Horizontal Shift of a function

[DarrenM]

I'm currently enrolled in College Algebra, and it is possible that I'm making too much out this; however, this is bugging me a bit and I can't quite get my head around it. I understand how to perform the various shifts and stretches of the graph of a function, but I'm trying to reach a better conceptual understanding of what the different effects are.

Horizontal shifts are baffling me a bit. For example, I understand that f(x) = (x-2)^2 is going to shift the parabola to the right. What I can't quite put my finger on is why? I'm not even sure if that question makes any sense, nor am I sure why this is sticking with me like this. Perhaps it's because it seems contrary to what it appears to do at first glance. A vertical shift f(x) = x^2 + c, with the constant outside the grouping symbols, is going to very obviously shift the graph of the function up (if c > 0) or down (if c < 0). That makes sense to me.

I can plug numbers in for x in f(x) = (x-2)^2, and I can observe the results, but the reasoning or logic behind it eludes me.

Any help here? Am I even making any sense? Am I over-analyzing a basic College Algebra class? I'm studying the material a great deal in an attempt to really understand the concepts rather than just memorizing the formulas, but I'll admit that I might be going overboard a bit...

 

[Hurkyl]

If you're having trouble with the concepts, then focus first on the math -- that's one of the main reasons we have it.

What is the mathematical condition expressing the assertion that the point (a,b) lies on the 'standard' parabola?

Now, what is the condition that says (c,d) is two units to the right of a point lying on the 'standard' parabola?

Note that last question can be rewritten as "what is the condition that says the point two units to the left of (c,d) is on the 'standard' parabola?"



Incidentally, once you've worked through the above exercises, repeat it, but replacing "two units to the right" with "two units up". And when I say that, I mean do this new problem in the same way, rather than using your current 'intuitive' understanding. This way, you can compare the reasoning with your intuition and see if and how they match up with each other.

 

[Architeuthis Dux]

First of all, I want to commend you for your dedication to truly understanding the concepts rather than just memorizing formulas. That will serve you well in your scientific studies.

Now, let's talk about horizontal shifts of a function. In order to understand why a horizontal shift occurs, we need to think about what the x-axis represents in a graph. The x-axis represents the input or independent variable of the function, while the y-axis represents the output or dependent variable. So, when we have a function f(x), the x-values we plug in are the inputs, and the corresponding y-values are the outputs.

Now, when we have a function like f(x) = (x-2)^2, what we are essentially doing is taking the original function f(x) = x^2 and subtracting 2 from the input value before squaring it. So, for example, if we plug in x=3 into the original function, we get f(3) = 3^2 = 9. But if we plug in x=3 into the shifted function, we get f(3) = (3-2)^2 = 1^2 = 1. So, the output is shifted to the left by 2 units.

In general, when we have a function f(x) = (x-a)^2, the a represents the amount of horizontal shift. If a is positive, the shift is to the right, and if a is negative, the shift is to the left.

I hope this explanation helps you understand the reasoning behind horizontal shifts. Keep up the good work in your studies!