Explainer: Graphs of Proportional Relationships

In this explainer, we will learn how to identify the graph of proportional relationships, determine the constant of proportionality (unit rate), and explain the meaning of each point on the graph.

Ratios are used to compare two numbers or quantities. Rates are ratios that compare two quantities of different nature (expressed in different units). For instance, we can compare the price paid with the quantity of items we bought. If 5 kg of carrots can be bought for $7, the ratio “price paid to mass of carrots” (in everyday language, we often say “weight,” but the proper word is “mass,” as you may have learned in science) can be expressed as the quotient $75kg.

This rate can be represented on a double-line diagram.

If the carrots are sold at a fixed unit price, then we know that the price paid is proportional to the mass of carrots (in kilograms). With the information we have, we can work out the unit price by finding the value of the quotient $75kg. It is the coefficient of proportionality between the mass of carrots and the price. We find $75=$1.40/.kgkg

Note that the unit for the unit rate is given by the units in the quotient: dividing dollars by kilograms gives a unit of dollars per kilogram. It gives the price in dollars for one kilogram.

Now that we have found the unit price for the carrots, we can easily find the price for 2,3,4, kilograms of carrots. We can place these values on our double-line diagram. The proportional relationship is clearly shown by the fact that, for each mass/price pair, the values of mass and price are vertically aligned.

Now, what if we would like to plot these pairs of values on a coordinate plane? Our double-line diagram can actually be used for the two axes, and we see that we get a straight line through the origin.

This is true for any proportional relationship between two quantities; the graph of a proportional relationship is always a straight line through the origin. Inversely, if the graph of a relationship between two quantities is a straight line through the origin, then the relationship is a proportional relationship.

Having such a graph allows us now to find the price of any mass of carrots. Let us use a coordinate plane with a grid and take for instance a mass of 2.4 kg. [Note that, in this coordinate plane, the scales for both axes are the same, in contrast to previously where 1 kg on the 𝑥-axis was represented by the same length as for $1.40 on the 𝑦-axis. The graph is of course still a straight line through the origin.]

An approximate value of the price of 2.4 kg of carrots can be found by reading the graph; we find it is slightly less than $3.40.

Let us now look at some questions to check our understanding of graphs of proportional relationships.

Example 1: Using a Graph to Identify a Proportional Relationship

Jennifer is using origami sheets to make flowers. The following table and graph show the number of flowers she makes and the number of origami sheets she uses. Is this a proportional relationship?

Number of Flowers1234
Number of Sheets24612

Answer

We are given here a table giving four pairs of values for two quantities, the number of origami sheets used by Jennifer and the number of flowers she made with these, and the corresponding graph. The question is whether these two quantities are proportional.

We actually only need either the table or the graph to answer this question. Recall that the graph of a proportional relationship is a straight line through the origin. Using the graph here, we quickly observe that the line is not straight because it bends after 6 sheets.

So, the answer is no, this relationship is not proportional.

Analyzing the data given in the table would of course yield the same answer, albeit requiring a little bit more work. Here, we need to check that all the ratios of one quantity to the other are equivalent. We find that this is the case for the first three pairs, 21=42=63=2, which is seen in the graph as the three points being aligned, but that the ratio with the fourth pair is different 124=3.

The blue dotted line in the diagram shows what the graph would look like if the number of flowers was proportional to the number of sheets. Note that the straight line passes through the origin. It also shows in the table the two values that are not in the same ratio as the others.

Example 2: Graphing a Proportional Relationship

Elizabeth started doing exercise in order to lose weight. She lost 150 grams per week. Which of the following graphs represents Elizabeth’s situation?

Answer

Elizabeth lost 150 grams per week. This means that, in two weeks, she lost 2 times 150 grams, that is, 300 grams; in 3 weeks, she lost 450 grams; and so on. This “150 grams per week” is a unit rate (it can be recognized from the “per” in the unit), and the weight lost is therefore proportional to the number of weeksElizabeth has been doing exercise.

This means that the graph should be a straight line through the origin and passing through the points (1 week, 150 g), (2 weeks, 300 g), (3 weeks, 450 g), and so on. All the graphs shown are straight lines. However, graph (b) does not pass through the origin, graph (a) passes through (1 week, 300 g), and graph (c) passes through (1 week, 100 g). Graph (d) is the correct one that passes through the origin and (1 week, 150 g).

The answer is then as follows: graph (d) gives the weight lost by Elizabeth as a function of the number of weeks she has been doing exercise.

Once we are able to identify a proportional relationship from its graph, we can use the graph to actually extract useful information in a given context. For this, we need to know the conventions for ordered pairs and axes in a graph. We will see in the next example how to interpret the meaning of a point on the graph of a proportional relationship.

Example 3: Understanding the Graph of a Proportional Relationship

Madison decided to make pizzas. The graph shows the number of tomatoes needed for a given number of pizzas.

Determine which of the following statements is true.

  1. The point (6,3) shows that to make 6 pizzas she needs 3 tomatoes.
  2. The point (5,10) shows that she needs 5 tomatoes to make 10 pizzas.
  3. The point (10,5) shows that she needs 5 tomatoes to make 10 pizzas.
  4. The point (8,4) shows that she needs 8 tomatoes to make 4 pizzas.
  5. The point (3,6) shows that to make 3 pizzas she needs 6 tomatoes.

Answer

The graph given is a straight line through the origin; therefore, the number of tomatoes needed to make a given number of pizzas is proportional to this number. We can use the graph given to find the number of tomatoes needed (read on the 𝑦-axis) to make a given number of pizzas (given on the 𝑥-axis). We are asked to find the true statement among the five statements given.

Let us go through all of them.

  1. The point (6,3) is not on the graph. Remember that an ordered pair always gives first the 𝑥-coordinate and then the 𝑦-coordinate. So, the statement is false.
  2. The point (5,10) is on the graph, but the 5 represents the number of pizzas and 10 the number of tomatoes needed to make these 5 pizzas, not the other way around. So, the statement is false.
  3. The point (10,5) is not on the graph. So, the statement is false.
  4. The point (8,4) is not on the graph. So, the statement is false.
  5. The point (3,6) is on the graph and means indeed that Madison needs 6 tomatoes to make 3 pizzas.

So, the answer is that statement E is true.

We are going to see in the next example how to use the graph of a proportional relationship to identify its unit rate.

Example 4: Identifying Unit Rates from a Graph

From the graph, find the constant of proportionality between the electricity bill and electricity consumption (i.e., the cost of electricity per kilowatt-hour).

Answer

The graph given shows the relationship between an electricity bill (in dollars) and the consumed electricity (in kilowatt - hours). We see that these quantities are proportional since the graph is a straight line through the origin. We want to find the constant of proportionality between the bill and the consumption, that is, the price of 1 kWh of electricity.

For this, we need to pick a well-defined point on the straight line. We observe that the line passes through the point (500 kWh, $60). Using this point, we find that the price for 1 kWh is $60500=0.12$/.kWhkWh

Note that the units have been written in the above calculation for learning purposes. To find a unit rate in dollars per kilowatt-hour, we need to divide dollars by kilowatt - hours. You may not be supposed to write the units in your workings, but, in any case, your final result must contain its unit.

Example 5: Understanding the Graph of a Proportional Relationship

The given graph shows the distance traveled against time. Which of the following statements is NOT true?

  1. In 45 minutes, the distance traveled is 7.5 miles.
  2. The speed is 10 miles per hour.
  3. The distance traveled in 7 hours is 70 miles.
  4. Any point of (𝑥,𝑦) coordinates on this graph means that the distance traveled in 𝑥 hours is 𝑦.
  5. It takes 10 hours to bike 1 mile.

Answer

The graph given is a distance–time graph. The graph is a straight line through the origin; therefore, the distance traveled is proportional to the travel time. We can use the graph given to find the distance traveled in miles (read on the 𝑦-axis) in a given number of hours (given on the 𝑥-axis). We are asked to find the false statement among the five statements given.

Let us go through all of them.

  1. 45 minutes is three-quarters of an hour, that is, 0.75 hours. The distance traveled in this time is given by the 𝑦-coordinate of the blue point in the graph. We find that it is indeed 7.5 miles. This statement is true.
  2. The speed is the unit rate of this distance–time relationship. It is the distance traveled in one hour. We see that the 𝑦-coordinate of the point on the graph whose 𝑥-coordinate is 1 is 10. Hence, 10 miles is traveled in 1 hour, which means that the speed is 10 miles per hour. This statement is true.
  3. The graph does not show the point with 𝑥-coordinate “7 hours.” However, we have just found that the speed (unit rate) is 10 miles/hour. Therefore, in 7 hours, the distance traveled is 7×101=70.hoursmileshourmiles Hence, this statement is true.
    [The units are given in the calculation for learning purposes. You probably do not have to write them in your workings, but it is good to have them in mind!]
  4. Any point (𝑥,𝑦) that belongs to the graph indeed means that 𝑦 miles are traveled in 𝑥 hours. Recall that, in an ordered pair, the first coordinate given is the 𝑥-coordinate, which is read on the horizontal axis. The second coordinate is the 𝑦-coordinate, read on the vertical axis. Hence, this statement is true.
  5. Does it take 10 hours to travel one mile? It seems here that the unit rate of “10” has been understood the other way around, namely, as 10 hours per mile. It is an interesting question, though, to find the unit rate when the relationship is taken in the other direction, namely, the time it takes to travel a given distance. Since 10 miles are traveled in 1 hour, the time to travel one-tenth of this distance (1 mile) will be one-tenth of 1 hour, that is, 110 of an hour, or 6 minutes (recall that an hour is 60 minutes). Hence, statement E is false.

The answer is that statement E is false.

Key Points

  1. Rates are ratios that compare two quantities of different nature (expressed in different units). For instance, if 5 kg of carrots can be bought for $7, the ratio of the price paid to the mass of carrots can be expressed as the quotient $75kg.
  2. If the carrots are sold at a fixed unit price, the unit price is $75=$1.40kgkg.
  3. The unit price is the coefficient of proportionality between the mass of carrots and the price.
  4. We can place any price of a given mass of carrots on a double-line diagram. The proportional relationship is clearly shown by the fact that, for each mass/price pair, the values of mass and price are vertically aligned.
  5. The graph of a proportional relationship is always a straight line through the origin.
    The graph here can be used, for instance, to find the price of 2.4 kg of carrots.

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