Comparing Linear Functions

Contributor: Ashley Nail. Lesson ID: 13902

How can you compare two lines on a graph? Maybe you could look at which direction they go or how steep the line is. In this lesson, you will use slope to compare different forms of linear functions.

categories

Functions, Pre-Algebra

subject
Math
learning style
Auditory, Kinesthetic, Visual
personality style
Lion, Otter, Beaver, Golden Retriever
Grade Level
Middle School (6-8)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:
  • How can you tell who is walking the fastest in a group of people?

Well, you could observe and compare whose feet are moving the quickest, who is in the front of the group, or who reached their destination the quickest.

  • Did you know you could use functions to make these comparisons?

All you would have to do is collect data on the distance each person is moving and the time it takes to move that distance. Like if someone walks 2 feet in 5 seconds.

You could compile your data in graphs, tables, or even make an equation. These are all forms of linear functions that you can then compare!

Imagine we have four linear functions all represented in different ways:

  1. The function named f is a graph.
  2. The function named g is a table.
  3. The function named h is an equation.
  4. The function named i is a graph.

functions

  • How can we compare these linear functions?
  • What observations can we make?

Here are some observations you can make about these functions:

  • They are all straight lines.
  • They all have different x and y intercepts.
  • The functions on the graph are pointed in different directions.
  • The functions all have different slopes.

Let’s focus on the last observation about the slopes. Remember that the slope of a linear function represents the rate of change, or how steep the line is.

Slope is expressed in a few different ways.

  • Which representation are you most familiar with?

slope

If you need more review on slope, visit our lesson found under the Additional Resources in the right-hand sidebar.

We can compare these functions by looking at their slopes!

When we compare the slopes of each function, we will be able to determine if the rate of change is increasing or decreasing. We will also be able to tell which function is increasing or decreasing faster.

Now, let’s look at each function in the example and find the slope or rate of change.

  1. First, look at function f.

Find the y-intercept and label the point A. Find another point and label it B.

f function

Then, you can use “rise over run” or find the “change in y over the change in x” to find the slope.

The slope of function f is positive 3.

  1. Next, let’s look at function g.

To find the slope from a table, we will find the change in y over the change in x.

Remember that two corresponding values in a function table make an ordered pair. You can pick any two ordered pairs from the table to find the slope.

g function

If you look at the table, you also see that the value of x increases by 1 and the value of g increases by 5.

The slope of function g is positive 5.

  1. Now, let’s look at function h.

To find the slope from an equation, you can either use the slope intercept form or find the x and y intercepts and then calculate the change in y over the change in x.

h function

The slope of function h is -2.

  1. Last, let’s look at function i.

We can find the slope of function i by finding the y-intercept and another point. Then, we calculate the change in y over the change in x, or the rise over run.

i function

The slope of function i is -3.

Now let’s compare the slopes of all of the functions.

slopes

Let’s graph all four example functions to better compare. We will graph using the points we used to calculate slopes.

slopes graphed

  • What can we determine about the functions by looking at this information?

Functions f and g have an increasing rate of change.

These two functions both have positive slopes.

Functions h and i have a decreasing rate of change.

These two functions both have negative slopes.

Function g is increasing at the fastest rate.

When looking at functions f and g (both positive), g is the most positive slope or the steeper line. Therefore, the rate of change is increasing the fastest.

Function i is decreasing at the fastest rate.

When looking at functions h and i (both negative), i is the most negative slope or the steeper line. Therefore, the rate of change is decreasing the fastest.

Function g has the fastest rate of change.

When looking at all functions, g has the greatest slope regardless of positive or negative. It is also the steepest of all the lines. Therefore, the rate of change is the fastest.

Function h has the slowest rate of change.

When looking at all functions, h has the smallest slope regardless of positive or negative. It is also the most flat of all the lines. Therefore, the rate of change is the slowest.

Watch Comparing Functions 8.F.2, from TenMarks Amazon, to see another example of comparing functions.

  • In this real-world example, why is it important to compare the two functions?

  • Are you ready to practice on your own?

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