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Reflecting Functions Vertically or Horizontally

Two types of trigonometry transformations act like reflections or flips that you see in graphing or geometry. One transformation changes all positive outputs to negative and all negative outputs to positive. The other reverses the inputs — positive to negative and negative to positive.

These two transformation types are

  • Reflecting up and down (outputs changed): –f (x)

  • Reflecting left and right (inputs changed): f (–x)

The preceding figure shows reflections of the function,


Reflecting downward puts all the points below the x-axis. Reflecting left makes all the input values move to the left of the y-axis. Even though it appears that the negatives should go under the radical, in fact, the negative in front of the x means that you take the opposite of all the negative x’s — which makes them positive.

A cash register can change inputs to the opposite (negative) numbers by taking coupon values that the cashier enters or scans in and changing them to negative values before doing the final computations. The graph of this process acts as a reflection downward from positive to negative.

Left and right reflections are a bit harder to describe in terms of a practical application. Try this one on for size: If a function tells you how many items a machine can produce in a certain number of hours, then inputting negative numbers helps you determine how far you have to back up — how many hours before a certain date and time — to produce that number of items by that date and time.

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