Function Transformations / Translations (page 1 of 4)

Sections: Basic rules, Additional rules, Moving the points, Working backwards from the graph

When you first started graphing quadratics, you started with the basic quadratic, y = x2:

Then you did some related graphs, such as:

 f(x) = –x2– 4x+ 5 f(x) = x2– 3x– 4 f(x) = (x+ 4)2

If you've been doing your graphing by hand, you've probably started noticing some relationships between the equations and the graphs. The topic of function transformation makes these relationships more explicit.   Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved

Let's start with the function notation for the basic quadratic: f(x) = x2. A function transformation takes whatever is the basic function f(x) and then "transforms" it or "translates" it, which is a fancy way of saying that you change the formula a bit and thereby move the graph around.

 For instance, the graph for x2 + 3 looks like this:

This is three units higher than the basic quadratic,  f(x) = x2. That is, x2 + 3 is f(x) + 3. We added a "3" outside the basic squaring function f(x) = x2 and thereby went from the basic quadratic x2 to the transformed function x2 + 3.

This is always true: To move a function up, you add outside the function:  f(x) + b is f(x) moved up b units. Moving the function down works the same way; f(x) – b is f(x) moved down b units.

 On the other hand, (x + 3)2 looks like this:

In this graph, f(x) has been moved over three units to the left: f(x + 3) = (x + 3)2 is f(x) shifted three units to the left.

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This is always true: To shift a function left, add inside the function's argument: f(x + b) gives f(x) shifted b units to the left. Shifting to the right works the same way; f(xb) is f(x) shifted b units to the right.

Warning: The common temptation is to think that f(x + 3) moves f(x) to the right by three, because "+3" is to the right. But the left-right shifting is backwards from what you might have expected. Adding moves you left; subtracting moves you right. If you lose track, think about the point on the graph where x = 0. For
f(x + 3), what does x now need to be for 0 to be plugged into f ? In this case, x needs to be –3, so the argument is –3 + 3 = 0, so I need to shift left by three. This process will tell you where the x-values, and thus the graph, have shifted. At least, that's how I was able to keep track of things....

 The last easy transformation is –f(x). Look at the graph of –x2:

This is just f(x) flipped upside down. Any points on the x-axis stay on the x-axis; it's the points off the axis that switch sides. This is always true: f(x) is just f(x) flipped upside down.

 For this next transformation, I'll switch to g(x) = x3:
 If I put –x in for x, I get (–x)3 = –x3:

This transformation rotated the original graph around the y-axis. Any points on the y-axis stay on the y-axis; it's the points off the axis that switch sides. This is always true: g(–x) is the mirror image of g(x), as reflected in the y-axis.

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 Cite this article as: Stapel, Elizabeth. "Function Transformations / Translations: Basic Rules." Purplemath. Available from     http://www.purplemath.com/modules/fcntrans.htm. Accessed [Date] [Month] 2016

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