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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 to describe the trends. 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 travel as they bike from home to school.
Look closely at the graphs 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. Categorical data belongs to one or more categories that have a fixed number of possible outcomes or values. Human blood groups are one example of categorical data.

This classification is based on whether certain antigens are present or absent on the surface of red blood cells. A person's blood type can be categorized into one of these groups.
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 are meant to show 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 shown 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. The speed eventually reaches a limit and does not change for a while. Finally, the speed of the car decreases at a constant rate until it becomes zero again.
Discussion

Increasing and Decreasing Intervals of Qualitative Graphs

A function is said to be increasing when, as the values increase, the values of also increase. On the other hand, the function is considered decreasing when, as increases, decreases. An increasing interval is an interval of the independent variable where the function is increasing. A decreasing interval is an interval of the independent variable when the function is decreasing.

Example

Think about a graph that shows how a soccer ball's height changes after it is kicked by a soccer player.
In the first part, the graph is increasing because the height of the ball increases as time passes. In the second part, the graph is decreasing because the height of the ball decreases as time passes.
Example

Interpreting Zosia's Speed-Time Graph

Consider 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, while decreasing parts indicate that the speed is decreasing.

Solution

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

The graph begins by increasing at a constant rate, remains unchanged for a while, 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 again in Part C. Finally, she slows down at a constant rate in Part D.

From this interpretation, statements II, III, and IV are correct.
Example

The Bike Race

Magdalena, Tadeo and Zosia decided to have 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 won the race?
b Based on the graph, which of the following comments can be made?

Hint
a Who completed the race in the least amount of 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. This can be shown on the graph by drawing a horizontal line through the points where each graph ends. The finishing 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. He finished the race first.
b Start with the first statement and 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 were at an equal distance from the start line at the same time. In other words, they were side by side and neck and neck in the race. The first statement is true.
Now consider 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 racers are moving. This can be understood by drawing small arrows above the graph. Steeper arrows indicate that the person is going faster.

At the beginning of the race, Zosia gradually increased her speed, which is visible in the graph as it gradually steepens. She then moved at a constant speed when the graph becomes a straight line. As she finished the race, she gradually reduced her speed, reflected in the graph as it beginning to flatten. Therefore, Statement II is false but Statement V is true.
The third and fourth statements are related to Tadeo's graph.
Although Tadeo takes the early lead, Magdalena eventually passes him. Towards the end of the race, Tadeo overtakes Magdalena again and wins the race. This means that Tadeo was not always in the lead during the race.
Finally, the intervals on his graph show that Tadeo started cycling at a steady pace, decided to speed up in the middle of the race, and continued at that fast pace until he crossed the finish line. 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 in 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 speeds up at an increasing rate. Her speed remains constant for a while. Finally, she begins to slow down at a steady rate until she comes eventually to a complete stop at the end of her run.

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 — time and Paulina's speed. Time is the independent variable and speed is the dependent variable, so the horizontal axis represents time and the vertical axis represents speed.

2
Sketch the Shape of the Graph
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The given 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 until she stops.

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 time Paulina starts running is considered to be and her starting speed is A constant rate means that the changes in one variable relative to another variable are always the same. This part of Paulina's run 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. This part of the graph 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 an 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 Paulina's speed throughout her run as described. Keep in mind that this is just a rough sketch. Due to a lack of numerical data, certain parts can appear longer, steeper, or flatter.

Example

Passing By the Library

Zosia is meeting up with her friends at the movie theater. She rides her bike from her house to the theater, passing by the library on the way. Zosia travels at a constant speed for the entire trip.

a Sketch a graph of Zosia’s distance to the movie theater 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 At the start of her bike ride, Zosia is meters from the movie theater. The function begins with its highest output value at time zero. When Zosia reaches the movie theater, her distance to the theater is
b Zosia starts out 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 at the theater.

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

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

  • Zosia is currently meters from the movie theater, so the function begins with its highest output value at time zero.
  • When Zosia reaches the movie theater, her distance to the theater 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 straight 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 movie theater is the dependent variable.
Label the horizontal and vertical axes accordingly.

In this exercise, the time it takes for Zosia to reach the movie theater is unknown. Only the distance is given. This value can be shown on the graph, too.

However, since a rough sketch is enough for the exercise, the label can also be omitted.

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

Consider some important features of the graph.

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

The amount of time it takes Zosia to arrive at the movie theater is still not given, but two output values are known — the distance from Zosia's house to the library is meters and the distance from the library to the movie theater is meters. These distances are the starting and ending points of the graph, respectively, and can be identified on the graph.

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

Example

The Temperature in the Theater

Zosia, Magdalena, and Tadeo are having a good time watching a movie in the theater. The movie starts and the temperature in the room remains constant for a while. Then, it starts to rise at a faster and faster rate until the air conditioning is then turned on. The theater then cools at a constant rate until it becomes colder than the initial temperature at the start of the movie.

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

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 curved?

Solution
In the given description, the temperature of the theater can be considered as a function of time because the temperature changes over time. Here, time is the independent variable and 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 theater remains constant for a while.
  • Part The temperature rises at a faster and faster rate.
  • Part The temperature decreases at a constant rate until it becomes colder than the initial room temperature.

Next, temporarily divide the graph into three intervals of time.

The average room temperature is around which is greater than As such, the starting temperature at 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.

Notice that the graph does not reach the axis. Without any data points, this axis can be understood to mean or that the theater is Who wants to stay in a movie theater that is literally freezing cold? Finally, get rid of the temporarily drawn lines that divide the graph into intervals.

This qualitative graph can represent the situation described at the beginning. Remember, this is only a rough sketch — some parts can be drawn longer or shorter, steeper or flatter, as long as they match the given situation.

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, then continues to increase at a slower, yet constant, rate. After that, it remains horizontal for a short time, then decreases at a sharp, constant rate.
His graph might be interpreted as follows.

Interpretation of Tadeo's Graph

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

Now consider the graph showing Magdalena's speed over time.
This graph consists of three distinct parts. In the first part, the graph increases at a constant rate. The second and third part indicate constant decreases, 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 ride. She then slows down at a constant rate. Towards the end, she concludes her journey by slowing down at a greater, but still constant, rate.

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