Learning Outcomes
- Graph coordinate pairs from a table of values
- Create a table of values from a graph
- Write a relation as a set of ordered pairs
- Identify the domain and range of a relation
- Identify local maxima and minima
- Identify lines of symmetry
KEY words
- A relation is a set of ordered pairs.
- The domain of a relation is the set of all [latex]x-[/latex]values.
- The range of a relationship is the set of all [latex]y-[/latex]values.
- A local maximum is the [latex]y-[/latex]value of a point where the graph turns to go back up after moving downwards.
- A local minimum is the [latex]y-[/latex]value of a point where the graph turns to go back down after moving upwards.
- A maximum is the highest [latex]y-[/latex]value on the graph.
- A minimum is the lowest [latex]y-[/latex]value on the graph.
Graphs and Tables
Graphs and tables have a symbiotic relationship. Ordered pairs can be written in the rows and columns of a table or graphed as points using the rectangular coordinate system. The same data can be displayed as a table or a graph.
Figure 1 shows a table of data points that when converted to [latex](x,y)[/latex] coordinates can be plotted on a rectangular coordinate system to become a graph.
| [latex]x[/latex] | [latex]y[/latex] |
|---|---|
| [latex]-4[/latex] | [latex]7[/latex] |
| [latex]-2[/latex] | [latex]6[/latex] |
| [latex]0[/latex] | [latex]5[/latex] |
| [latex]2[/latex] | [latex]4[/latex] |
| [latex]4[/latex] | [latex]3[/latex] |
| [latex]6[/latex] | [latex]2[/latex] |
| [latex]8[/latex] | [latex]0[/latex] |
Figure 1
A set of ordered pairs is called a relation. From figure 1, the relation is [latex]\{(-4,7),(-2,6),(0,5),(2,4),(4,3),(6,2),(8,0)\}[/latex]. The set of all values of the independent variable [latex]x[/latex] is the domain of the relation, while the set of all the values of the dependent variable [latex]y[/latex] is the range of the relation. For the data in figure 1, domain = [latex]\{-4,-2,0,2,4,6,8\}[/latex] and range = [latex]\{0,2,3,4,5,6,7\}[/latex].
Relation
A relation is a set of ordered pairs [latex](x,y)[/latex].
The domain of a relation is the set of all [latex]x[/latex]-values.
The range of a relation is the set of all [latex]y[/latex]-values.
We can also convert from coordinates on a graph to data points in a table.
example
Use the graph to complete the table. Then state the domain and range of the relation.
| [latex]x[/latex] | [latex]y[/latex] |
|---|---|
| [latex]0[/latex] | |
| [latex]7[/latex] | |
| [latex]0[/latex] | |
| [latex]2[/latex] | |
| [latex]-4[/latex] | |
| [latex]4[/latex] |
Solution
| [latex]x[/latex] | [latex]y[/latex] |
| [latex]0[/latex] | [latex]4[/latex] |
| [latex]7[/latex] | [latex]-3[/latex] |
| [latex]-1[/latex] | [latex]0[/latex] |
| [latex]3[/latex] | [latex]2[/latex] |
| [latex]-4[/latex] | [latex]-5[/latex] |
| [latex]5[/latex] | [latex]4[/latex] |
domain = [latex]\{-4,-1,0,3,5,7\}[/latex] and range = [latex]\{-5,-3,0,2,4\}[/latex]
We only listed [latex]4[/latex] once because it is not necessary to list it every time it appears in the range.
TRY IT
List the domain and range for the following table of values, then graph the data points on a coordinate system.
| x | y |
|---|---|
| [latex]−3[/latex] | [latex]4[/latex] |
| [latex]−2[/latex] | [latex]4[/latex] |
| [latex]−1[/latex] | [latex]4[/latex] |
| [latex]2[/latex] | [latex]4[/latex] |
| [latex]3[/latex] | [latex]4[/latex] |
The graphs we have seen so far in this section have been of relations with a finite number of points. Some relations have an infinite number of points, with the points lying so close together on a graph they form a curve or a line. Since the relation has an infinite number of points, it is impossible to write every ordered pair in a table. However, a few points can be tabulated to show a pattern.
example
Use the graph to state the domain and range of the relation. Then complete the table of values.
| [latex]x[/latex] | [latex]y[/latex] |
|---|---|
| 3 | |
| 3 | |
| -3 | |
| -3 |
Solution
The domain is the set of all possible [latex]x[/latex]-values. In this graph the [latex]x[/latex]-values start at [latex]x=-3[/latex] and end at [latex]x=3[/latex]. Both end points are included in the domain. This set of values can be written using interval notation or using set-builder notation.
[latex]\text{Domain}:\left[-3,3\right][/latex] or [latex]\{x|-3≤x≤3,\;x\in\mathbb{R}\}[/latex].
The range is the set of all possible [latex]y[/latex]-values. In this graph the [latex]y[/latex]-values start at [latex]y=-3[/latex] and end at [latex]y=3[/latex]. Both end points are included in the range. This set of values can be written using interval notation or using set notation.
[latex]\text{Range}:\left[-3,3\right][/latex] or [latex]\{y|-3≤y≤3,\;y\in\mathbb{R}\}[/latex].
To complete the table we look for the points on the graph with the given [latex]x[/latex] or [latex]y[/latex] values.
| [latex]x[/latex] | [latex]y[/latex] |
|---|---|
| [latex]0[/latex] | 3 |
| 3 | [latex]0[/latex] |
| -3 | [latex]0[/latex] |
| [latex]0[/latex] | -3 |
Notice that this graph has an infinite number of lines of symmetry. A line of symmetry is any line that cuts the graph in half, with each half being a mirror image of the other. In a circle, every diameter is a line of symmetry.
LINE OF SYMMETRY
A line of symmetry is any line that cuts the graph in half, with each half being a mirror image of the other.
example
Use the graph to state the domain and range of the relation. Then complete the table of values by identifying the points on the graph. In addition, describe any lines of symmetry.
| [latex]x[/latex] | [latex]y[/latex] |
|---|---|
| -2 | [latex]0[/latex] |
| -1 | [latex]-3[/latex] |
| 0 | [latex]0[/latex] |
| 1 | [latex]-3[/latex] |
| 2 | [latex]0[/latex] |
Solution
The domain is the set of all possible [latex]x[/latex]-values. In this graph the [latex]x[/latex]-values start at [latex]x=-\infty[/latex] and end at [latex]x=\infty[/latex]. Since [latex]\infty[/latex] can never be reached, the end points are not included in the domain. This set of values can be written using interval notation or using set notation.
[latex]\text{Domain}:\left[-\infty,\infty\right][/latex] or [latex]\{x|-\infty≤x≤\infty,\;x\in\mathbb{R}\}[/latex].
The range is the set of all possible [latex]y[/latex]-values. In this graph the [latex]y[/latex]-values start at [latex]y=-4[/latex] and end at [latex]y=\infty[/latex]. The lowest value of [latex]y=-4[/latex] is included in the range, but [latex]y=\infty[/latex] is not. This set of values can be written using interval notation or using set notation.
[latex]\text{Range}:\left[-4,\infty\right)[/latex] or [latex]\{y|\;y≥-4,\;y\in\mathbb{R}\}[/latex].
To complete the table we look for the points on the graph with the given [latex]x[/latex]values.
| [latex]x[/latex] | [latex]y[/latex] |
|---|---|
| -2 | [latex]0[/latex] |
| -1 | [latex]-3[/latex] |
| 0 | [latex]0[/latex] |
| 1 | [latex]-3[/latex] |
| 2 | [latex]0[/latex] |
This graph has the [latex]y-[/latex]axis as a line of symmetry.
Notice that this graph has three turning points: a local maximum of [latex]0[/latex] at [latex]x=0[/latex] and two local minima of [latex]-4[/latex] at [latex]x[/latex]-values close to [latex]-1.4[/latex] and [latex]1.4[/latex]. The exact [latex]x[/latex] values are impossible to tell from the graph. The local maximum of 0 is not the overall maximum of the graph, which is [latex]\infty[/latex]. Rather is it the maximum value in an area around [latex]x=0[/latex]. On the other hand, the local minima value of [latex]-4[/latex] is also the overall minimum value of the graph.
local maxima and minima
- A local maximum is the [latex]y-[/latex]value of a point where the graph turns to go back up after moving downwards.
- A local minimum is the [latex]y-[/latex]value of a point where the graph turns to go back down after moving upwards.
TRY IT
Use the graph to state the domain and range of the relation. Then state any local maxima or minima.
