Analyzing Functional Relationships by Graphing Worksheets
To this point in math, we have examined two different types of relationships that may exist between two different factors or variables. You have begun to examine statistical relationships that may exist. Those types of relationships are not precise. When we plot statistical relationships on a graph they may display a trend, but the data is rather scatter and we are doing our best to understand. Functional relationships, on the other hand, are exact and solid predictors of future outcomes. When we plot data that exhibits this behavior on a graph you will see a clearly discernable line that follows some type of pattern. In this section of our site, we will examine data that has been plotted to describe a relationship between two variables. The goal with the worksheets and lessons is to examine what these lines indicate to us and what it means about the connection that exists between those two variables. Once we have a good handle on this, we can use it to better understand just about any system, whether it is in a physical or theoretical state.
Aligned Standard: Grade 8 Functions - 8.F.B.5
- Explaining Graphs Step-by-Step Lesson- This is a great activity for students to understand what real graph lines actually mean.
- Guided Lesson - Now you are asked to explain graphs. It was super easy when you were just plotting points, now you have to tell what they mean.
- Guided Lesson Explanation - Here is another example of how you can show the answers in many ways.
- Independent Practice - A very neat way to lock down problems that involve graphs.
- Matching Worksheet - Match the graphs to their overall meaning and feelings.
- Graphing Functions Five Pack - Tell us the type of functions that you see here. It's math and vocabulary in one.
- Answer Keys - These are for all the unlocked materials above.
You will find out very quickly how far along students are with understanding what graphs actually mean with this skill.
- Homework 1 - The graph has three stages that go along with the story we have been presented with.
- Homework 2 - Describe the graph of the function between x = 4 and x = 8. We can plot this function in graph. Since there is no value for y, they are just vertical linear graphs at those x coordinates.
- Homework 3 - If the rate of change is constant. So it is a linear function.
It is all about understanding what the graphs mean and what they tell us about the variables.
- Practice 1 - Describe the graph of the function. Does the graph display a linear relationship?
- Practice 2 - Jimmy walks to the park in early morning and runs around. After running in the park, he goes home using a bicycle. During which activity is Jimmy moving at the fastest rate?
- Practice 3 - Interpret the the data chart to find a relationship.
Math Skill Quizzes
We look at the data from all types of angles here.
- Quiz 1 - Describe the graph of the functions between x = 6 and y = 5.
- Quiz 2 - Interpret the lines that you see.
- Quiz 3 - Break apart the graph and explain the trend that you see.
How to Analyze Functional Relationships by Graphing
Analyzing a functional relationship is not as difficult as it may seem, and yet many Grade 8 students often find it daunting. Using the right examples backed up by the right theoretical understanding can make analyzing functional relationships easier.
A relation is simply a set of all the ordered pairs. Remember, not all relations are functional relationships. A function exists when each x-value (independent value, input) paired with exactly one y-value (dependent value, output). This sort of pairing is referred to as a functional relationship. To understand a functional domain, you need to understand what a function is. A function is when each member of the domain is associated with just one of the members of the range.
There is another way of analyzing a functional relation. If you can make a straight vertical line that can touch two or more points on the graph simultaneously, it means that the relationship is not functional. We call this the Vertical Line Test. This test is relatively simple. You draw a vertical line (up and down) and if the vertical line hits the graph more than once, it is not a function. That is because that would indicate a place where an x value has two y values. The best way to run this test is to take your pencil and lie it down on you paper pointing directly up and down. Move that pencil from the very left of that graph to the very right of that graph. If you hit the graph twice or more, as you are moving your pencil, than a functional relationship does not exist.