Transformation (Movement) of Functions

In this section we are going to look at moving functions around. Specifically we will be looking at translations, stretching and shrinking, and reflections of functions. We will discuss six things that we can do to the basic function. Any of these actions could be combined. As you read through this section try the various functions on your calculator. If you have a TI calculator simply go to the y= screen and enter the functions. (Be certain to adjust your window so that the function is shown on the screen.)

Given a function f(x)

• We can vertically translate (raise or lower) the whole function by some amount.
• If we move the function up or down we have: f(x) + k

  Example: Basic function f(x) = log(x)

  • Vertical Translation up by 3 units: log(x) + 3
  • Vertical Translation down by 2 units log(x) - 2

• We can horizontally translate (left or right) the whole function by some amount.
• If we move the function left or right we have: f(x - d).

  Example: Basic function: f(x) = e^x

  • Horizontal translation 4 units to the right:e^{(x - 4)}
  • Horizontal translation 2 units to the left:e^{(x + 2)}

• We can vertically stretch or shrink a function by some amount. We will call this vertical stretching/shrinking. If we stretch or shrink the function we multiply it by an external constant (we will worry about sign later) f(x) = a f(x) (where |a| > 1 for stretch and 0< |a|< 1 for shrink).

  Example: Basic function f(x) = Sq Root(x)

  • Vertical stretch by a factor of 2: 2*Sq Root(x)
  • Vertical shrinking by one-half: 1/2 Sq Root(x)

• We can horizontally stretch or shrink a function by some amount. We call this horizontal stretching/shrinking. If we stretch or shrink the function we multiply it internally by a constant. f(x) becomes f( cx ) where |c| > 1 shrinks the function horizontally and 0< |c| <1 stretches the function.

  Example: f(x) = log(x)

  • Horizontal shrink: f(2x) = log(2x)
  • Horizontal stretch: f(0.5 x ) = log( 0.5x)
  • In the figure that follows horizontal stretching/shrinking has been illustrated using the sine function (sin on your calculator). To try these functions on your calculator you MUST put the (TI) calculator in radian mode (it WILL NOT graph correctly in degree mode). I used the sine function because it very nicely illustrates the point.

• We can reflect a function across the x axis. If we do this f(x) becomes -f(x) (We multiply by a minus sign. Past examples have called this an external change).

  Example: Basic function f(x) = ab^x

  Reflection across the x axis: -f(x) = -ab^x

• We can reflect a function across the y axis. If we do this f(x) becomes f(-x). (This time the minus is an internal change).

  Example: Basic function: f(x) = e^X Reflection across the y axis: f(-x) = e^-x

The figures shown below are designed to provide concrete examples of what has been discussed in the previous material. Hopefully they will help you visualize the various effects.