Value Table

Before attempting to plot a graph, we should prepare a table of values. This is a table that will contain all the data that need to appear on the graph, and it will be very helpful in drawing it. Before plotting any graph (for example, of a constant function), it is recommended to prepare a table of values. Now let's understand how to do this.

Suppose we want to plot the graph of a linear function
$Y=X+2$

Maybe later you can draw this function by heart, but even if it's not completely correct. That's exactly why we'll use the table of values.

First, we'll draw the table of values for $X$ and $Y$.

Now we fill in the table. To fill it, we select any number we want for the value of $X$, and for $Y$ we'll get a solution for the value of $Y$.

It is advisable to use small numbers to facilitate the calculation.

For example, we put $X=0$ and we get $Y=2$, then we can put $X=1$ and we get $Y=3$.

Now we want to draw a function from data

$Y=X+2$

Using a data table.

To do this, we will draw the X-axis and the Y-axis.

Then we'll go over each pair of values in the table and plot them on the axes. Each pair of values we find represents a point on the graph. Let's start with the first value:

$X=0,Y=2$

This can be briefly listed as follows: $\left(0,2\right)$

It is located at . It looks like this:

We'll move on to the next point:

$X=1,Y=3$

Or in short $\left(1,3\right)$

We will also mark it on the table.

Similarly, we plot the rest of the other points we find using the graph.

It will look like this:

Now we can draw an imaginary line connecting all the points we've got:

This is a drawing of the graph of the function.

$Y=X+2$
Now let's look at an example of the reverse process. We'll try to understand from the table of values, which linear function it represents.

Look at the following table of values:

Next, you'll learn how to mathematically calculate what the linear equation is (that is, what the function is) that corresponds to these values. At this step, we can use our logic to deduce this on our own.

Note that for every value of $X$, the corresponding value of $Y$ is exactly $11$ times smaller.

Therefore, we can conclude that the straight line is

$Y=X-1$

Now we want to plot this function, using the given table of values. Similar to the previous exercise, we will draw the two axes, and on top of them, we'll plot the points we obtained.

After marking the points, we'll draw the line between them, and this is the graph of the function. Note that it can be continued as much as we want on both sides. It is mandatory to draw with a ruler for greater precision.

Observe the following table:

Next, we'll learn how to mathematically construct the corresponding graph from this table. In this step, we can deduce once again from our logic what the equation is that fits the data table. We will try to show what the relationship between $X$ and $Y$ is. Note that for each pair $X$, $Y$, the value of $Y$ is exactly $2$ times that of $X$.

Therefore, we can conclude that the appropriate equation is
$Y=2X$

Now we want to plot this straight graph, using the data table. As in previous occasions, we'll mark all the points we received on the axes:

Now we'll draw the straight line that goes through the points with a ruler.

This is our straight graph.