A table of values contains two lists of numbers written alongside each other. The first list contains the chosen input values, which are often the 𝑥 coordinates. The second list contains the outputs obtained when this first list is put into a given equation, often the y coordinates. Together, the pairs of numbers in both lists make up coordinates that can be plotted as a graph.
In this example below, we have a table of values for y = 2𝑥 + 1.
There are two lists of numbers shown in the table of values.
The first list is the given 𝑥 values. Here we chose 0 to 4. We can choose any numbers we like to go in the 𝑥 row. We chose 0 to 4 because they are easy to substitute into equations because they are small numbers.
The y values have been calculated using the equation y = 2𝑥 + 1. This equation tells us to multiply the 𝑥 values by 2 and then add one to them to find the y values.
For example, when × = 3, we multiply 3 by 2 and then add 1.
3 × 2 + 1 = 7 and so, we write 7 in the y row below the 𝑥 value of 3.
Doing this calculation for each 𝑥 value results in a different number which is written in the y row alongside the 𝑥 value it came from.
A table of values is made of two rows, the first labeled as 𝑥 and the second as y. In the 𝑥 row, consecutive numbers are chosen from between the smallest and largest 𝑥 coordinates on the axes given. The numbers in the y row are calculated by substituting these 𝑥 values into the given equation.
Here is a set of axes for plotting y = 2𝑥 – 3.
The smallest 𝑥 value is -10 and the largest 𝑥 value is 10.
We only need two points to draw a straight line but it is a good idea to choose three to five different 𝑥 values for the table of values. This allows us to better notice any mistakes and to help us draw a more accurate line through the points.
To choose what 𝑥 values to use for the table of values, it is best to choose small positive whole numbers if possible.
In this example, we will choose 0, 1, 2, 3 and 4.
The numbers in the y row are calculated by substituting these 𝑥 values into the equation y = 2𝑥 – 3. This tells us to multiply each 𝑥 value by 2 and then subtract 3.
The following table shows the calculations for the table of values.
| 𝑥 Coordinate | Calculation for y = 2𝑥 – 3 | y Coordinate |
|---|---|---|
| 𝑥 = 0 | y = 2 × 0 – 3 | y = -3 |
| 𝑥 = 1 | y = 2 × 1 – 3 | y = -1 |
| 𝑥 = 2 | y = 2 × 2 – 3 | y = 1 |
| 𝑥 = 3 | y = 2 × 3 – 3 | y = 3 |
| 𝑥 = 4 | y = 2 × 4 – 3 | y = 5 |
A table of values contains pairs of 𝑥 and y values which form pairs of coordinates that can be plotted as points. The 𝑥 coordinate tells us how far right the point is and the y coordinate tells us how far up the point is. If the coordinates are negative, then the point is left or down respectively. Once each point is plotted, simply draw a line through them.
For example, here is the table of values for y = 3𝑥 – 5.
We have the following coordinates:
The coordinates are plotted on the graph as shown.
The following table explains how to plot coordinates:
| Coordinate | How to Plot it |
|---|---|
| Positive 𝑥 value | Move to the right |
| Negative 𝑥 value | Move to the left |
| Positive y value | Move up |
| Negative y value | Move down |
Each of these 2 numbers describe one point.
Once each point is plotted, the line is graphed by drawing a straight line through them all to the very edge of the axes.
A table of values is linear if as 𝑥 increases by a constant amount, the y values all increase by a constant amount. If the y values increase by the same amount from one number to the next, then the coordinates will form a straight line when plotted. If the y values increase by different amounts each time, then the table of values is non-linear.
If something is linear, this means that it forms a straight line.
For example, here is a table of values where the y values increase by 2 every time that 𝑥 increases by 1.
Since the y values increase by 2 from one number to the next, this table of values is produced from a linear expression.
Alternatively, we can plot the 𝑥 and y values on a set of axes and look at it to see if it forms a straight line or not.
We can see that the coordinates form a straight line and so, the table of values produces a linear graph.
If the equation that produced the table of values contains only y and 𝑥 as the variables, then it will produce a linear graph. If it contains 𝑥 2 or any other powers of 𝑥, then it is non-linear.
This table of values is formed from the equation y= 2𝑥 + 1 and because there are no higher powers of 𝑥, the equation is a linear one.
The equation of a straight line is y = m𝑥 + c, where m is the gradient and c is the y-intercept. The gradient is the amount y increases every time that 𝑥 increases by 1 in the table. The y-intercept is the y value that accompanies the 𝑥 = 0 value in the table.
Here is an example of finding the equation from a table of values.
The first step is to find the gradient. This is how much the y values increase every time that the 𝑥 values increase by 1.
We can see that the y values increase by 3 every time the 𝑥 values increase by 1. The gradient is 3.
The second step is to find the y-intercept. This is the y value that is below the 𝑥 = 0 value. Below 𝑥 = 0 is y = -5. Therefore the y-intercept is -5.
The equation of the line is given by y = m𝑥 + c, where m is the gradient and c is the y-intercept.
The gradient of 3 is written in front of the 𝑥 and the y-intercept of -5 is just written afterward.
The equation of the line is y = 3𝑥 – 5.
Now try our lesson on Right Angles where we learn how to identify right angles in a variety of situations.
