To understand which graph best represents the function [tex]\( f(x) = 25(1.45)^x \)[/tex], let's break down the components of this function and analyze its behavior.
1. Initial Value: The function starts with [tex]\( f(0) = 25 \)[/tex].
- This means when [tex]\( x = 0 \)[/tex], the number of members equals [tex]\( 25 \)[/tex].
2. Growth Factor: The term [tex]\( (1.45)^x \)[/tex] represents exponential growth.
- Specifically, [tex]\( 1.45 \)[/tex] indicates a 45% annual increase.
3. Understanding Exponential Growth: Exponential functions of the form [tex]\( f(x) = a \cdot b^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex]) exhibit the following characteristics:
- The graph will always be increasing if [tex]\( b > 1 \)[/tex].
- The growth rate gets faster as [tex]\( x \)[/tex] increases due to the compounding effect.
4. Behavior at Key Points:
- At [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 25 \times 1^0 = 25 \)[/tex].
- At [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 25 \times 1.45 = 36.25 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] increases rapidly due to the exponential nature.
5. Intercepts and Growth:
- The graph should begin at [tex]\( (0, 25) \)[/tex].
- It should curve upward, becoming steeper as [tex]\( x \)[/tex] increases (representing the 45% growth rate).
By these characteristics, the graph will start at [tex]\( y = 25 \)[/tex] when [tex]\( x = 0 \)[/tex] and will steadily rise at an accelerating rate due to the exponential growth factor of 1.45.
Thus, the graph best representing the function [tex]\( f(x) = 25(1.45)^x \)[/tex] will be an exponentially increasing curve starting at the point (0, 25) and rising rapidly.
