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Using Graphs to Determine the Constant of Proportionality

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  • 0:04 Constant of Proportionality
  • 1:18 Proportional Graph
  • 1:59 Finding the Constant
  • 3:16 Example
  • 3:48 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Because the constant of proportionality only exists for linear relationships, you can use the graph of a proportional relationship to figure out what your constant of proportionality is. Learn how in this lesson.

Constant of Proportionality

In math, the constant of proportionality tells you how two different sets of values relate to each other. This is always a linear relationship because a proportional relationship is always one where your ratios are always the same. This is only true for linear relationships.

For example, think of the model toys that some people build. These are miniature replicas of a real-world item. When you compare all the measurements of the toy model to the measurements of the real-world item, you'll find that all these ratios will be the same. If the toy model is 1/12 the size of the real-world object, then each measurement of the toy model is multiplied by 12 to get to the real-world measurement. Here, you have a linear relationship between your toy model measurements and your real-world measurements. You also have a proportional relationship. Your constant of proportionality is 12 since you have to multiply by 12 each time when going from toy model size to real world size.

In math, the formula for a proportional relationship is:


proportional graph


The k is your constant of proportionality. The x is the value you multiply your constant by, and your y the resulting values. For the toy model example, the x values are the model measurements, the y values are the real-world measurements, and k is 12.

Proportional Graph

So, what does a proportional relationship look like when it's graphed? It looks like a straight line, like this:


proportional graph


This particular graph is actually a very useful one you can use in cooking. You can use this graph to help you figure out how to halve your recipes. Say you have a recipe that makes 24 cookies, but you only want to make 12 for a date night. You can find the amount given in the recipe on the x-axis and then find what it converts into on the y-axis. For example, 5 tablespoons convert into 2.5 tablespoons. You know that this has to be a proportional relationship because your ratios have to be the same for your cookies to turn out the same. If you add more of one ingredient over another, your cookies won't turn out the same.

Finding the Constant

Now that you have the graph of your proportional relationship, you can actually use this to help you find what your constant of proportionality is. Because this graph is used to halve your recipes, what do you expect the constant of proportionality to be? It could be 1/2 because you have to multiply your ingredient amounts by a half to find out what you should put in to make half a recipe.

Let's see if that's what you get.

To find your constant of proportionality from your graph, you're basically finding the slope of your line. You want to locate two points with easy coordinates, then you count the number of squares you need to go up followed by the number of squares needed to go to the right to get from your first point to the second point, your first point being to the left of your second point.

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