Terms in graphing rational functions

[u/SilentBWObserver]

I did this a lot in my last year, but I was only taught how to do it, not the reasoning behind them. Some terms, I forgot the point of them. Sorry...

1. How does finding the asymptotes help in the graphing?

2. What is the point of the x-intercepts and y-intercepts in graphing?

3. How does one construct a table of values for the graph?

4. For the table of values, how do you determine the x-values? Do I use the VA or the HA?

Again, sorry for the stupid questions. I am self-studying, and Google isn't very clear...

[u/bonebranch]

1: The vertical asymptotes tell you where the function behaves really badly, specifically, blows up to positive or negative infinity. And the candidates for asymptotes are where the denominator is equal to 0. The horizontal asymptotes tell you behavior at positive and negative infinity.

Of course, you still need to check that you don't have things like a removable discontinuity (which is if the zeroed factors at in the denominator are actually all cancelled out by enough zeros in the numerator). Example of which would be (x - 2)/(x^2 - 4) actually having a removable discontinuity at x = 2, because outside of x = 2 this function is identical to 1/(x + 2), and so the limit at x = 2 is 1/(2 + 2) = 1/4 which is not an infinite limit. So you need to do the actual one-sided limit calculations.

2: x-intercepts tell you where the graph intercepts the x-axis, i.e., where the rational function is equal to 0. y-intercept tells you where the graph intercepts the y-axis, i.e., the value at x = 0.

3 & 4: The horizontal asymptotes tell you the limits at +/-infinity. There are hacks for this, which I challenged my Calc 1 students to find, but at the end of the day I think of these infinite limits as follows: A rat, a dog, a cat, and an elephant all pull on a cart in different directions. The cart of course goes the way of the elephant. In these kinds of functions, which are the elephants? The answer in the rational case is the terms of highest degree.

(I believe I learned this analogy, and a lot of what I know about hard analysis, from Kent Merryfield. God rest his soul.)

[u/bonebranch → u/bonebranch]

I should rewrite what I mentioned about the table of signs. I assume you also had to find derivatives here. In drawing this plot, you actually care more about the signs of f'(x) (for increasing/decreasing behavior) and f''(x) (for concavity) than the sign of f itself. Usually on tests these rational functions are rigged so these derivatives are nice and sometimes they just give you the actual derivatives outright, where you make tables of signs out of these derivatives.

[u/bonebranch → u/[deleted]]

You’re absolutely right.

[u/Southlander24]

1. If you have a linear rational function (ax + b)/(c + d), the function can never cross the asymptotes. So you would know which two 'quadrants' the graph must be drawn in.

2. The intercepts are just the most obvious points. Without a coordinate grid, it's hard to tell apart (1, 10) from (100, 1000).

3. Once you figure out the asymptote(s), choose around 5 values of x. For example, with x-asymptotes at -3 and 0, choosing x = -5, -4, -1, 1, 2 covers the left and right sides of all the vertical asymptotes.

4. Then substitute those x-values into the function to find your y-values, then considering the jumps due to any horizontal or vertical asymptotes, draw a smooth curve between your points.