Transformation

Learning Outcomes
  • Understand transformations of exponential functions including stretching, compressing, and reflecting.
  • Graph transformed exponential functions.
  • Write equations for transformed exponential functions.
Transformations Overview

  • Transformations include shifts, stretches/compressions, and reflections of the parent function ( f(x) = b^x ).

  • Stretching vertically occurs when the function is multiplied by a constant ( |a| > 1 ) (e.g., ( g(x) = 3(2^x) ) stretches ( f(x) )).

  • Compressing vertically occurs when ( 0 < |a| < 1 ) (e.g., ( h(x) = \frac{1}{3}(2^x) )).

  • Reflections are changes in the direction of the function, affecting its behavior about the axes:

    • Reflection about the x-axis: Multiply the function by -1 (e.g., ( g(x) = -2^x )).
    • Reflection about the y-axis: Multiply the input by -1 (e.g., ( h(x) = 2^{-x} )).
Stretched and Compressed Functions
  • If ( a > 1 ), the function is stretched.
    • Example: ( g(x) = 3(2^x) ) → stretched by a factor of 3.
  • If ( 0 < a < 1 ), the function is compressed.
    • Example: ( h(x) = \frac{1}{3}(2^x) ) → compressed by a factor of 1/3.
Key Features of Exponential Functions
  • Parent Function: ( f(x) = b^x ) with:
    • ( y )-intercept at ( (0, 1) ) if ( b > 1 ).
    • Horizontal asymptote at ( y = 0 ).
    • Range: (0, ∞).
    • Domain: ( (-\infty, ∞) ).
Graphing Transformed Functions
  1. Step 1: Identify the transformation:
    • For example, ( g(x) = 4(\frac{1}{2})^x ) stretches vertically by 4.
  2. Step 2: Find key points and shape:
    • Use points such as ( (0, 4), (-1, 8), (1, 2) ) for plotting.
  3. Step 3: Draw the smooth curve connecting the key points.
Reflections Overview
  • Reflecting about x-axis:
    • If ( f(x) = b^x ), then ( g(x) = -b^x ) reflects across the x-axis.
  • Reflecting about y-axis:
    • If ( f(x) = b^x ), then ( h(x) = b^{-x} ) reflects across the y-axis.
Example of Reflection
  • Original function: ( f(x) = 2^x )
  • Reflected about the x-axis: ( g(x) = -2^x )
  • Reflected about the y-axis: ( h(x) = 2^{-x} )
General Equation for Transformations
  • The general form is: [ f(x) = a b^{(x+c)} + d ]
    • Where:
    • ( c ) shifts horizontally.
    • ( d ) shifts vertically.
    • ( |a| > 1 ) indicates vertical stretching, ( 0 < |a| < 1 ) indicates vertical compression.
Writing Functions from Descriptions
  • Describe transformations, e.g.,
    • Shifted left ( c ) units, stretched by ( |a| ) if ( |a| > 0 ), reflected about x-axis if ( a < 0 ):
    • Use this description to derive the function equation.
Specific Example: Stretching and Shifting
  • From ( f(x) = e^x ):
    • Stretched by 2, reflected across y-axis, and shifted up 4: [ f(x) = 2e^{-x} + 4 ]
    • Domain: ( (-\infty, \infty) ), Range: ( (4, \infty) ), Asymptote: ( y = 4 ).