Sign charts are quite indispensable in this class, so I figured that it would be a good idea to explain why sign charts work.
First, let’s review what the purpose of a sign chart is. Given any function, a sign chart tells you everywhere the function is positive and everywhere it is negative. This is useful primarily for solving inequalities, and later on this will be useful for graphing functions (in more ways than you might initially guess).
Let’s walk through an example here. Consider the function f(x) = \frac{2(x-3)(x-1)^2}{(x+6)^2}. We’ll note that this function is equal to zero when x=3 or x=1. Additionally, the function is undefined at x=-6.
Plotting these points on a numberline looks like this. Here, I’m denoting the zeros with closed circles and the undefined points with open circles. Now suppose for a moment that the function takes a negative value somewhere to the left of -6, say, at x=-8. Is it possible for our function to become positive anywhere to the left of -6? No. Why is this?
Well, remember that these points we’ve plotted make up all of the zeros and undefined points on the function, and think about what’s special about these points. With any rational function like this, the only points of discontinuity are going to be at the zeros of the denominator, and so, these are the only places that the function can “jump” from one side of the x-axis to the other, that is, switch from negative to positive or positive to negative. The function can’t do this anywhere else, because it is continuous everywhere else.
So we’ve established why zeros of the denominator—points of discontinuity—are possible switching points. What about zeros of the numerator? This one’s more obvious. If our function is continuous in a region, then the only places that it can cross the x-axis are where it first touches the x-axis, and these are precisely the zeros of the numerator. And so, we see, the points we’ve chosen make up all the possible places that the function can change sign. Let’s apply this knowledge.
Our number line is split up by these potential switching points into several intervals [emphasize intervals], and because we know that the function can’t change sign within these intervals, it suffices to pick a point inside each interval, plug it into the function, and observe the sign at that specific point.
On this interval here, we can pick -10. This gives us -196.625, so we mark this interval with a minus sign. On the next interval, we can use 0, which is especially convenient to plug in, and gives us -0.1666, so again, we place a minus sign. Continuing in this fashion, we find the signs of the remaining intervals are as shown.
Notice in this example, that the function did not change sign just because we were at a possible switching point. You have to be sure to check all of the intervals you know you need to care about, instead of just figuring out one interval and then assuming alternating signs for the rest. This is an error I not so uncommonly see.
Let’s see what the function f(x) actually looks like, to check our work.
[Graph f(x)]
Here, we see that indeed, the function is negative everywhere to the left of -6, negative between -6 and 1, once again negative between 1 and 3, and then positive from 3 onwards. Additionally, as predicted, we have zeros at 1 and 3, and the function is undefined with a vertical asymptote at -6. This is quite a lot of information to have gleaned from such a simple process, and this hints as to how sign charts will be so useful with regards to graphing.