Using Slope and y-Intercept to Graph Lines


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Using Slope and y-Intercept to Graph Lines (page 1 of 2)

Given two points (x₁, y₁) and (x₂, y₂), the formula for the slope on the straight line going through these two points is:

...where the subscripts merely indicate that you have a "first" point and a "second" point (that is, that you have two points). This formula for slope is sometimes referred to as "rise over run", because the fraction consists of the "rise" (change in y, going up or down) divided by the "run" (change in x, going to the right). If you've ever done roofing, built a staircase, graded landscaping, or installed outflow piping, you've probably encountered this "rise over run" concept. The point is that slope tells you how much y is changing for every so much that x is changing. This will probably make more sense with a picture, so let's look at the line y = ( ²/₃ )x – 4.

If, say, x = –3, then y = ( ²/₃ )(–3) – 4 = –2 – 4 = –6, so the point (–3, –6) is on the line. If x = 0, then y = ( ²/₃ )(0) – 4 = 0 – 4 = –4, so the point (0, –4) is on the line. Now that we have two points on the line, we can find the slope:

By the way, you may already have known that the slope of the line was ²/₃, because the line equation is y = ( ²/₃ )x – 4, and the slope is always the number multiplied on the x (as long as you have the equation in "y=" format!).

Now that we've found these two points, let's look at them on the graph: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
In going from the first point to the second, we can form a right triangle:
The distance between the y-values (that is, the height of the triangle) is the "y₂ – y₁" part of the slope formula. The distance between the x-values (that is, the length of the triangle) is the "x₂ – x₁" part of the slope formula. Note that the slope is ²/₃, or "two over three". To go from the first point to the second, we went "two up and three over". This always works for slope.
To get to the "next" point, we can go up another two (to y = –2), and over to the right another three (to x = 3):

(If you're not sure that this works, put 3 in for x, and verify that you get –2 for y.)

Let's try another line equation: y = –2x + 3. We've learned that the number on x is the slope, so m
= –2 for this line. If, say, x = 0, then y = –2(0) + 3 = 0 + 3 = 3. Then the point (0, 3) is on the line. With this information, we can find more points on the line. First, though, you might want to convert the slope to a fraction (so you can more easily do the "up and over" thing). Just remember that any number is a fraction if you put it over "1", so, in this case, it is more useful to say that the slope is m = ^–2/₁. That means that we will be going "down two and over one" for each new point.