Moving Points

Here's an example of what that looks like:

• Given the following graph of f (x), graph the transformation −f (x + 1) − 3

(In situations like this, they must give you clearly-plotted points, either labelled with coordinates or else obviously at integer x- and y-values. Otherwise, there's no reasonable way for you to locate any points. So if, in an exercise like this, a point looks like it has nice neat coordinate values, you can safely assume that it does indeed have those nice values.)

Note: What I'm going to show you below is not what you would hand in. You would not show all this work in your homework, and certainly not on a test. But this displays the thinking that should be going through your head as you transform each point. If you're not sure of yourself, this might be what your scratch-paper would look like.

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If you're not sure of the order in which to do the transformation's steps, then stick to working from the inside out (from inside the argument, to anything that's multiplied onto the function, to anything done outside the function), like I did above. Figure out the exact sequence of steps that each point will need to make. Then pick a point to move, and trace out the sequence of steps with your pencil tip, drawing in the translated point once you reach its final location. Then, once you've moved all of the points, you can draw in the transformation.

Note: For the above transformation, they could have factored the minus sign from out in front of the translation expression, and then presented the transformation to me as being −[ f (x + 1) + 3 ]. Then the point movements would have been "left one, up three, and then flip across the x-axis". But the end result would have been exactly the same.

(It can be helpful to use a pencil [say, on the test] to trace the moving points. Once you get a point's final transformed location, you can erase the pencilled in-between steps.)

If your book or instructor uses this other way of expressing transformations, or if you get an exercise that's formatted this way, don't panic. Just work things out logically and step-by-step — and, if you prefer, multiply things back out to get a transformation expression more like what we'd started this page with.