Which function represents a vertical stretch of an exponential function?
[tex]f(x)=3(\frac{1}{2})^x[/tex]
[tex]f(x)=\frac{1}{2}(3)^x[/tex]
[tex]f(x)=(3)^{2 x}[/tex]
[tex]f(x)=3^{(\frac{1}{2} x)}[/tex]
Answer (2)
Vertical stretch occurs when the function's output is multiplied by a constant greater than 1.
Analyze each function to determine if it represents a vertical stretch.
f ( x ) = 3 ( 2 1 ) x has a coefficient of 3 > 1, representing a vertical stretch.
The function that represents a vertical stretch is f ( x ) = 3 ( 2 1 ) x .
Explanation
Understanding Vertical Stretch We are given four exponential functions and need to identify the one that represents a vertical stretch. A vertical stretch occurs when the function's output is multiplied by a constant greater than 1.
Analyzing Each Option Let's analyze each option:
f ( x ) = 3 ( 2 1 ) x : This function has a coefficient of 3 multiplying the exponential term ( 2 1 ) x . Since 3 > 1, this represents a vertical stretch.
f ( x ) = 2 1 ( 3 ) x : This function has a coefficient of 2 1 multiplying the exponential term ( 3 ) x . Since 2 1 < 1 , this represents a vertical compression (or shrink).
f ( x ) = ( 3 ) 2 x : This can be rewritten as f ( x ) = ( 3 2 ) x = 9 x . This is a standard exponential function with a base of 9, not a vertical stretch.
f ( x ) = 3 ( 2 1 x ) : This can be rewritten as f ( x ) = ( 3 2 1 ) x = 3 x . This is also a standard exponential function with a base of 3 , not a vertical stretch.
Conclusion Based on the analysis, 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 example, if you start with a certain amount of bacteria that doubles every hour, and you want to model the population growth with an initial amount that is three times larger, you would use a vertical stretch of the exponential function. Similarly, in finance, if you invest an amount that grows exponentially, a vertical stretch can represent the effect of starting with a larger initial investment.
The function that represents a vertical stretch of an exponential function is f(x) = 3igg(\frac{1}{2}igg)^x . This function has a coefficient greater than 1, indicating a vertical stretch. The other options either represent vertical compressions or do not include a coefficient greater than 1.
;
