| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount}} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount}} |
| {{ 'ml-lesson-time-estimation' | message }} |
Here are a few recommended readings before getting started with this lesson.
Consider the following pair of graphs. Apart from the shape of the graphs, what difference can be established between them? If needed, press the button to show a vertical line. Then, move the line horizontally and watch for the number of times it intersects each graph along the way.
In the previous exploration, each graph represented a certain relationship between the variables x and y. In fact, the value of y depends on the value of x. Next, the definition of relation is developed along with one way of visualizing it.
Ignacio, the younger brother of Kriz, loves watching Kriz's college volleyball games. He recorded Kriz and their teammates' names, ages, and heights. Ignacio is such a curious kid that he wants to describe some relations based on these values. Then he will share them with his big sibling.
Represent the relation between Kriz's teammates' ages and heights using a table of values, a mapping diagram, a set of ordered pairs, and points in a coordinate plane.Table of Values:
Ages | 18 | 20 | 23 | 20 | 21 | 19 |
---|---|---|---|---|---|---|
Heights (m) | 1.70 | 1.96 | 1.85 | 1.75 | 1.91 | 1.87 |
Mapping Diagram:
Ordered Pairs: {(18,1.70), (20,1.96), (23,1.85), (20,1.75), (21,1.91), (19,1.87)}
Coordinate Plane:
To make a mapping diagram, place all the ages in one set and all the heights in another set. Then, connect each age with its corresponding height using an arrow. To write the relation as a set of ordered pairs, place the ages in the first component and the corresponding heights in the second component.
Using the information that Ignacio gathered, a relation between the ages and heights can be made and represented using different visualizations.
First, organize the relation using a table of values. Place the ages in the first row and the heights in the second row.
Ages | 18 | 20 | 23 | 20 | 21 | 19 |
---|---|---|---|---|---|---|
Heights (m) | 1.70 | 1.96 | 1.85 | 1.75 | 1.91 | 1.87 |
A comparison for a function is to think of it as a machine. The inputs are the materials placed in the machine, and the outputs are the objects created. In the following applet, there are four preset inputs. This particular machine only accepts numbers between -100 and 100 to be plugged in as materials. See what happens!
It is essential to determine whether a relation describes a function or not. For example, imagine that a programmer wrote the following relation for a printer's software.
If a user selects color=0, then they cannot be sure if the document will be printed in black or red. In this case, the programmer needs to write a relation that is a function. Determining if a relation is a function can be done by using either a mapping diagram or the Vertical Line Test. The appropriate method to be used depends on how the relation is represented.
Ignacio enjoyed displaying the relations between the ages and heights of Kriz's teammates so much that he wanted to go a step further and continue analyzing relations. He then asked Ramsha and Diego the ages and heights of their family members and wrote that data, including each player's values, in the following sets.
Note that (18,1.75) means that some member of Ramsha's family is 18 years old and 1.75 meters tall.
Only set R represents a function.
By analyzing the two data sets, some other conclusions can be drawn.
Of course, based on the given information, it does not have to be the exact situation. The conclusions were done simply to show one way of interpreting the data recorded.
Only the first graph represents a function.
At first, Ignacio was glad that his mistake only caused the graph to change domains and ranges. He only realized that it was no longer a function when he applied the vertical line test. He better change it back before someone needs to use the database!