To graph linear equations, you need a solid foundational understanding of linear equations. Most importantly, you should remember that slope-intercept form is the most useful form to graph linear equations with.
As a refresher, slope-intercept form is written as y = mx+b, where m represents the slope and b is the y-intercept of our line. This means that all we really need to graph a line is a single point and the slope!
If the point on the line changes, then the entire line will shift up, down, left, or right.
If the slope changes, then the line will change how steep it is. You can think of slope in terms of having to ski up or down a hill.
A negative slope means your line will be going down from left to right and you can ski down the hill easily.
A positive slope means your line will be increasing from left to right and it would be challenging to ski up the hill.
The real question is: how do we figure out where to start graphing our line?
Slope-intercept form is the most useful form, but the y-intercept should be the first thing you consider when graphing a line.
For instance, if I have the line y=2x-4, my y-intercept is -4, meaning that I should find the point on the y-axis with a value of -4.
Once you’ve found the y-intercept and drawn the point at the correct value, all that is left is to determine how the slope will affect the graph.
The old saying that slope = rise over run is particularly useful. Since our line y=2x-4 has a slope of 2, we can interpret this as 2/1. This means that our slope means we must rise 2 units.
And we must run 1 unit.
This shows us that we end up at the point (1, -2) when we go 1 unit to the right. Therefore, we can continue this pattern and extend our line in both directions infinitely! Of course, just draw some arrows on the end of the line to show this, otherwise you’ll just be drawing it for the rest of your life (and who would want that?).
This method of graphing lines is great when we have a linear equation in slope-intercept form, but what happens when we are given a line that isn’t in this form? It’s simple, just get that equation into slope-intercept form and use the same process described earlier!
For example, if we have the standard form of a linear equation such as x-5y=-15, we must first simplify this equation to solve for y.
Subtract x from both sides to find:
-5y = -x-15
Divide both sides by -5 to find:
y = ⅕x+3
Et voila! After two steps, we have the same form of a linear equation. This tells us that the slope is ⅕ and the y-intercept is 3.
Using the same method from earlier has us plot the point (0, 3) as the y-intercept and then rise 1 unit and run 5 units.
Now, we have graphed a line that was given in standard form! This line has a much less steep slope than our previous example, but the process for graphing was identical. You can apply this technique for any line, as long as you always remember to correctly find the slope-intercept form of the line.