MathReads
Get Premium
Dashboard

Dashboard Sign out
PA
Pre-Algebra View details
6. Qualitative Graphs
Lesson
Exercises
Tests
arrow_back
eCourses /
Pre-Algebra /
Chapter 8
6. 

Qualitative Graphs

Qualitative graphs are used to display categorical data, helping to understand relationships and trends between different groups. This lesson focuses on the importance of recognizing data types, especially categorical data, and how to represent them through qualitative graphs. Sketching and interpreting these graphs allows for a clearer understanding of how data points relate to one another. These skills are useful in various fields, including market research, social sciences, and any situation where data categories are being analyzed. The lesson provides practical guidance on how to visually present and interpret data in a way that makes trends and comparisons easier to understand.
Show more expand_more
Lesson Settings & Tools
11 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Qualitative Graphs
Slide of 11
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
expand_more

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
expand_more

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.

Level 1
Level 2
Level 3
Qualitative Graphs
Exercises

Kriz sketches graphs that illustrate how a bird's vertical and horizontal distances change over time relative to Kriz themselves.

Kriz then draws another graph by combining these two graphs.

Which statements are true?

The given qualitative graphs show the position of the bird relative to the position of Kriz. In other words, we are keeping an eye on the bird's flight in both vertical and horizontal directions relative to Kriz's position. Let's try to visualize what this flight might look like.

Now let's think about the given statements one at a time.

Statement I

By looking at the applet, we can see that the bird begins to descend slowly, makes a dive, and then continues gliding. This means that the first statement is true.

Statement II

The second statement is about the vertical distance. To better analyze the vertical movement of the bird, let's draw small arrows parallel to the vertical distance-time graph.

We see that all arrows point downward, meaning they have a negative slope. Therefore, the bird's vertical distance decreases during its flight. At first, the arrows are not very steep. They suddenly become very steep, suggesting a sharp change in altitude. Finally, the arrows begin to flatten out again. Therefore, we can conclude that Statement II is also true.

Statement III

Finally, we will interpret how the bird's flight changes its horizontal distance from Kriz. In this case, we will consider the horizontal distance-time graph.

We can see that the bird starts out moving horizontally at a quick rate because the arrows are relatively steep. After that, the arrows become nearly horizontal — that is to say, the bird barely travels horizontally at all. This must be when the bird dives! After that, its horizontal distance increases quickly until it reaches its maximum distance. This means that Statement III is also true.

Maya and Dominika are runners competing in a race. The graph illustrates their distances from the finish line over time.

Which of the following sentences might the sports announcer say while describing the race between these two athletes?
Which athlete finishes first?

We are given a graph showing the distance from the finish line over time of two athletes in a race.

In this graph, the vertical axis represents the distance to finish line and the horizontal axis represents time. Remember that the slope of a line is the ratio of the vertical change to the horizontal change. Slope = vertical change/horizontal change In this case, the vertical change is the change in distance and the horizontal change is the change in time. Since the ratio of the change in distance to the change in time is called speed, the slope of the graphs indicates how fast each athlete runs. Speed = change in distance/change in time Let's describe the speed of both athletes throughout the race.

Analyzing Maya's Graph

We can draw little arrows on Maya's graph to get an idea of how fast she is running. The steeper the arrows, the faster Maya is running.

At the beginning of the race, Maya's distance from the finish line decreases at a constant rate. This means that she is running at a constant speed. Her speed is enough to stay ahead of Dominica as the race starts. She runs at that speed for most of the race. ✓ I. & Maya has the early lead and runs at a & constant speed for a long time. After that, her distance to the finish line decreases at a faster rate, which suggests that her speed increases. Although she increases her speed, we know that she finishes the race behind Dominica because Maya needs more time to finish the race than Dominika does. This means that the last statement is false. * IV. & Maya runs the race at a constant speed for & a long time without much effort. Towards & the end of the race, she starts to increase & her speed and comes in first. Now let's look at the graph of Dominika's race.

Analyzing Dominika's Graph

This time we draw will little arrows on Dominika's graph to get an idea of how fast she is running. The steeper the arrows, the faster Dominika is running.

At the beginning of the race, Dominika's distance from the finish line decreases at a non-constant rate that shows that her speed is increasing more and more. Running at an increased speed for a long time suggests that Dominica has made an immense effort. After some time, her efforts pay off and she manages to pull ahead of Maya. ✓ II. & Dominika is picking up speed and manages & to get ahead of Maya. At a certain point, her graph becomes a straight line until it reaches the horizontal axis. This shows that Dominika's speed becomes constant as she approaches the end of the race. Therefore, the sports announcer would probably say the third statement as well. ✓ III. & Dominika sprints at the beginning of the & race. As she reaches the end, she reduces & her speed and maintains a constant pace. As a result, we can expect that the sports announcer might say the first three statements.


From the given graph, we can see that Dominika's line is the first to reach the horizontal axis. This means that Dominika finishes the race first.


Davontay's father drives to the gas station and fills up his tank. Then he drives to the market. Which graph displays the relationship between the amount of fuel in the gas tank of his car and time?

We are asked to determine which of the given qualitative graphs displays the relationship between the amount of fuel in the car's gas tank and time. Let's analyze the options one at a time, starting with Graph A.

Graph A

The graph starts decreasing at a steady rate, remains constant for a short time, and then decreases at a steady rate again.

In this graph, the decreasing parts indicate that the car is in motion, while the constant part indicates that the car is not using any gas, implying that it is not moving. We can interpret the graph as follows using this observation: Davontay's dad went to the market, did his shopping, and then came back home. This does not match the given situation.

Graph B

The graph starts by decreasing at a steady rate, remains constant for a short time, and then increases at a steady rate.

In this graph, the decreasing part indicates the car is moving, the flat part indicates the car is stationary, and the increasing part indicates fuel being added to the tank. We can interpret the graph as follows using this observation: Davontay's dad went to the market, did his shopping, and then filled up his tank. This does not match the given situation.

Graph C

Graph C has two decreasing parts and one increasing part.

In this graph, the decreasing parts indicate the car is moving, while the increasing part indicates fuel being added to the tank. This graph matches the given situation. Davontay's father drives to the gas station and fills up his tank, then he drives to the market. The answer is C.

Graph D

The difference of this graph from Graph C is that the last part is constant. This suggests that the car is parked at the gas station for a while after the tank is filled. This does not coincide with the given situation.

The graphs illustrate the movement of LaShay's and Diego's cars over time, starting simultaneously from the same LaShay's house.

Which of the following statements are true regarding the movements of the cars?

We are given two graphs that look the same. However, note that LaShay's graph shows the distance of her car from home as a function of time, while Diego's graph shows the speed of his car as a function of time.

Since we do not have numbers on either axis, we cannot talk about specific values of time, distance, or speed. Instead, we can make qualitative statements.

LaShay's Graph

Let's take a look at LaShay's graph. For the distance function, the output values tell us how far from home she is.

LaShay starts moving away from home at a constant rate. When the graph is constant, the car's distance from home is not changing, which suggest that it has come to a stop for a while. LaShay then turns around and comes back home. After staying at home for a time, her car moves away from home again at a constant rate. Our interpretation aligns with the first statement, so it is true. ✓ I. & LaShay moves away from home at a constant & rate, pauses briefly, then returns home. After a & while at home, she moves away again at a & constant rate.

Diego's Graph

For the speed function, the output values tell us how fast Diego is moving.

Diego starts from zero and speeds up at a constant rate. When the graph becomes a horizontal line, he is maintaining his speed for a while before slowing down until he comes to a complete stop. At this point, it is not possible to tell whether he has returned to LaShay's house or not. We can conclude that Statement II is false. * II. & Both LaShay and Diego make the same & movements throughout their journeys. The horizontal part of the graph on the horizontal axis means that Diego is stationary for a while, probably somewhere far from his starting point. Finally, he speeds up again. Therefore, the last statement is true. ✓ III. & Diego leaves LaShay's house, speeds up at a & constant rate, and maintains that speed for a & time. He then slows down until he stops. He & stays stationary for some time, then speeds up & again. As a result, both Statements I and III are correct.

Qualitative Graphs

Recommended exercises

To get personalized recommended exercises, please login first.
Return to lesson.
Loading content