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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. 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 becategorizedinto one of these groups.
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.
Consider 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, while decreasing parts indicate that the speed is decreasing.
The graph is a qualitative graph where the $x-$axis represents time and the $y-$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. |
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 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. |
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.
The given 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 time Paulina starts running is considered to be $0$ and her starting speed is $0.$ 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.
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.
Some important features of the graph can be described as follows.
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 $1200$ label can also be omitted.
Consider some important features of the graph.
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 $300$ meters and the distance from the library to the movie theater is $900$ 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.
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.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 curved?
From the description, the graph should consist of three parts.
Next, temporarily divide the graph into three intervals of time.
The average room temperature is around $68_{∘}F,$ which is greater than $0_{∘}F.$ As such, the starting temperature at $t=0$ 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.
Notice that the graph does not reach the $t-$axis. Without any data points, this axis can be understood to mean $T=0,$ or that the theater is $0_{∘}F.$ 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.
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. |
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. |