Convert between Tables, Graphs, Mappings, and Lists of Points 04:03 minutes

Video Transcript

Transcript Convert between Tables, Graphs, Mappings, and Lists of Points

Emilia and Oscar are playing a board game called The Joyous Journey when Oscar gets a call. Seizing the opportunity, Emilia rolls the dice and deftly moves her game piece. When Oscar returns from his call, he notices something fishy SOMEHOW, Emilia ended up on a space where she gets a bonus card! Oscar asks Emilia where her turn started and what she rolled to land on this space. Conveniently, Emilia doesn't seem to remember those details anymore, but she does remember she rolled a 3. Oscar decides to examine the game board a bit more carefully.

Mapping Diagram

We can use a mapping diagram to help us visualize the situation. We know Emilia ended up on space 7 and that she moved forward. If Emilia started at the first space and rolled a 3, she would have ended up at space 4, but space 4 sends a player back to the start so she would still be at the first space. We can show this relation by drawing an arrow that starts at our input, 1, and ends at our output, 1.
So Emilia didn't start at the first space, what about the second? Since she rolled a 3, she would have gone to space 5. But space 5 moves the player 3 additional steps forward. This brings us to space 8. Space 8 moves a player back 2 steps, which finally ends our journey at space 6 where you have to take a though-card. So an input of 2 gives us an output of 6. We can show this by drawing an arrow beginning at the input, 2, and ending at the output, 6. Starting at space 3 and rolling a 3 also leads to space 6.
And since space 4 sends a player back to space 1, it’s not possible that this space was Emilia's starting point. Space 5 isn't a possibility because it sends a player 3 spaces forward to space 8, and space 8 sends a player 2 spaces back. Emilia's chances are running out. Oscar looks at the last possible starting point, space 6; rolling a 3 would lead to space 9.
All input-values, also called x-values, are called the domain. And all output-values, also called y-values, are called the range. In this example, the domain is {1,2,3,6} and the range is {1,6,9}. Since there is exactly one output for every possible input this relation is a function.

Putting the Data in Table and Graph Form

This is not the only way to show these things, however. We can group these inputs and outputs into ordered pairs, which also allows us to put the data in table and graph form. For our first input, we have 1 and from our mapping diagram, we see that the output is also 1. The ordered pair for this is (1, 1). Our second input, 2, has an output of 6. This ordered pair is (2, 6). Think you're getting the hang of it? Our third input, 3, also has an output of 6, making its ordered pair (3, 6). And our final ordered pair is (6, 9).
To show this relationship in a table, we write all the x-values to the left and the corresponding y-values to the right. Our graph, however, is arranged a bit differently. Our inputs are our x-values and our outputs are our y-values. This gives us our points on the graph.

Oscar's upset because it's obvious Emilia cheated to get to the lucky space. But did Oscar play fair?