Describe the Functional Relationship Between Quantities

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• 0:01 Functional Relationships
• 1:05 Graphs to Functions
• 2:23 Example Problems
• 3:45 Lesson summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll learn how to look at a graph, table, or function written mathematically and verbally describe the functional relationship shown. For example, you'll learn to describe whether the function is increasing or decreasing.

Functional Relationships

You probably know a function is something you write out with numbers, show in a table, or plot on a graph. But you can also describe a functional relationship, or the relationship between the inputs and outputs of a given function, with words. Sometimes, you can specify exact numbers and specific quantities, but even if you can't do that, you can describe the general behavior of the function.

For example, think about the function y = 3 - x. Here's a table showing some input and output values of that function:

Input (x) Output (y)
1 2
2 1
3 0
4 -1
5 -2
6 -3

If you wanted to describe the relationship between the numbers in the left-hand column and the numbers in the right-hand column, you might say that for every 1-unit increase in x, there is a 1-unit decrease in y. Or if you didn't have the exact numbers, and all you had was a graph, you could also say, as the inputs get bigger, the outputs get smaller.

You've just described a functional relationship between two sets of quantities! You can also describe other aspects of a functional relationship with words. For example, you can say whether the output is changing at a constant rate, or whether the rate of change itself is changing. If the rate is changing, is it speeding up or slowing down?

From Graphs to Functions

If you're looking at a graph, you don't need to see exact numbers to be able to describe the relationships you're seeing. You can simply look at patterns of behavior. And in fact, even if you do have exact numbers, learning to recognize these patterns of behavior will help you recognize functional relationships more quickly.

Here are a few of the biggest players. If you see a graph where the line is slanting up from left to right, it means that the function is increasing; in other words, as x increases, y increases. If you see a graph where the line is slanting down from left to right, it means the opposite: the function is decreasing, or as x increases, y decreases.

If the graph is a straight line, the rate of change is constant. If the graph is curved, the rate is changing. If you have a changing rate, if the graph is getting steeper, then the rate of change is increasing. If it's getting flatter, the rate of change is decreasing.

Using these relationships, you can come up with a verbal description of a function's behavior even if you don't know any specific numbers and can't put it into a table. In fact, even if you did have numbers on those graphs, it's still very handy to be able to look at the picture and instantly see what the general behavior of the function is like - it saves you a lot of time plugging in numbers.

Example Problems

Now, let's look at an example. Here's a function. Can you describe its behavior?

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