This lesson will make use of slopes and y-intercepts on a graph. If you need to review these topics, click here for slope
(linear equations slope.doc) or click here for y-intercepts. (linear equations intercepts.doc)
Let's start with a line
that has a positive slope
Suppose we are given the function y = 2x + 5. Because this is in slope-intercept form of a line, we can see that the slope is 2 and the y-intercept is 5.
This is enough information to graph the function. Because we know the y-intercept is 5, we can start by placing a point at (0, 5) on the coordinate system.
The slope of 2 gives us the additional information we need to complete the graph. Remember that slope is “rise over run” or “the change in y over the change in x”. In any case, we always need to think about slope as a fraction. Since we know the slope is 2, we should think of it in the fraction form of . This says that every time we change the y-value by 2, we must also change the x-value by 1. Most of the time this is done by moving up 2 (changing in the y-direction) and over 1 (changing in the x-direction) and then placing the new point. You can see on the graph below that the new point is at (1, 7).
Once you know two points on a graph, you can connect those two points with a line and you then have the graph of the equation.
To find other points on the line, simply choose a starting point, and then move right one space and up two spaces to satisfy the slope of . You should know that you can also move to the left one space and down two spaces and still find another point on the line. Why would this work? Think about this. When you move to the left one you are changing the x-value by -1. When you move down two you are changing the y-value by -2. This will give a slope of which is what our line indicates. Look at the graph below to verify moving left 1 and down 2 will give you the point (-1, 3) which is on the graph.
Let’s now consider what should happen if we have a negative slope
Look at the function . The y-intercept for this graph will be at the point (0, -3). This will be our starting point for finding other points on the graph.
Since the slope is there are several ways to approach this problem. We know that we will be moving in the y-direction two spaces and in the x-direction three spaces. We just need to figure out whether those moves are left, right, up or down.
We need to remember that when there is a negative fraction, as there is in this case, it can be written in one of three ways.
Let’s find two more points.
- This indicates that if we move down 2, we should move right three because of .
- It also says that if we move up 2, we should move left 3 because of .
The graph of is shown below. You should now verify these three points on the graph.
- Using , we start from (0, -3) move right (x-direction) 3 and down (y-direction) 2. This will put a new point at (0+3, -3-2) = (3, -5)
- Using , we again start from (0, -3) move left (x-direction) 3 and up (y-direction) 2. This will put a new point at (0-3, -3+2) = (-3, -1)