Exponential Growth and Decay Functions

Exponential Growth and Decay

• Exponential growth describes a quantity increasing over time at a constant percentage rate.
• Exponential decay describes a quantity decreasing over time at a constant percentage rate.
• Both are modeled by the function y = a * b^x, where:
  • 'a' is the initial value (when x = 0).
  • 'b' is the growth/decay factor. If b > 1, it's growth; if 0 < b < 1, it's decay.
  • 'x' is the time variable.

Writing Exponential Equations with Rational Exponents

• Rational exponents represent fractional powers. Example: x^(1/2) = √x.
• Exponential functions can include rational exponents in their base or exponent. For example:
  • f(x) = 2^(x/2)
  • f(x) = 3^√x
• These functions still follow the general form of exponential growth or decay but involve non-integer exponents.
• The growth or decay factor is found by the exponent's fractional part, considering the base's fractional powers.

Transformations of Exponential Functions

• Transformations of exponential functions work similarly to transformations of other functions.
• Vertical Shifts: Adding a constant 'c' to the function shifts it vertically. f(x) + c shifts up 'c' units, and f(x) - c shifts down 'c' units.
• Horizontal Shifts: Adding or subtracting a constant 'c' inside the function's exponent shifts it horizontally. f(x-c) shifts right 'c' units, and f(x+c) shifts left 'c' units.
• Vertical Stretches/Compressions: Multiplying the function by a constant 'a' stretches or compresses it. a*f(x) is a vertical stretch if |a| > 1, and a compression if 0 < |a| < 1.
• Reflections: Multiplying the function by -1 reflects it across the x-axis (f(-x)). Reflecting across the y-axis involves changing the input x to -x.
• Combining transformations alters the function's position, shape, and orientation. For example, f(x−2) + 3 represents a horizontal shift to the right by 2 units and a vertical shift up by 3 units.

Writing Exponential Functions

• To write an exponential function, you need to identify key features:
  • Initial value (a): The value of the function when the input (x) is zero.
  • Growth/decay factor (b): The factor by which the output is multiplied in each unit of change in the input. This is often determined by knowing a second point on the graph.
  • Time variable (x).
• Example: If an initial population of 100 grows by 5% every year, use the formula f(x) = a * b^x and define a = 100, and b =1 + 0.05 = 1.05. f(x) = 100 * 1.05^x.
• Often, you're given multiple points on a graph (or data) and need to solve for these factors. Using two points allows the creation of a system of equations and solving using substitution or elimination.