Which is a stretch of an exponential growth function?
$f(x)=\frac{2}{3}\left(\frac{2}{3}\right)^x$
$f(x)=\frac{3}{2}\left(\frac{2}{3}\right)^x$
$f(x)=\frac{3}{2}\left(\frac{3}{2}\right)^x$
$f(x)=\frac{2}{3}\left(\frac{3}{2}\right)^x$
Answer (1)
Identify exponential growth functions: f ( x ) = a x where 0"> a > 0 and 1"> b > 1 .
Check if the function is a stretch: 1"> a > 1 .
Analyze each function and determine a and b values.
The function f ( x ) = 2 3 ( 2 3 ) x is a stretch of an exponential growth function because 1"> a = 2 3 > 1 and 1"> b = 2 3 > 1 .
The answer is f ( x ) = 2 3 ( 2 3 ) x .
Explanation
Understanding Exponential Growth and Stretches We are given four exponential functions and we need to identify which one represents a stretch of an exponential growth function. An exponential growth function has the form f ( x ) = a x where 0"> a > 0 and 1"> b > 1 . A stretch occurs when the function is multiplied by a constant 1"> a > 1 .
Analyzing Each Function Let's analyze each function:
f ( x ) = 3 2 ( 3 2 ) x : Here, a = 3 2 and b = 3 2 . Since b < 1 , this is not an exponential growth function.
f ( x ) = 2 3 ( 3 2 ) x : Here, a = 2 3 and b = 3 2 . Since b < 1 , this is not an exponential growth function.
f ( x ) = 2 3 ( 2 3 ) x : Here, a = 2 3 and b = 2 3 . Since 1"> b > 1 and 1"> a > 1 , this is an exponential growth function and a stretch.
f ( x ) = 3 2 ( 2 3 ) x : Here, a = 3 2 and b = 2 3 . Since 1"> b > 1 but a < 1 , this is an exponential growth function but a compression.
Conclusion Therefore, the function that represents a stretch of an exponential growth function is f ( x ) = 2 3 ( 2 3 ) x .
Examples
Exponential growth functions are used to model various real-world phenomena, such as population growth, compound interest, and the spread of diseases. A stretch in this context means the initial value or starting point is amplified. For example, if you invest money in a bank account with compound interest, the stretch factor represents how much your initial investment is multiplied. Understanding stretches helps in predicting how quickly these phenomena will grow or decline.
