Writing an Equation from a Graph

1. 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:

formula to know

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).

big idea

At the x-intercept, , and at the y-intercept,

2. When the y-Intercept Is Visible

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 , where b is the y-coordinate of the y-intercept. The only other thing we will need to solve for is the slope.

EXAMPLE

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.

	Use the slope formula and substitute in the two points: and
	Evaluate the numerator and denominator
	The slope

Now that we have the slope, , and the y-coordinate of the y-intercept, 2, we can write the equation of the line in slope-intercept form:

3. 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.

EXAMPLE

Write the equation to the line graphed below:



Since we cannot easily determine the y-intercept graphically, let's identify two points that we can see on the graph and calculate the slope of this line. We'll choose the points (3, 5) and (4, 2).

	Use the slope formula and substitute in the two points: and
	Evaluate the numerator and denominator
	Simplify
	The slope

Now that we found the slope of the line to be -3, we can say that the equation so far is . We just need to solve for b. To do this, we can take any point on the line, plug in 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 .

	Using the equation that we have so far, plug in a point on the line, for instance, (4, 2). Substitute 4 for x and 2 for y
	Multiply -3 and 4
	Add 12 to both sides
	The y-coordinate of the y-intercept

Now that we have the slope, -3, and the y-coordinate of the y-intercept, 14, we can write the equation of the line in slope-intercept form:

The equation of the line is .

hint

If the original graph was extended vertically, we would have been able to see the y-intercept at (0, 14).