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In mathematics, recognizing and interpreting trends in a graph is as important as making sense of the numbers on it. For graphs that focus on general trends and lack numerical data, qualitative statements can be made. This lesson will describe and analyze qualitative graphs using real-life examples.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Interpreting Graphs

The graph shows how fast Magdalena, Tadeo, and Zosia are biking from home to school.
Look closely at each graph and describe how fast each cyclist is going during their ride.

Discussion

Data Types

Data that can be quantified or represented by numbers is known as numerical data, while data without numerical values is known as categorical data.

Concept

Categorical Data

Categorical data, also called qualitative data, is data that can be split into groups. Such data belongs to one or more categories that have a fixed number of possible outcomes or values. An example of categorical data is when countries are identified as a part of a continent.

Every country on Earth is generally accepted to lie within one of these seven continents. For that reason, any country can be categorized as belonging to one of these categories.

Discussion

What Is a Qualitative Graph?

The word qualitative refers to the characteristics of something, rather than its numerical value. As a result, qualitative information tends to be subjective. Graphs without specific numbers on the grid are known as qualitative graphs and aim to understand general relationships between variables.

Concept

Qualitative Graph

A qualitative graph is a graph used to represent the relationships between quantities without using specific numbers on the axes.
As observed in the example graphs, the axes have labels but lack numerical values. The first graph shows how the speed of a car changes over time. At time zero, its speed is zero. The speed of the car increases at a constant rate. Then the speed reaches a limit and does not change. Finally, the speed of the car decreases at a constant rate until its speed becomes zero.

Example

Interpreting Zosia's Speed-Time Graph

Examine the graph that shows Zosia's speed on her way to school.

Which statements are true?

Hint

In this case, increasing parts of the graph indicate that the speed is increasing, and decreasing parts indicate that the speed is decreasing.

Solution

The graph is a qualitative graph where axis represents time and axis represents Zosia's speed. The graph consists of increasing, decreasing and constant parts.

The graph begins by increasing at a constant rate, then remains unchanged for a while, then increases again at a constant rate, and finally decreases at a constant rate. These parts in the graph can be interpreted as follows.

Interpretation of The Graph

Zosia increases her speed at a constant rate in Part A. Then her speed stays the same for a while during Part B. After that, she increases her speed one more time in Part C. Finally, she slows down at a constant rate in Part D.

Considering this interpretation, statements II, III, and IV are correct.

Example

Three Friends' Bike Race

Magdalena, Tadeo and Zosia organized a friendly bike race after school. They rode nearly three miles in the exciting competition. The graph below depicts the distance covered by each racer in miles as a function of time in minutes.
a Who wins the race?
b Examining the graph, which of the following comments can be made?

Hint

a Looking at the graph, who completed the race in less time?
b The graphs show the relationship between distance covered and time elapsed. A point on the graph shows how far the bicyclists are from the starting point at that moment. The steepness of a graph indicates how fast the bicyclists are moving.

Solution

a When the race is over, each friend is miles from the starting point because this is a mile race. This can be shown on the graph by drawing a horizontal line through the points where each graph ends. The order can be found by drawing a vertical line from the end point of each graph to the axis.
If the finishing times of Tadeo, Magdalena and Zosia are respectively, then Tadeo wins the race because his time, is the smallest.
b Start by the first statement. Take a look at Tadeo's and Magdalena's graphs.
The graphs intersect at two different points. This means that at each of the two points, Tadeo and Magdalena are at an equal distance from the start line at the same time. The first statement is true.
Now examine Zosia's graph to determine whether the second and fifth statements are true.

Since this is a distance-time graph, the steepness indicates how fast the bicyclists are moving. This can be understood by drawing small arrows above the graph. Steeper arrows indicate Zosia is going faster.

At the beginning of the race Zosia gradually increases her speed because the graph gets steeper. She then moves at a constant speed because the graph becomes a straight line. As she finishes the race, she gradually reduces her speed because the graph becomes flatter. Therefore, II is false but V is true.
The third and fourth statements are related to Tadeo's graph.
Although Tadeo takes the early lead, Magdalena passes him after a certain time. Towards the end of the race, Tadeo overtakes Magdalena and wins. This means that Tadeo was not always ahead during the race.
Finally, Tadeo starts cycling at a steady pace, decides to go faster in the middle of the race, and continues at that fast pace until the end of the race. Therefore, the fourth statement is true.
As a result, I, IV, and V are true.

Discussion

Sketching Qualitative Graphs

The key elements of a situation are visually represented through qualitative graphs. When a situation is described verbally, a rough sketch of its graph can be drawn. Consider the example.

Example Situation

Paulina begins to run at a steady rate. While jogging downhill, she increases her speed at an increasing rate. After that, her speed remains constant for some time. Following that, she slows down at a steady rate throughout the rest of her run and comes to a complete stop.

The qualitative graph of such a situation can be drawn in two steps.
1
Draw the Axes
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In the given situation, two variables can be identified. These are time and Paulina's speed. Time is the independent variable and speed is the dependent variable. In this case, the horizontal axis is labeled with time and the vertical axis is labeled with speed.

2
Sketch the Shape of the Graph
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The situation consists of four different parts, or intervals.

  • Part Paulina runs at a constant rate.
  • Part She speeds up at an increasing rate.
  • Part She runs at a constant speed.
  • Part She slows down at a steady rate and stop.

The graph can be temporarily split into four parts. These parts do not have to be placed at equal intervals because this is a rough sketch.

The starting point of the graph is the origin because the moment Paulina starts running is considered to be and her speed is A constant rate means that the changes in one variable relative to another variable are always the same. This part can be represented by a line segment with a positive slope starting from the origin.

Her speed then increases faster and faster as she runs downhill. The graph in this case can be drawn as a curve that becomes steeper.

After that phase, she runs at a constant speed. That is, her speed remains the same for a certain amount of time. This can be represented by a horizontal line segment.

Finally, her speed decreases at a constant rate for the rest of the run. This part should be a decreasing line segment and continue until it touches the horizontal axis.

As a final step, get rid of the temporary auxiliary lines.

This qualitative graph represents the situation described verbally at the beginning of the page.

Example

Passing By the Library

On the weekend, Zosia will go to the cinema with her friends. She rides her bike from her home to cinema passing by the library on the way, and traveling at a constant speed for the entire trip.

a Sketch a graph of Zosia’s distance to the cinema as a function of time.
b Sketch a graph of Zosia’s distance to the library as a function of time.

Answer

a Example Graph:
b Example Graph:

Hint

a Zosia is meters from the cinema, so the function begins with its highest output value at time zero. When Zosia reaches the cinema, her distance to the cinema is
b Zosia is meters from the library, so the function begins with its output value at time zero. Additionally, the distance from the library increases and reaches its maximum when she arrives the cinema.

Solution

a Before sketching a graph, think about the given information and how Zosia's distance to the cinema will change.

Some important features of the graph can be described as follows.

  • Zosia is currently meters from the cinema, so the function begins with its highest output value at time zero.
  • When Zosia reaches the cinema, her distance to the cinema is and the ride is over. Therefore, the last point on the graph should be on the horizontal axis.
  • Zosia’s speed is constant throughout her ride, so the graph must be a line.
  • It must be a decreasing line because the distance is decreasing.
A rough sketch can be drawn using these inferences. In this case, time is the independent variable and the distance to the cinema is the dependent variable.
Label the horizontal and vertical axes accordingly.

In this problem it is unknown how long it takes Zosia to go to the cinema. Only the distance is known. This value can be shown on the graph, too. Considering the features depicted previously, the graph will look like the following.

Since a rough sketch is enough, the label can be omitted.

b In this task, think about how Zosia's distance from the library changes as she goes to the cinema.

List some important features of the graph.

  • Zosia is meters from the library, so the function begins with its output value at time zero.
  • The distance decreases to meters when Zosia arrives at the library.
  • The distance from the library then increases and reaches its maximum when she arrives the cinema.
  • The absolute values of the slopes of the two lines are equal since she rides at a constant speed.
A rough sketch can be drawn using these inferences. In this case, time is the independent variable and the distance to the library is the dependent variable.
Label the horizontal and vertical axes accordingly.

Once again, the duration of Zosia's journey to the cinema is not known. However, two output values are known, with meters and meters representing the starting and ending points of the graph, respectively. These values can be displayed on the graph.

Since this is a rough sketch, the numbers can be omitted.

Example

The Temperature of the Cinema Hall

Zosia, Magdalena, and Tadeo are having a good time watching a movie in the cinema. After the movie starts, the temperature in the cinema hall remains constant for a while. Following this, it starts to rise at a faster and faster rate. The air conditioning is then turned on and the hall is cooled at a constant rate until it becomes cooler than the initial room temperature.

Sketch a qualitative graph to represent the temperature of the hall.

Answer

Example Graph:

Hint

The graph should consist of three parts: one part decreasing, one part increasing and one part constant. Which part of the graph should be a curve?

Solution

In the given description, the temperature of the hall can be considered as a function of time because the temperature changes over time. In such a case, time is the independent variable and the temperature is the dependent variable.
Start by labeling the horizontal and vertical axes of the coordinate plane accordingly.

From the description, the graph should consist of three parts.

  • Part The temperature in the cinema hall remains constant for a while.
  • Part It rises at a faster and faster rate.
  • Part It decreases at a constant rate until it becomes cooler than the initial room temperature.

Next temporarily divide the graph into three time intervals.

The room temperature is usually around which is greater than For this reason, the starting point should be a positive value, not Since the temperature remains the same for a certain amount of time, Part should be a horizontal line segment as shown in the diagram.

The temperature begins to rise more and more rapidly. This part should be a curve that gets steeper as time goes by because the rate of change increases over time.

For the last part, the temperature decreases at a constant rate and it becomes cooler than the initial room temperature. A constant rate implies that its graph should be a straight line. Since the temperature decreases, the last part is represented by a line segment with a negative slope. Ensure the graph extends below the starting value.

Finally, get rid of the temporarily drawn lines.

This qualitative graph represents the situation described at the beginning. Remember, this is only a rough sketch. Some parts can be drawn longer, steeper or flatter.

Closure

Interpreting Qualitative Graphs

In one of the previous examples, Zosia's speed-time graph was interpreted. As the lesson comes to a close, the remaining two graphs will be interpreted. Take a look at the given qualitative graph.
Tadeo's speed-time graph initially increases at a constant rate, and then it continues to increase at a slower, constant rate. After that it remains horizontal for a short time, then decreases at a constant rate.
The interpretation of the graph can be as follows.

Interpretation of Tadeo's Graph

Tadeo starts his journey by increasing his speed at a constant rate. After a certain time, he decides to ease off, decreasing the acceleration. After finding a steady pace for a while, he concludes the ride by slowing down at a constant rate until he comes to a stop.

Lastly, take a look at Magdalena's graph showing speed over time.
This graph consists of three parts. In the first part, the graph increases at a constant rate. The second and third part indicate a constant decrease, but the third part decreases more rapidly than the second part.

Interpretation of Magdalena's Graph

Magdalena starts her bike ride by increasing her speed steadily. She maintains that constant rate for about half of the journey time. Then, she slows down at a constant rate. Towards the end, she concludes her journey by slowing down at a greater, constant rate.