How to handle rational functions:
Rational functions can be tricky. You need to worry about when functions do not exist! What does that even mean? Any rational function can be expressed as f(x) = g(x) / h(x); where g(x) and h(x) are polynomial functions. There will be values of x that cause h(x) to equal zero. When h(x) equals zero, the function does not exist at that value of x, simple as that. This lack of existence can be an asymptote or a “hole”. A vertical asymptote is a vertical line that our function never crosses. As x approaches this vertical line, our function approaches either infinity or negative infinity. “Holes” are simply holes in our function. Our function can cross a hole but does not exist at the hole.
Let’s walk through an example with a 5-step process that will allow you to graph any rational function you encounter.
Consider the following rational function:
Step 1: Simplify the top and bottom:
This may require use of the quadratic formula.
Step 2: Identify the vertical asymptotes and holes:
We know our function does not exist when x = -3 or x = -2 because the denominator equals zero when x holds these values. There is a hole at x = -3 because our denominator AND our numerator both equal zero at this value. There is a vertical asymptote at x = -2 because only our denominator equals zero at x = -2.
Holes: x = -3
Vertical Asymptotes: x = -2
Step 3: Identify “Asymptote Behavior”
If we’re going to graph this function it is important that we know what y does as x approaches our asymptote. In our function, there is only 1 asymptote, but sometimes there are more. We can approach a vertical asymptote from the left (negative side) or the right (positive side).
We know our function is going to “blow up” as we approach an asymptote, so we simply must determine if we’re going to negative infinity or positive infinity.
Let’s start with the left side. Just pick a number that’s really close to and less than -2 (like -2.01). If x = -2.01, then we see that our denominator is negative and our numerator is positive, therefore our function is negative and must approach negative infinity.
Now, we’ll do the same thing for the right side, but the number we pick must be close to and greater than -2. Let’s pick x = -1.99. When we plug this value in, we find that the denominator is positive, and the numerator is positive. Therefore, our function is positive and must approach infinity.
That’s it! We can simply ignore the hole and just remember that our function cannot exist there. You can just draw a small hole at this point when you graph the function.
Step 4: Identify “End behavior”
Now we must figure out what the function does as x approaches infinity and negative infinity. This will also allow us to find any horizontal asymptotes. Like step 3, we can take a similar “plug it in” approach. Let’s ignore the parts of the equation that create the hole (in this case it is the (x+3) term in the numerator and denominator). This gives us the following equation (this isn’t the most mathematically sound approach but bear with me).
The key idea here is that infinity+4 is basically equal to infinity and infinity+2. If you add or subtract any real number from infinity, then you will still pretty much have infinity. This is either very confusing or very simple, don’t overthink it. The idea of infinity is hard to comprehend and difficult to understand in terms our normal day to day life. Anyways, we can “plug in” x = infinity and x = -infinity and see what we get for our value of y.
Boom! We have our end behavior and our horizontal asymptote! Our horizontal asymptote is y=2. Remember that functions can never cross asymptotes, so we know what direction our function will approach the horizontal asymptote from, and we are done!
Step 5: Graph the function!
If you have completed the previous 4 steps, then the graphing should be a piece of cake. You draw your asymptotes with dotted lines, remember the holes, and draw your curved lines.
Hope this is helpful!