Slope and Intercepts
Before we begin to write equations from a graph, let's review some important information about the graphs of lines: slopes and intercepts. Being able to easily identify slopes and intercepts from a graph will be extremely useful when writing the equation to the line.
The slope of a line can be found by using any two points on a graph. All we need are the x– and y–coordinates from two points to plug into our equation for slope:
Intercepts are where the line crosses a particular axis. The x-intercept is the location where the line crosses the x-axis, and it has coordinates of (x, 0). The y-intercept is the location where the line crosses the y-axis, and it has coordinates of (0, y). The big idea here is that at the x-intercept, y = 0, and at the y-intercept, x = 0.
Using the y-intercept
If the y-intercept of a line is displayed on the graph, we will want to write the equation in slope-intercept form, because part of the equation contains the y-intercept itself, or at least the y-coordinate of it. Slope-intercept form is y = mx + b, where b is the y-coordinate of the y-intercept. The only other thing we will need to solve for is the slope.
Find the equation of the line graphed below:
Here, the y-intercept is clearly displayed on our graph. The line intercepts the y–axis at the point (0, 2). This means that our b-value is 2 in our equation written in slope-intercept form. Now we just need to calculate slope. To calculate slope, we can identify another point on our line and plug the coordinates into the formula for slope. Let's use the point (2, 5), which is also on our line.
Now that we have the slope, we can write the equation of the line in slope-intercept form:
When the y-intercept is Not Visible
In our first example, we saw how helpful it was to use the y-intercept on the graph to design our equation. Sometimes, the y-intercept is not visible on the graph, but we can calculate it algebraically. The first thing we'll want to do in this situation is locate any two points on the graph and calculate the slope. From there, we will be able to plug in fixed values for everything else in slope-intercept form, thus allowing us to solve for b.
Write the equation to the line graphed below:
Since we cannot easily determine the y-intercept graphically, let's identify two points on the graph and calculate the slope of this line. We'll choose the points (3, 5) and (4, 2).
Now that we know the slope of the line is –3, we know that the equation so far is y = –3x + b. We just need solve for b. To do this, we can take any point on the line, and use the coordinates of that point for x and y in the equation, and solve for b. To make things easier, we can choose either of the two points we have already identified when we calculated the slope. Let's use the point (4, 2). This means that we will plug in 4 for x and 2 for y in the equation y = –3x + b.
The equation of the line is: