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

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

• Determine to which positions Emilia could have started.

  Hints

  Use the board game.

  An input can't have two different outputs, but it is possible for two different inputs to have the same output.

  Solution

  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.

  Oscar looks at the last possible starting point, space $6$; rolling a $3$ would lead to space $9$.

• Explain how to represent the input and output of a function with a graph.

  Hints

  Keep the order of pairs in a coordinate system in mind.

  Here is a table representing the possible moves Emilia could have made in her board game with Oscar.

  From the table, we can see that starting at position $3$ leads to an ending position of $6$.

  An example of an ordered pair is $(3,6)$.

  Solution

  For a graph we draw a coordinate system with a $x$- and a $y$-axis.

  You can see such a coordinate system in the picture.

  The input is represented by the $x$-values and the output is represented by the $y$-values. So we can draw each ordered pair $(x,y)$ into the coordinate system.

• Plot the given points in a graph.

  Hints

  Keep the order of the coordinates in mind.

  For example, with $(3,7)$, you have the line $x=3$ parallel to the $y$-axis and the line $y=7$ parallel to the $x$-axis. The intersection of those lines is the point $(3,7)$.

  The horizontal axis is the $x$-axis, while the vertical axis is the $y$-axis.

  The number of pairs given by the table is five.

  Solution

  From the table,

  $\begin{array}{c|c} x&y\\ \hline 1&1\\ 3&2\\ 5&3\\ 7&3\\ 8&4 \end{array}$

  we get the ordered pairs $(1,1)$, $(3,2)$, $(5,3)$, $(7,3)$, and $(8,4)$.

  To graph these points, keep in mind that the first coordinate of an ordered pair is the $x$-coordinate. So for example, go to $x=3$ and then two steps up to get the point $(3,2)$.

  For instance, to differentiate between the points, or ordered pairs, $(5,3)$ and $(3,5)$:

  • $(5,3)$ lies on a line through $x=5$ parallel to the $y$-axis.
  • $(3,5)$ lies on a line through $x=3$ parallel to the $y$-axis.

  You can see the point from the table plotted in the graph above and to the right.

• Decide which graph corresponds to which mapping diagram.

  Hints

  You can write each pair of input $x$ and output $y$ as an ordered pair.

  The relation where each $x$ is assigned to the same $y$ is called a constant relation.

  The corresponding points lie on a line parallel to one coordinate axis.

  Keep the order of the coordinates in mind:

  The first coordinate (the input) corresponds to the $x$-axis, while the second corresponds to the $y$-axis.

  Solution

  You can write down ordered pairs for each relation given by a mapping diagram.

  Let's start with the diagram on the right. You see the corresponding graph above and to the left, and can check that they match by checking the ordered pairs given by the diagram.

  Next, let's look at the relation where each input is assigned to the same output, namely $3$. The corresponding graph is the one where all the points lie on a line parallel to the $x$-axis.

  Now the first mapping diagram from the left gives us the pairs $(1,2)$, $(2,2)$, $(3,4)$, $(4,4)$, and $(5,6)$.

  The diagram where each input is assigned to itself gives us the graph where all points lie on a line from bottom left to top right.

• Match the image with its corresponding definition.

  Hints

  In a mapping diagram, the input is the domain and the output the range.

  For a graph, we use a coordinate system.

  An ordered pair in general is given by $(x,y)$.

  Solution

  To represent any relation between an input and an output we can use different imagines.

  From left to the right we have:

  The mapping diagram

  On the left side we have the domain, the set of all inputs, and on the right the range, where we find all possible outputs. We use arrows to show the corresponding relationships.

  Ordered pairs

  Using the relations represented by the mapping diagram, we can also write ordered pairs: the first position is the input $x$ and the second the output $y$.

  Table

  $\begin{array}{c|c} x&y\\ \text{input}&\text{output}\\ \hline 1&1\\ 2&6\\ 3&6\\ 6&9 \end{array}$

  Here you see the corresponding table: row by row you see the ordered pairs. Again you can recognize the input on the left and the output on the right side.

  Graphs

  To represent the relation in a more visual way we use graphs: for this we draw each ordered pair $(x,y)$ into a coordinate system, where the $x$-axis stands for the input while the $y$-axis represents the output.

• Determine the coordinates of the ordered pairs.

  Hints

  Keep the order of the coordinates in mind.

  The first coordinate is the $x$-coordinate, which corresponds to the $x$-axis, or the input.

  For example, the order pairs of this graph, from left to right:

  • $(1,5)$
  • $(3,4)$
  • $(5,3)$
  • $(7,2)$

  Solution

  Let's start with the topmost graph:

  • For $y=3$, we get $x=2$.
  • For $x=4$, we get $y=5$.
  • For $x=5$, we get $y=5$.
  • The last point is $(7,2)$

  For the middle graph, we get the point :

  • $(1,5)$.

  For the lower graph, we get:

  • $(2,1)$
  • $(3,3)$
  • $(4,5)$
  • $(5,7)$