How do you graph a simple rational function?

Graphing Simple Rational Functions: No Sweat!

Rational functions. The name itself can sound a bit intimidating, right? But trust me, graphing these guys, especially the simpler ones, isn’t as scary as it seems. Think of it like following a recipe – break it down into steps, and you’ll be serving up beautiful graphs in no time.

So, what is a rational function? Simply put, it’s a fraction where the top and bottom are both polynomials. We’re talking things like f(x) = (x + 1) / (x – 2). Now, the trick to graphing these lies in spotting their key features: the asymptotes that act like invisible fences, the intercepts where the graph crosses the axes, and the overall way the function behaves.

First things first: the domain. This is basically all the x-values you’re allowed to plug into the function. The one big no-no? Making the denominator zero. Division by zero is a mathematical black hole, so we want to avoid it! To find those forbidden x-values, just set the denominator equal to zero and solve. For instance, with f(x) = 1/(x-2), you can’t have x = 2, because that would make the bottom zero. So, the domain is everything except 2. Easy peasy.

Now, let’s talk about those “invisible fences,” or asymptotes. These are lines that the graph gets really close to but never quite touches. Imagine the graph trying to sneak past, but the asymptote keeps pulling it back. There are three main types: vertical, horizontal, and slant.

• Vertical Asymptotes: These are the easiest to spot. They happen where the denominator equals zero after you’ve simplified the function. So, if you can cancel out any common factors between the top and bottom, do it! If a factor cancels, it doesn’t create an asymptote, but a “hole” in the graph (more on that later). Vertical asymptotes are always written as x = some number, like x = 3.

• Horizontal Asymptotes: These tell you what the function does way out on the edges of the graph – as x gets super big (positive or negative). Here’s the rule of thumb: compare the degrees (highest power of x) of the top and bottom polynomials.

  • If the top’s degree is smaller than the bottom’s, the horizontal asymptote is always y = 0 (the x-axis).
  • If the degrees are the same, the horizontal asymptote is y = (leading coefficient of top) / (leading coefficient of bottom). Basically, divide the numbers in front of the highest powers of x.
  • If the top’s degree is bigger than the bottom’s, forget about a horizontal asymptote. You might have a slant asymptote instead.

• Slant (Oblique) Asymptotes: These are a bit fancier. They show up when the degree of the top is exactly one more than the degree of the bottom. To find it, you have to do polynomial long division (remember that from algebra?). Divide the top by the bottom, and the answer you get (ignoring the remainder) is the equation of the slant asymptote. It’ll look like y = mx + b. And remember, you can only have either a horizontal or a slant asymptote, not both!

Next up: intercepts. These are where the graph crosses the x and y axes. Think of them as the graph’s way of saying “hello” to the axes.

• x-intercepts: To find these, set the numerator (the top part) of the function equal to zero and solve for x. These are the roots of the function. Just double-check that those x-values are actually allowed in the function’s domain (i.e., they don’t make the denominator zero).

• y-intercept: This one’s even easier. Just plug in x = 0 and see what you get for f(0). That’s your y-intercept!

Okay, now for the fun part: plotting and sketching. Here’s where the graph really comes to life.

Let’s do a quick example: f(x) = (x + 1) / (x – 2).

Graphing rational functions might seem a bit tricky at first, but with a little practice, you’ll get the hang of it. Just remember to take it one step at a time, and don’t be afraid to experiment with different test points. Before you know it, you’ll be a rational function graphing pro!