.

In a **linear relationship**, the data points on a graph form a straight, best-fit line. We call the line created from the data points a curve, even if this curve is a straight line. For a linear relationship, we can represent this curve mathematically by using the point-slope form of the **equation of a line**:

Here, *x* and *y* are the coordinates for any point on the curve, the slope (*m*) is a measure of the steepness of the curve, and the *y*-intercept (*b*) is the point where the curve passes through the *y*-axis of the graph.

In a **direct relationship**, as one variable increases, the other increases, or as one decreases, the other decreases. When the variables are increasing or decreasing at the same rate and the curve passes through the graph's origin, we have a special form of direct relationship called a **directly proportional** relationship. Since this makes the *y*-intercept equal zero, our equation of a line formula changes to the following:

Here, the slope is a constant that never changes, called the **constant of proportionality**.

**Slope** is an important concept in both linear and direct relationships. While we generally define slope as a measure of the steepness of a curve, it can more accurately be described as rise over run, which can be shown mathematically as follows:

In a physics class, a slope represents not only just a number but also some physical property. By interpreting what the slope means, we can learn more about that physical property.

After you've completed this lesson, you'll be able to:

- Describe linear and direct relationships
- Identify the equations of lines that coincide with these relationships
- Recall what the constant of proportionality is
- Explain how slope can be used to establish relationships between data points

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Physics: High School18 chapters | 212 lessons

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