Linear and Non-Linear Relationships

What is a relation?

Simply put, a relation is a collection of ordered pairs. Relations can be represented algebraically, graphically, and numerically.

One ordered pair has an x-value and a corresponding y-value, as seen below:

(4, 10)

A set of ordered pairs, could be shown as an equation, in a table of values, or as a graph.

We will cover relations over 4 different sub-modules, so there is no need to go into more depth right now but this page can provide you with a brief overview.

What is a linear relation?

A linear relation is a relation that, when graphed, results in a straight line.

Equations of linear relations take the following form:

The first two graphs below are linear relations, but the last two graph is not a linear relation.

Linear relation

Not a linear relation

Equations of the form y = mx + b

Let’s take a closer look at the form y = mx + b.

As stated before, m is the slope of the line. The slope can be thought of of the steepness of a line.

The y-intercept, b, is the y-value (y = a number) when x = 0. You can think of y-intercept as the y-value of the point at which the line crosses the y-axis.

All linear relations take the form of y = mx + b or can be manipulated to take that form. Below are some examples of linear and non-linear equations.

<--- Examples of linear equations

The 4th and 6th equations require some algebraic manipulation to make the equation look more like the typical form which we will do in the video below.

Examples of non-linear equations -->

You’ll notice that many of these non-linear relations have x and y variables that are raised to a power other than 1. Though we don’t write the 1, in our examples of true linear relations, the x’s without a power or exponent can be thought of as having an invisible 1 in the exponent.

Next, we will look at another method of determining whether a function is linear or not, but first we have to look at slope (m) in greater detail.

Slope

As stated previously, slope can be thought of as the steepness of the line. Consider the following graphs:

You can see from the screenshots, that the equations of the two graphs are as follows:

From these equations and from the graphs, we can see that both linear relations have a y-intercept, or b value, of y = 2. The only difference between the two graphs is the slope, or m value.

The first relation has a slope of 0.25 or ¼, whereas the second has a slope of 7. We notice that the second graph (with a slope of 7) has a much “steeper” line, whereas the first relation’s graph isn’t as steep.

This makes sense when we think of slope as “steepness”, because the relation with a bigger numerical slope ( 7 > 0.25 ) is steeper.

In a more mathematical way, slope is defined as follows:

The triangle symbol is called “delta”, and means “change in”; when reading this equation, we say that slope is “the change in y divided by the change in x”.

Let’s look at slope more closely together:

Before, we go on, we will also look at a couple of special slopes.

From these two videos, we have the following key understandings:

The slope of a line can be thought of as the steepness of a line
Slope is defined as “the change in y divided by the change in x”
Slope can be calculated with any two points on a line
The slope of a line (or linear relation) is the same no matter where you are on the line.
Slope can be negative or positive
Slope can be zero, which results in a horizontal line
Slope can be undefined, which results in a vertical line

Determining whether a relation is linear or not using slope

So far, we have the following three methods of determining whether a relation is linear or non-linear:

Visually by looking at the graph of the relation
Looking at the equation, we should only see x (or x¹), not x³, x⁷, or x^-2 or anything else.
Knowing that all linear relations can be expressed with the form y = mx + b. We might have to manipulate the equation a little bit to make it look like this, like we did in the previous section.

Another way we can determine whether a relation is linear or non-linear is using a table of values and slope.

Let’s take a moment to discuss tables of values.

If we are given a table of values that summarizes points of a relation, we can use our understanding of slope to determine if the relation is linear or not. We know that linear relations have a constant slope. Given the following 2 tables of values, we will determine if each relation is linear or not.

The method we used in the video of subtracting one y-value from the previous y-value is called finding the first differences.

Summary

To summarize, we have the following three methods to determine whether a relation is linear or not:

Graphically

Is the graph a straight line with a constant slope? If yes, we have a linear relation.

Using the equation

Is the equation of the form y = mx + b, or can it be manipulated to take that form? If yes, we have a linear relation.

Using a table of values to find first differences

When we compute the first differences using the table of values, are we getting the same value for all of the first differences? If yes, we have a linear relation.

Check Your Understanding

(1) Use the graph to determine whether the relation is linear or not. Comment briefly on the graph to support your decision.

Linear or non-linear?

(2) Determine whether the following equations represent linear or non-linear relations. You may need to manipulate some equations algebraically.

(3) Use the following tables of values to determine if the relation is linear or non-linear.

*See solutions to provided practice problems on Solutions Page*

More Resources

First Differences: Linear or Non-linear?

Simplifying Linear Expressions

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