Algebra 2B - Exponential Functions Flashcards

Exponential Functions

• General form: ( y = ab^x + k )
• Example transformation: ( y - 2 = 3^{\frac{2x}{7}} ) simplifies to ( y = \frac{9}{7} * 3^x + 4 )
• Initial value is derived from function output when ( x = 0 ).
• Horizontal asymptote indicates the function's end behavior, for example, ( f(x) = 3 ) for the function approaches.
• Domain of exponential functions is all real numbers ((-∞, ∞)).
• Range of exponential functions typically starts above the horizontal asymptote, e.g., ( (-4, ∞) ).

Behavior of Functions

• As ( x ) approaches ( -∞ ), functions can stabilize towards a horizontal asymptote, while as ( x ) approaches ( ∞ ), they can tend toward ( ∞ ).
• Examples of specific functions:
  • ( f(x) = 2^x - 5 ), horizontal asymptote at ( y = -5 ).
  • The function ( f(x) = 1.25(0.3)^x + 7 ) is a decreasing function.
  • The function ( f(x) = 2 * 3^x + 4 ) is increasing, passing through point ( (0, 6) ).

Logarithmic Functions

• Logarithms are the inverses of exponential functions, such as ( f(x) = 2^x ) having its inverse as ( f^{-1}(x) = log_2(x) ).
• Relationship between logarithmic and exponential forms: ( \log_x y = z ) means ( x^z = y ).
• Basic conversions include changing ( a^b = c ) to ( \log_a c = b ).

Evaluating Logarithms

• Common values: ( log_{10} 10,000 = 4 ), ( log_2 8 = 3 ), ( log_3 81 = 4 ).
• Approximations can be calculated, e.g., ( log 250 \approx 2.4 ), ( log 3.7 \approx 0.6 ).
• Finding the base of logarithms from equations relates to their original exponential forms.

Transformations of Exponential Functions

• Vertical translations adjust the graph up or down, while horizontal translations shift it left or right.
• Dilations modify the steepness of the graph:
  • If ( g(x) = 3f(x) ), the graph is vertically stretched by a factor of 3.
  • Horizontal stretches/compressions can be represented as ( g(x) = f(\frac{1}{bx}) ).
• Reflections occur when ( g(x) = -f(x) ), indicating a flip across the x-axis or y-axis based on transformation rules.

General Concepts

• Asymptotes describe the flatness of the graph on one or two sides, helping understand long-term behavior.
• Initial values give insights into where the function starts, guiding calculations for growth or decay.
• The context of function transformations can incorporate combinations of shifts, reflections, and dilations, producing diverse graphs from basic forms.