Graphing Non-Proportional Linear Relationships

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  • 0:00 Linear Equations
  • 1:06 Proportional vs.…
  • 2:19 Graphing…
  • 3:50 Example
  • 4:49 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will define proportional and non-proportional linear relationships. We will look at how to distinguish between them and the steps involved in graphing non-proportional linear relationships.

Linear Equations

Suppose it's a beautiful summer day, and Mary and Sally both decide to take advantage of the weather and sell lemonade to make some money. Mary is lucky, because her mom already has all the supplies she needs, so she doesn't have to pay any money to get started.

Sally, however, has to make a quick trip to the store to buy supplies since she doesn't have any on hand. She ends up spending $10 on supplies. Both girls charge $2 per cup of lemonade. Based on this, we can represent the amount of money that each girl makes using the following equations:

Mary: y = 2x

Sally: y = 2x - 10

Where x is the number of cups of lemonade each girl sells.

In mathematics, both of these equations are linear equations. A linear equation is an equation that can be put in the form y = mx + b, and the graph is a line. Are you with me so far? Good, because this is where it gets interesting!

Proportional vs. Non-Proportional

We can separate linear equations into two categories: proportional and non-proportional relationships. A linear equation y = mx + b:

  • is a proportional linear relationship between y and x if b = 0 (so y = mx),
  • is a non-proportional linear relationship between y and x if b ≠ 0.

What do you think about each girl's equation? Proportional or non-proportional? If you're thinking that Mary's equation is proportional and Sally's is non-proportional, then you're right!

Mary's equation, y = 2x, is of the form y = mx + b, where m = 2 and b = 0. Since b = 0, the relationship between y and x is proportional.

Sally's equation, on the other hand, is of the form y = mx + b, where m = 2 and b = -10. Since b ≠ 0, the relationship between y and x is non-proportional.

In this lesson, we want to concentrate on non-proportional linear relationships and how to graph them, so let's get started.

Graphing Non-Proportional Equations

When it comes to graphing non-proportional linear relationships, it's helpful to recognize a couple of characteristics of the equation y = mx + b.

  1. In this equation, m is equal to the slope of the line. That is, m is the rate of change of y with respect to x and is equal to the change in y divided by the change in x from one point on the line to another.
  2. b is the y-intercept, or the point at which the line intercepts the y-axis on the graph.

Since b ≠ 0 in a non-proportional relationship, we know that the graph will not cross the y-axis at the origin. This fact enables us to distinguish between the graph of a non-proportional and proportional linear relationship.

  • The graph of a proportional linear relationship will be a line that passes through the origin
  • The graph of a non-proportional linear relationship will be a line that doesn't pass through the origin.

Okay, now let's take a look at the steps we take to graph a non-proportional linear relationship. Then we'll take a crack at graphing Sally's equation using these steps.

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