The Right y-Intercept

We just wrote an equation for a line with a slope of 3 going through the origin. Let's slide our old line up the y-axis so it has a y-intercept of 2. We have to do this carefully, so the slope of the line will still be 3.

The yellow fly used to be at point (1, 3). The fly is still sitting on the same part of the line, but now it is at point (1, 5). The fly's x-coordinate has not changed, but its y-coordinate has gone up by 2. Has the same thing happened to the other points? Let's make another table to find out.

x	old y	new y
0	0	2
2/3	2	4
1	3	5
4/3	4	6

In every row of the table, the new y-coordinate is 2 more than the old y-coordinate. Let's write this as an equation:

new y = old y + 2

But we already know that the old y-coordinate is the same as 3 times the x-coordinate. So we can rewrite our equation as:

new y = 3*x + 2

Now we have an equation for our new line. Let's drop the "new" and just say

y = 3*x + 2

What do the numbers in this equation mean? Well, the slope of the line is 3, and its y-intercept is 2. So if we tell Joan, "Graph y = 3*x + 2," that's the same as telling her, "Graph a line with a slope of 3 and a y-intercept of 2."