Unit
Linear Equations and Their Graphs
MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
To find the slope of a line given the graph of a line, find any two points on the line. Choose points in which the coordinates are whole numbers to ensure accuracy. One you have the two points, you can either use the slope formula and plug in the values for x1, y1 and x2, y2, or you can use to the slope triangle. Draw a triangle using the two points on the line by using the segment between the two points as the hypotenuse and drawing a vertical and horizontal line from one point to the other as the legs of the triangle. The legs of the triangle will help you count the vertical change (numerator of the slope) and vertical change (denominator of the slope). The direction of the line -- whether it is upward sloping or downward sloping -- will indicate whether or not the slope is positive (upward sloping) or negative (downward sloping).
This is a problem where I’m asked to find the slope of the line, but I’ve got to be careful because there’s no like official points. A lot of times students get confused when they look at a line like this and they say, it doesn’t have any points, but you guys remember, a line is like all kinds of point. In fact it’s like infinitely many points squished together and all of those points are solutions to the equation that the line is graphing.
But so when you think of it that way, you can find any points you want to, most of the time, people look for what corners of the graph like what rectangular corners the line goes through exactly. So I’m going to look really carefully and see which corner it goes through, it goes through that one exactly and it goes through that one exactly, and those two points will help me to determine what the slope is.
Once I’ve found two points on the line, you could either use this equation where you do y2 take away y1, on top of x2 minus x1, or you can count using a slope triangle. For visual people like me, the slope triangle helps. And what you do for the slope triangle, is you draw in the little triangle on the graph that is defined by those two points, and then label how much you’re changing vertically and horizontally, so here I’m just doing a vertical change of 1, 2, 3, 4 and a horizontal change of 2, so my slope number is going to be 4 on top of 2, vertical over horizontal change.
You guys probably know that can be reduced to just plain old 2 over 1, or even just the integer 2. And so that’s neat because I went from using that line in the triangle which looks kind of confusing, I had to use a fraction, but I found out that the slope is pretty easy to work with, it’s just 2.
One thing I want to show you guys about the slope triangle method is that I would have gotten the same answer if I used two different points, like for example, what if I used that guy and this guy right here, it’s a different point that the line goes through exactly. Now my slope triangle would be something like that, but again you can see my vertical change is 2, my horizontal change is 1.
I could have even used a different point, I could have used one that’s way up here, and I would have a vertical change of 6, horizontal change of 3. I did that in my head because these are all equal ratios, they’re all fractions that reduced to 2.
The point being, no matter what point s you pick on your line, when you write them as a slope ratio like this, and reduce the fraction, you’re always going to get the correct slope number even if you use different points.