Learning Outcome
- Graph linear functions using tables
- Identify important features of graphs of linear functions
A helpful first step in graphing a function is to make a table of values. This is particularly useful when you do not know the general shape the function will have. You probably already know that a linear function will be a straight line, but let us make a table first to see how it can be helpful.
When making a table, it is a good idea to include negative values, positive values, and zero to ensure that you do have a linear function.
Make a table of values for [latex]f(x)=3x+2[/latex].
Make a two-column table. Label the columns x and f(x).
x | f(x) |
---|---|
Choose several values for x and put them as separate rows in the x column. These are YOUR CHOICE – there is no “right” or “wrong” values to pick, just go for it.
Tip: It is always good to include 0, positive values, and negative values, if possible.
x | f(x) |
---|---|
[latex]−2[/latex] | |
[latex]−1[/latex] | |
[latex]0[/latex] | |
[latex]1[/latex] | |
[latex]3[/latex] |
Evaluate the function for each value of x, and write the result in the f(x) column next to the x value you used.
When [latex]x=0[/latex], [latex]f(0)=3(0)+2=2[/latex],
[latex]f(1)=3(1)+2=5[/latex],
[latex]f(−1)=3(−1)+2=−3+2=−1[/latex], and so on.
x | f(x) |
[latex]−2[/latex] | [latex]−4[/latex] |
[latex]−1[/latex] | [latex]−1[/latex] |
[latex]0[/latex] | [latex]2[/latex] |
[latex]1[/latex] | [latex]5[/latex] |
[latex]3[/latex] | [latex]11[/latex] |
(Note that your table of values may be different from someone else’s. You may each choose different numbers for x.)
Now that you have a table of values, you can use them to help you draw both the shape and location of the function. Important: The graph of the function will show all possible values of x and the corresponding values of y. This is why the graph is a line and not just the dots that make up the points in our table.
Now graph [latex]f(x)=3x+2[/latex].
Using the table of values we created above, you can think of f(x) as y. Each row forms an ordered pair that you can plot on a coordinate grid.
x | f(x) |
[latex]−2[/latex] | [latex]−4[/latex] |
[latex]−1[/latex] | [latex]−1[/latex] |
[latex]0[/latex] | [latex]2[/latex] |
[latex]1[/latex] | [latex]5[/latex] |
[latex]3[/latex] | [latex]11[/latex] |
Plot the points.
Since the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.
![A line through the points in the previous graph.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/04/18232426/image006.gif)
Let us try another one. Before you look at the answer, try to make the table yourself and draw the graph on a piece of paper.
Example
Graph [latex]f(x)=−x+1[/latex].
In the following video, we show another example of how to graph a linear function on a set of coordinate axes.
These graphs are representations of linear functions. Remember that a function is a correspondence between two variables such as x and y.