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Physics: High School18 chapters | 211 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.

Explore how we tell when two variables are in quadratic or inverse relationships in this lesson. Once you understand the basics, we'll go over a couple of examples where these relationships show up in a physics class.

When you're working a physics lab for a class, you'll often find yourself making graphs of a couple variables. You change one of the variables yourself, and track the corresponding change in the other. For instance, you might be tasked with placing a sealed container filled with gas in a pot of water, and measuring the change in pressure of the gas as the water is heated. Here, the temperature of the water is the variable you are changing, and the gas pressure is the second variable you are tracking. When you make a graph of pressure vs. temperature you are exploring the relationship between these two variables.

In an introductory physics course, there are four different common relationships between variables you are bound to run into: they are linear, direct, quadratic, and inverse relationships. Here, we'll go over both quadratic and inverse relationships, and a couple examples of places they pop up in a physics course.

A **quadratic relationship** is a mathematical relation between two variables that follows the form of a quadratic equation. To put it simply, the equation that holds our two variables looks like the following:

Here, *y* and *x* are our variables, and *a*, *b*, and *c* are constants. If you didn't have this equation, and only had some data points for a graph, you'd be able to tell it's a quadratic relation if the graph's curve forms a **parabola**, which on a graph looks like a dip or a valley. Even if we have an equation like this where *b* and *c* both equal zero, it's still considered quadratic. If that happens, we get the simplest form of a quadratic relationship:

This works because it turns out that it's the *x* squared component that's absolutely necessary for a relationship to be quadratic. So, unlike *b* and *c*, *a* must never equal zero because that will remove the *x* squared from the formula since zero multiplied by anything is zero.

One of the first places you'll encounter a quadratic relation in physics is with projectile motion. This makes sense if you think about how a projectile travels through the air over time. Imagine you're tossing a baseball straight up in the air. Let's try visualizing this with a height vs. time graph. Over time the ball goes up to a maximum height, and then back down to the starting height again when you catch it.

We can see our graph creates an upside-down parabola, which is the sort of thing you might expect from a quadratic relation. To make absolutely sure the relation between height and time is quadratic, we'll also look at the vertical equation for projectile motion that deals with position and time:

Does it look familiar? Let's try rearranging the equation a bit:

You might not know this yet, but in this equation the only two variables are height (*y*) and time (*t*). Everything else is constant over the course of a single throw. So, the equation we're seeing here is really in the exact form of a quadratic equation:

You might remember that in a direct relationship, as one variable increases, the other increases, or as one decreases, the other decreases. In an **inverse relationship**, instead of the two variables moving in the same direction they move in opposite directions, meaning as one variable increases, the other decreases.

Often in a physics course, the type of inverse relationship you'll run across is an inversely proportional relationship. For **inversely proportional** relationships, we specify that as one variable increases the other decreases at the same rate.

When we graph an inversely proportional relationship we see that the graph forms a **hyperbola**, which looks roughly like two parabolas facing back to back on a graph. Because of the way a hyperbola is shaped, it's not too uncommon to only see half of it when graphing data points, such as when you only have positive numbers in your data set. An inversely proportional relationship can be represented by the following equation, where *k* is a constant:

We can see an example of inverse proportionality in physics with Boyle's law. Robert Boyle investigated the behavior of gas when held at a constant temperature. He found that if you change the pressure of a gas at a constant temperature, the volume will change to keep the product between the two constant. In other words:

This formula can also be written with pressure and volume on opposite sides of the equals sign, giving us an inversely proportional equation:

In fact, the definition of **Boyle's law** states that at a constant temperature the volume and pressure of a gas are inversely proportional to each other.

In a physics lab, you will often be tasked with exploring the relationship between two variables by making graphs. Two of the more common types of relationships you will run into are quadratic and inverse relationships. A **quadratic relationship** is a mathematical relation between two variables that follows the form of a quadratic equation:

Here, *y* and *x* are the variables we are exploring the relationship between, and *a*, *b*, and *c* are constants. It's all right for the *b* and *c* constants to equal zero, but *a* must never equal zero in a quadratic relationship. If *a* equaled zero, we would lose the *x* squared part of the formula, which is necessary for the relationship to remain quadratic. This means an equation of the following form shows a quadratic relationship even though it's missing the *b* and *c* parts:

If you don't have an equation to look at, you'll be able to tell you have a quadratic relation if the graph's curve forms a **parabola**, which on a graph looks like a dip or a valley. In an **inverse relationship** as one variable increases the other decreases. Often you'll come across a special type of inverse relationship called an inversely proportional relationship.

For two variables to be considered **inversely proportional** as one variable increases the other must decrease at the same rate. In an inversely proportional relationship we see that the graph forms a **hyperbola**, which looks roughly like two parabolas facing back to back on a graph. Finally, an inversely proportional equation will take the following form:

This lesson on quadratic and inverse relationships is designed to help you to:

- Write and explain the equations for quadratic and inverse relationships
- Identify the graph for a quadratic or an inverse relationship

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Physics: High School18 chapters | 211 lessons

- What is Physics? - Definition, History & Branches 5:56
- Math Review for Physics: Algebra 7:45
- Math Review for Physics: Trigonometry 5:16
- Elements of the SI: Base & Derived Units 7:41
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- Unit Conversion and Dimensional Analysis 10:29
- Significant Figures and Scientific Notation 10:12
- Linear & Direct Relationships 6:19
- Quadratic & Inverse Relationships 7:26
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