Transformational Form of an Exponential Function

							Transformational Form of an Exponential Function

• When we worked with quadratic equations, we used different
  methods to express the function. One method that we used to
  get the image of a given graph was mapping notation. In this
  section of the course, we will look at how mapping notation can
  be used to generate an image graph of an exponential function
  under different transformations.

  The standard base table             When the points in the table of
  of values for the exponential       values are graphed, we get the
  y = 2x is given by:                 base function:


       x      y
       -3




                    →
              1
              8
       -2     1
              4
       -1     1
              2
       0      1

       1      2

       2      4

       3      8



  (a.) Describe the shape of the function.
         The function is in the shape of an exponential curve.
  (b.) Identify the domain and range of the function graphed.
         D = {x | x ∈ }
             {
         R = y | y > 0, y ∈       }
• All transformations and image graphs generated in this section
  of the course will be applied to base exponential functions of
  the form y = 2x . Because of this, we will look at the change in
  the ordered pairs which will then be used to graph the image
  function.
      Vertical Translation of an Exponential Function

Vertical Translation
y − k = 2x

   • The first transformation that we will look at is a Vertical
     Translation. As we’ve seen with quadratics, the Vertical
     Translation (VT) affects the base graph y = 2x by translating or
     shifting the graph up or down.
   • Example:
     Let’s look at the following functions. We will use a mapping rule
     to generate a table of values for the image function and then
     graph all three functions on the same axes.

      (a.) y = 2x       (b.) y − 3 = 2x        (c.) y + 5 = 2x

                        (x, y) → (x, y + 3)    (x, y) → (x, y − 5)

                        VT = 3                 VT = −5

           x        y         x     y                x     y
          −3        1        −3    25               −3     39
                                                         −
                    8               8                       8
          −2        1        −2    13               −2     19
                                                         −
                    4               4                       4
           −1       1        −1     7               −1      9
                                                          −
                    2               2                       2
           0        1        0      4               0     −4

           1        2        1      5               1     −3

           2        4        2      7               2      −1

           3        8        3     11               3      3
• The focal points of the functions are (0, 1) , (0, 4) and (0, −4) .
  It is also important to note that the VT also gives the number
  value for the equation of the horizontal asymptote.
• When there is y − 3 in the equation, then there is a VT of 3
  units. This will result in a graph that is shifted 3 units up from
   y = 2x .
• When there is y + 5 in the equation, then there is a VT of −5
  units. This will result in a graph that is shifted 5 units down from
   y = 2x .
Summary

          Equation:                          Mapping Rule:
          y − k = 2x         VT = k          (x, y) → (x, y + k)

  • The Vertical Translation is the OPPOSITE of the value added to
    y in the equation.
  • The Vertical Translation is the SAME as the value added to y in
    the mapping rule.
  • When k > 0 , the graph is shifted k units UP from y = 2x .
  • When k < 0 , the graph is shifted k units DOWN from y = 2x .
  • The focal point is (0, k) .
  • The Horizontal Asymptote is y = k .