How do you translate a linear function horizontally?

Shifting the Line: A More Human Take on Translating Linear Functions Horizontally

So, you’re looking to move a line, huh? In math, we call that a transformation. And one of the most basic transformations is a translation – basically, just sliding something around. Now, vertical translations? Those are usually pretty straightforward. But horizontal translations… that’s where things can get a little quirky. Let’s break it down, shall we?

First things first: linear functions. Remember those? Think of them as your basic straight lines. The most common way to write one is f(x) = mx + b. The m is your slope (how steep the line is), and b is where the line crosses the y-axis (the y-intercept). The simplest linear function? That’s just f(x) = x. Easy peasy.

Okay, so what is a horizontal translation? Simple: it’s taking that line and sliding it left or right along the x-axis. The line doesn’t change its angle or anything; you’re just picking it up and moving it. Think of it like sliding a piece of paper on a table.

Now, here’s the key: how do you actually do it? This is where the rule comes in, and it can be a bit of a head-scratcher at first. To shift a linear function horizontally, you mess with the x inside the function. The golden rule?

• g(x) = f(x – h)

g(x) is your new, translated function. And h? That’s the magic number that tells you how far and which way to shift.

Here’s the thing that always trips people up:

• If h is positive (like, say, 3), you shift the line to the right by h units.
• If h is negative (like -2), you shift the line to the left by |h| units (so, 2 units in this case).

Yep, it’s backwards! Subtracting moves it right, adding moves it left. I know, I know – math can be weird sometimes. Just gotta roll with it.

Let’s make this crystal clear with some examples. I always find that helps.

“But why does this work?” I hear you ask. Good question! Think about it this way: when you replace x with (x – h), you need to plug in a bigger number to get the same result as the original function. That “bigger number” means you’ve shifted everything to the right. It’s like compensating for the h you subtracted. The opposite happens when you add h – you need a smaller number, so you shift left.

Now, here’s a fun fact: with linear functions, horizontal and vertical shifts are actually related! Because the slope is constant, you can often achieve the same result by shifting up or down instead of left or right. It’s just a different way of looking at the same thing. Pretty neat, huh?

Of course, translations are just the beginning. You can also flip lines (reflections), stretch them, or squish them. Math is full of ways to play around with shapes!

So, there you have it. Horizontal translations of linear functions, demystified. Remember the g(x) = f(x – h) rule, keep that sign convention straight, and you’ll be shifting lines like a pro in no time! It might seem a little strange at first, but with a bit of practice, you’ll get the hang of it. And hey, if I can do it, anyone can!