Comparing Linear Functions

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Audio:

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.

• How can you compare these linear functions?
• What observations can you 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?

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

You can compare these functions by looking at their slopes!

When you compare the slopes of each function, you can determine if the rate of change is increasing or decreasing. You can also tell which function is increasing or decreasing faster.

Now, 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.

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.

2. Next, look at function g.

To find the slope from a table, you 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.

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.

3. 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.

The slope of function h is -2.

4. Last, look at function i.

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

The slope of function i is -3.

Now, compare the slopes of all functions.

Graph all four example functions to compare better. You will graph using the points used to calculate slopes.

• What can you 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 fastest.

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 fastest.

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 the video below 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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