Graphs and Functions
Objective: When done with this lesson, you will be able to determine whether or not a graph is a function, identify the independent and dependent variables in a situation, and sketch a reasonable graph of a relationship between two known variables in a situation.
Approximate completion time: 4 hours
"Do you bathe because your body odor bothers you or does your body odor bother you because you do not bathe?"
Most people would prefer you bathe regularly for whatever reason.
As you use coordinate systems, you should know that most people prefer you follow some "conventions". A convention is something we all agree we will do the same–not because it's the best way, but because we just agree to do it the same.
The convention we use in coordinate systems is that we try to always place the independent variable on the horizontal axis and the dependent variable on the vertical axis. Sometimes these variables are also called manipulated and response variables.
Without even knowing it, you probably already have a good understanding about "dependent" and "independent" variables. Let's try it out.
Variable pairs Making sense of it... Dependent variable # of corn plants growing
# of corn seeds plantedThe # of corn plants growing depends on the # of corn seeds planted # of corn plants growing age of the watermelon
size of the watermelonThe size of the watermelon depends on the age of the watermelon. size of the watermelon # of items bought
total grocery billThe total grocery bill depends on the # of items bought. total grocery bill So what are the independent variables for those examples? You guessed correctly if you were thinking "# of corn seeds planted", "age of the watermelon", and "# of items bought".
Sometimes, especially in science, the independent variables are called the manipulated variables. One could manipulate how many corn seeds are planted or one could change the number of items bought, hence they are called "manipulated". Note that the idea of manipulating time, as in the watermelon example, does not quite make sense.Try your hand a few more examples. Click on the variable you think is the independent variable. If you click on the wrong variable you'll see an explanation why it isn't correct. (If you were correct, you might click on the other variable to see the explanation and deepen your understanding.)
Once you are accustomed to "dependent" and "independent" variables, you're ready for graphing ideas!
Example: Sketch a graph of your 'hunger level' throughout the day.
Example: Sketch a graph of your 'hunger level' throughout the day. The graph above looks nice. How hungry I am depends on the time day. So the variables look like they are in the correct locations. As I look at the graph left-to-right, I can see the hunger levels go up and down throughout the day. This graph is correct. The graph above looks strange. With 'hunger level' on the horizontal axis the graph is implying "the time of day depends on how hungry I am". While the variables are related, the time of day is the more independent of the two. So this graph is not correct. You will practice some graphing of your own in your assignment.
That graph in option 2 in the last example was problematic for more than one reason. Look at this graph and think about it. The variables are on the correct axes (the horizontal should be time and the vertical should be hunger level), but something is strange. Can you see it? (Move your mouse over the graph for a hint.)
How hungry would I be at 11am? Is it possible to have multiple hunger levels for that one time? Such a graph seems impossible. As a matter of quick and easy testing, we use the "vertical line test".
If a vertical line can be drawn somewhere on the graph such that it interects the graph in more than one place, then the graph fails the vertical line test and is not a "function". If a vertical line cannot be drawn anywhere on the graph such that it interects the graph in more than one place, then the graph passes the vertical line test and is called a "function".
This is not a function.
This is a function.
When a graph passes the vertical line test we call it a function for many reasons. The main reason is because that's what mathematicians have defined it:
Definition A function is a relationship between two variables for which there is exactly one dependent value for each independent value.
Think about what that definition is saying in context of a hardware store. The two main variables in a store are items and prices. We all know that the price depends on the item. So we would say "Prices are a function of the item". That means that each independent value (each item in the store) has only one dependent value (price). It would be silly to have multiple prices for one item in the store. Yet it is not unreasonable to have many items with the same price. (That is why we don't mind if a function fails the "horizontal line test".)
Now it's time for you to practice.
You can download some practice problems here.
