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Here are a few recommended readings before getting started with this lesson.
Data that can be quantified or represented by numbers is known as numerical data, while data without numerical values is known as 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 becategorizedas belonging to one of these categories.
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.
Examine the graph that shows Zosia's speed on her way to school.
Which statements are true?In this case, increasing parts of the graph indicate that the speed is increasing, and decreasing parts indicate that the speed is decreasing.
The graph is a qualitative graph where $x-$axis represents time and $y-$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. |
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 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. |
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.
The situation consists of four different parts, or intervals.
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 $0$ and her speed is $0.$ 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.
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.
Some important features of the graph can be described as follows.
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 $1200$ can be omitted.
List some important features of the graph.
Once again, the duration of Zosia's journey to the cinema is not known. However, two output values are known, with $300$ meters and $900$ 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.
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.Example Graph:
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?
From the description, the graph should consist of three parts.
Next temporarily divide the graph into three time intervals.
The room temperature is usually around $20_{∘}C,$ which is greater than $0_{∘}C.$ For this reason, the starting point should be a positive $T-$value, not $0.$ Since the temperature remains the same for a certain amount of time, Part $1$ 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 $T-$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.
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. |
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. |