Understanding Exponential Functions: Growth vs. Decay

TLDR; Exponential functions with different initial values and base values result in varying growth and decay graphs. The base value dictates the rate of increase or decrease.

⚙️ Exponential Functions Overview

All equations are in the form "A x B to the power of X", indicating they are exponential functions.

The initial value 'A' represents the function value when X = 0 and is also the y-intercept of the function.

If the base value 'B' is between 0 and 1, it results in exponential decay, with the graph decreasing from left to right.

If 'B' is greater than 1, it leads to exponential growth, with the graph increasing from left to right.

📈 Initial Value and Y-Intercept

The initial value 'A' is the starting amount or value before exponential growth or decay starts.

The graph with 'A' = 4 and 'B' > 1 represents exponential growth, as it has a y-intercept of +4 and goes uphill from left to right.

Another graph with 'A' = 3 and 'B' = 0.7 represents exponential decay, as it has a y-intercept of +3 and goes downhill from left to right.

📉 Identifying Exponential Decay

Equations with base values between 0 and 1 represent exponential decay.

Graphs with base values of 0.7 and 0.5 both represent exponential decay, as they have the same initial value 'A' and go downhill from left to right.

🔍 Differentiating Exponential Decay Graphs

The red graph represents faster exponential decay compared to the blue graph due to its steeper decrease from left to right, indicating a smaller value of 'B'.

The closer the base value 'B' is to zero, the faster the exponential decay.

📊 Identifying Exponential Growth

Equations with base values greater than 1 represent exponential growth.

Graphs with base values of 1.5 and 1.2 both represent exponential growth, as they have the same initial value 'A' and go uphill from left to right.

📈 Comparing Growth Rates

The black graph represents faster exponential growth compared to the green graph due to its steeper increase from left to right, indicating a larger value of 'B'.

The value of 'B' dictates the rate of increase or decrease in the exponential graph.

🧮 Understanding Exponential Functions

Different initial values and base values result in varying characteristics of exponential functions, affecting the rate of growth or decay.

The characteristics of exponential functions help in identifying the type of graph they produce.