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Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

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Lesson Transcript

Instructor:
*John Sepanski*

John has taught 6th Grade Mathematics through Geometry and has a Master's degree in Education

In this lesson, we will learn about modeling linear relationships by converting domain and range values into ordered pairs and then constructing a graph.

Do you text a lot? Maybe you've got a friend who texts a whole lot. I've got a friend named Zack who texts all of the time. I mean a whole lot! The other night I sat and watched him for a little while and determined that he was sending about eight texts every five minutes! At that rate, it means he would send 24 texts every 15 minutes. That would mean that he could send almost 100 texts an hour! That would be about 2,304 texts a day! That would be 16,128 texts a week! That's a whole lot of texts! But it's also a great model of a linear relationship.

A **linear relationship** is a special type of function that produces a straight line when it is graphed. Linear relationships are relationships that involve a constant rate of change. Therefore, in this example, we must assume that Zack is constantly sending texts at a constant rate. That means that he could not type any faster or slower. He has reached his actualization as the Textmaster General of the world!

In the relationship, one element of the domain is entered, the function is performed and one element of the range is produced. The domain is the input, and the range is the output. Let's look at the example of Zack and all of that texting.

In this case, the time (in minutes) is our domain, and the number of Zack's texts is our range. If you remember, I figured out that Zack sends eight texts every five minutes. This value can be placed on a graph at the ordered pair (5, 8). If we think about it, that would mean that when we insert a ten into our machine, it would produce a 16: (10, 16). This would mean that in ten minutes, Zack could send 16 texts. We could plot that point on our graph as well!

Likewise, if we continue to determine the number of Zack's texts, he would send 24 texts in 15 minutes and 32 texts in 20 minutes. Wait a minute! Zack's gone and texted himself way off our graph! How could I show all future texts that Zack can make? Oh wait, I've got an idea!

I could use a ruler and connect the dots making a line - that would show all of the texts that Zack could send in any number of minutes. Would it? I think we have to add something else. If we added an arrow on the end of the line, it would indicate that this pattern would repeat forever, and it certainly looks to me like Zack is going to go on to text forever! I guess we should fill in the line back to zero.

What about the negative numbers? Can Zack text a negative number of texts in negative minutes? I know it's annoying, but I don't think it's negative! Let's get rid of the negative quadrants. In this case, all our data falls in quadrant one because every element of our input, or domain, and every element of our output (range) lies within the set of positive rational numbers. We've completed our model of a linear relationship between two quantities.

A **linear relationship** between two quantities will produce a graph of a straight line. The line represents every possible solution for the range of the function. In order to create the line, we use the function equation and evaluate the range, or output, values based upon several of the domain, or input, values. If we were to evaluate the number of texts received over time, we could use the following equation: *t* = 3*m*, where *t* represents the number of texts received, and *m* represents the number of minutes.

In order to evaluate the function, we can write it in function notation, input some domain values and evaluate. We can convert these domain and range values into ordered pairs. And then we can use those ordered pairs to graph a line. We then place arrows on the end of the line to indicate that it is infinite.

Finally, we consider our domain and range and realize that we do not need to use negative minutes or texts, so we constrict our model to just the first quadrant. And that's how we model a linear relationship between two quantities. Now all we've got to do is figure out how to get Zack to put down the phone!

Now that the lesson's over, put your knowledge to the test:

- Describe a linear relationship between two quantities
- Define range and domain
- Model a linear relationship on a graph

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Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

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