Exponential Functions Flashcards

Exponential Growth and Decay

• Exponential growth is represented by functions where the output increases at a constant percentage rate, such as f(x) = 500(2)^x or f(x) = 4(1.5)^x.
• The equation f(x) = 500(2)^x indicates an initial value of 500, and a multiplicative rate of change of 2.
• Exponential decay occurs when quantities decrease at a rate proportional to their current value, exemplified by f(x) = 5,000(0.4)^x, where 40% remains after each time period.

Graphical Representations

• Graphical representations of exponential functions reveal their behavior:
  • Functions like f(x) = 4(1.5)^x show increasing y-values for increasing x-values, indicating exponential growth.
  • Functions such as f(x) = (1/4)(4)^x exhibit behavior characteristic of exponential decay.
• The specific graph of f(x) = 3(2/3)^x represents decay, while f(x) = 3(4)^x shows growth.

Key Points on Functional Characteristics

• For functions f(x) = 4(3)^x and f(x) = 5(4)^x, understanding how to derive equations that pass through specific points (like (2, 36) or (2, 80)) assists in real-world applications of exponential models.
• Initial values and factors of increase or decrease considerably influence the shape of exponential functions.

Multiplicative Rate of Change

• The multiplicative rate of change for exponential functions reflects how quickly the function's output is changing, with examples of rates such as:
  • A rate of change of 5 indicates rapid growth.
  • In cases of decay, values like 0.4 or 0.97 reflect slower diminishment in quantity.

Sequences and Common Ratios

• Geometric sequences are characterized by a consistent multiplication factor known as the common ratio, such as -3 in the sequence -2, 6, -18, 54.
• An explicit formula for a geometric sequence can be identified, such as the height sequence of a bouncing toy represented by f(x) = 64(3/4)^(x-1).

Contextual Applications

• Real-world problems, such as population growth, investment values, or collector's items, can be modeled using exponential functions to predict future values based on initial quantities and growth rates.
• Equations like y = 18(1.15)^x help estimate the future worth of limited-edition items or changing town populations.

Summary of Graph Characteristics

• Determining whether a graph exhibits exponential growth or decay requires analysis of its slope and y-intercept; the domain and range also provide insight into the function's behavior over time.
• Visualizing the trends in population dynamics, bacterial decay, or financial growth helps clarify the importance of rate of change in exponential functions.