Which function represents a vertical stretch of an exponential function?
$f(x)=3\left(\frac{1}{2}\right)^x$
$f(x)=\frac{1}{2}(3)^x$
$f(x)=(3)^{2 x}$
$f(x)=3^{\left(\frac{1}{2} x\right)}$
Answer (1)
Vertical stretch of an exponential function has the form c " , w h ere c > 1$.
f ( x ) = 3 ( 2 1 ) x has 1"> c = 3 > 1 , representing a vertical stretch.
f ( x ) = 2 1 ( 3 ) x has c = 2 1 < 1 , representing a vertical compression.
f ( x ) = ( 3 ) 2 x and f ( x ) = 3 ( 2 1 x ) are exponential functions without a vertical stretch.
The function that represents a vertical stretch is f ( x ) = 3 ( 2 1 ) x .
Explanation
Analyzing the Problem We are given four functions and asked to identify which one represents a vertical stretch of an exponential function. A vertical stretch of an exponential function a x is of the form c " , w h ere c > 1$. Let's analyze each option.
Analyzing Option 1
f ( x ) = 3 ( 2 1 ) x : This function has the form c " , w h ere c=3 > 1 an d a = \frac{1}{2} . T h u s , t hi sre p rese n t s a v er t i c a l s t re t c h o f t h ee x p o n e n t ia l f u n c t i o n (\frac{1}{2})^x$.
Analyzing Option 2
f ( x ) = 2 1 ( 3 ) x : This function has the form c " , w h ere c=\frac{1}{2} < 1 an d a = 3$. This represents a vertical compression (or shrink) of the exponential function 3 x .
Analyzing Option 3
f ( x ) = ( 3 ) 2 x : This function can be rewritten as f ( x ) = ( 3 2 ) x = 9 x . This is an exponential function with base 9. There is no vertical stretch.
Analyzing Option 4
f ( x ) = 3 ( 2 1 x ) : This function can be rewritten as f ( x ) = ( 3 2 1 ) x = ( 3 ) x . This is an exponential function with base 3 . There is no vertical stretch.
Conclusion Therefore, the function that represents a vertical stretch of an exponential function is f ( x ) = 3 ( 2 1 ) x .
Examples
Vertical stretches of exponential functions are used in various real-world scenarios, such as modeling population growth or radioactive decay. For instance, if a bacterial population doubles every hour, the growth can be modeled by an exponential function. A vertical stretch could represent an initial population size larger than 1. Similarly, in finance, compound interest can be modeled using exponential functions, and a vertical stretch could represent an initial investment amount.
