Most of the graphs you’ll see on the SAT are linear (straight line) graphs. For any other shape, you should concentrate on one point at a time; don’t get hung up trying to “understand” such graphs.
Here are three tricks to make linear graphs easier to deal with:
1) Slope Trick
Q. What is the slope of the line that contains the points (2,7) and (4,5)?
First of all, on the SAT, the word “line” means “straight line.” You might say “I see a curved line,” but the SAT won’t.
In math class, you learned that the slope = change in y/change in x, or rise/run. Then you learned the Point Slope Form to determine the slope:
y – y1 = m(x – x1)
You may or may not find that daunting. In any case, there’s a much easier way than memorizing and solving an equation with 5 variables. First, write the 2 points vertically:
…and then just write the changes in y and x to the right and left of those points:
+2 (4,5) -2
That is, the y values went from 7 to 5, which is -2, and the x values went from 2 to 4, which is +2. So the change in y/change in x is -2/+2, or -1.
Yes, the logic is the same as that of the formula, but it’s a lot harder to mess up.
2) Distance Trick
This is another trick in which common sense beats algebra. For the distance d, you were taught this formula:
d = √[(y2 – y1)2 + (x2 – X1)2]
You were probably also taught that this formula is based on the Pythagorean Theorem (a2 + b2 = c2), which is a much easier formula to use. The trick here is just to substitute the change in y for a, and the change in x for b. Then solve the Pythagorean Theorem for c, which is the distance.
Q. What is the distance between the points (7,6) and (-1,5)?
First, get the change in y and change in x as you did for the first trick:
-8 (-1,5) -1
Use -1 for a and -8 for b (actually, it doesn’t matter if you switch them. You can also ignore the negative signs since you’re going to square them anyhow).
a2 + b2 = c2
12 + 82 = c2
1 + 64 = c2
√65 = c (the distance)
In the diagram, you can see how the change in y and change in x are the legs of a right triangle for which the hypotenuse is the distance.
3) “Human Graphing Calculator” Trick
A Human Graphing Calculator (HGC) can look at a graph and immediately tell you the approximate formula. And if you provide a formula, an HGC can quickly show you the graph.
Maybe HGC’s really exist. If so, they have a pretty freaky gift. But for linear graphs, you can be an HGC. You probably already have the tools you need.
Remember, any straight line can be expressed by the formula y = mx + b, where m is the slope, and b is the y-intercept. The y-intercept is easy – that’s where the line crosses the y-axis (where x = 0). And you should know what different slopes look like. If the slope is zero, the line is horizontal. The line goes up (from left to right) if the slope is positive; a line with a negative slope goes down. You can estimate the magnitude of the slope by how steep the line is. If it rises steeply, the slope is greater than 1. If it falls slowly (shallowly), the slope is between 0 and -1.
That’s it! If you’re given y = -3x + 5, you know the line will fall steeply and intercept the y-axis at 5. So, when you see a problem involving a straight line graph, you should be confident.