How do you translate a rational function?

Translating Rational Functions: A User-Friendly Guide

Rational functions. They might sound intimidating, but they’re really just fractions where the numerator and denominator are polynomials. And like any function, you can move them around – translate them, in math speak. Think of it like sliding a picture on your phone screen. This article is all about how to do that with rational functions.

The Parent Function: Where It All Begins

Let’s start with the simplest rational function out there: f(x) = 1/x. It’s the “parent” function, the foundation for everything else. Picture its graph: it’s a hyperbola, sort of like two curves bending away from each other. Notice those invisible lines that the curves get closer and closer to but never actually touch? Those are called asymptotes. This parent function has one vertical asymptote at x = 0 and one horizontal asymptote at y = 0. Keep these asymptotes in mind; they’re key to understanding how translations work.

Vertical Translations: Up, Up, and Away (or Down, Down, Down)

Vertical translations are super straightforward. We’re talking about shifting the whole graph up or down. Just add or subtract a number, k, to the function:

• f(x) = 1/x + k: This moves the graph k units up if k is a positive number.
• f(x) = 1/x – k: And this moves the graph k units down if k is positive.

The horizontal asymptote is the one that feels this change. It shifts right along with the graph, landing at y = k. The vertical asymptote? It doesn’t budge.

Example: Imagine our parent function, f(x) = 1/x. Now, picture f(x) = 1/x + 3. That’s the same graph, just lifted 3 units higher. So, the horizontal asymptote is now at y = 3. Simple as that!

Horizontal Translations: Sliding Left and Right

Now for the sideways shuffle! Horizontal translations move the graph left or right. To do this, replace x with (x – h):

• f(x) = 1/(x – h): Shifts the graph h units to the right if h is positive.
• f(x) = 1/(x + h): Shifts the graph h units to the left if h is positive.

This time, the vertical asymptote is the one that moves. It shifts to x = h. The horizontal asymptote stays put at y = 0.

Example: Think about f(x) = 1/(x – 2).

Putting It All Together: The General Form

Ready to combine these moves? Here’s the general form of a translated rational function:

f(x) = a/(x – h) + k

Let’s break it down:

• a: This stretches or squishes the graph vertically. If |a| is bigger than 1, it’s a stretch. If it’s between 0 and 1, it’s a squish. And if a is negative? Flip the graph upside down!
• h: This is our horizontal translator.
• k: And this is our vertical translator.

The vertical asymptote? It’s hanging out at x = h. The horizontal asymptote? You’ll find it at y = k.

Example: Check out f(x) = 2/(x + 1) – 4.

• The “2” stretches the graph vertically.
• The “+ 1” shifts the graph 1 unit to the left, so the vertical asymptote is at x = -1.
• The “- 4” drops the graph 4 units down, putting the horizontal asymptote at y = -4.

Graphing: From Equation to Picture

Okay, how do you actually draw one of these translated rational functions?

Domain and Range: What’s Allowed, What’s Possible

The domain is all the x-values that you’re allowed to plug into the function. For a translated rational function, it’s everything except the x-value of the vertical asymptote. So, it’s (-∞, h) ∪ (h, ∞).

The range is all the possible y-values you can get out of the function. That’s everything except the y-value of the horizontal asymptote. So, it’s (-∞, k) ∪ (k, ∞).

Wrapping Up

Translating rational functions is all about moving their graphs around by tweaking their equations. Master the roles of a, h, and k in f(x) = a/(x – h) + k, and you’ll be able to understand and graph these functions like a pro. Knowing how translations affect those asymptotes, the domain, and the range? That’s the key to really getting it. So go forth and translate!