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Function Transformations / Translations (page 2 of 4)

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

The transformations so far follow these rules:

• f(x) + a is f(x) shifted upward a units
• f(x) – a is f(x) shifted downward a units
• f(x + a) is f(x) shifted left a units
• f(x – a) is f(x) shifted right a units
• –f(x) is f(x) flipped upside down ("reflected about the x-axis")
• f(–x) is the mirror of f(x) ("reflected about the y-axis")

There are two other transformations, but they're harder to "see" with any degree of accuracy. If you compare the graphs of 2x², x², and ( ¹/₂ )x², you'll see what I mean:

The parabola for 2x² grows twice as fast as x², so its graph is tall and skinny. On the other hand, the parabola for the function ( ¹/₂ )x² grows only half as fast, so its graph is short and fat. You can tell, roughly speaking, that the first graph is multiplied by something bigger than 1 and that the third graph is multiplied by something smaller than 1. But it is generally difficult to tell exactly what a graph has been multiplied by, just by looking at the picture.

The other more-difficult type of transformation is displayed below:

As you can see, multiplying inside the function (inside the argument of the function) causes the graph to get thinner or fatter. This looks a lot like the other multiplication transformation, and is about impossible to identify from a graph. It helps to look at the zeroes of the graph (if it has more than one). For instance, looking at y = x² – 4, you can see that multiplying outside the function doesn't change the location of the zeroes, but multiplying inside the function does:

So the "left", "right", "up", "down", "flip", and "mirror" transformations are fairly straightforward, but the "multiply" transformations, also called "stretching" and "squeezing", can get a little messy. Hope that they aren't frequently required of you.

Typical homework problems on this topic ask you to graph the transformation of a function, given the original function, or else ask you to figure out the transformation, given the comparative graphs.

• Thinking of the graph of f(x) = x⁴, graph f(x – 2) + 1

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The graph of f(x) looks like this:

Looking at the expression for the transformation, the "+1" outside tells me that the graph is going to be moved up by one unit. And the "–2" inside the argument tells me that the graph is going to be shifted two units RIGHT. (Remember that the left-right shifting is backwards from what you might expect.)

Then my graph looks like this:

When they are having you graph by moving other graphs around, they can't be terribly critical of your drawing, since you're not supposed to be making a T-chart and computing exact points. But do try to make your graph look reasonable.

You can always "cheat", by the way, especially if you have a graphing calculator, by quickly graphing (x – 2)⁴ + 1 and verifying that it matches what you've drawn. But you do need to know how to do function transformations, because there are ways to ask the questions that don't allow you to cheat....

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