Which function represents a vertical stretch of an exponential function?
y = 2^x
y = 3 \cdot 2^x
y = 2^{3x}
Answer (2)
A vertical stretch of a function f ( x ) by a factor of k is given by k c d o t f ( x ) , where 1"> k > 1 .
The function y = 3 c d o t 2 x is a vertical stretch of y = 2 x by a factor of 3.
The function y = 2 3 x can be rewritten as y = ( 2 3 ) x = 8 x , which represents a horizontal compression of y = 2 x .
The function that represents a vertical stretch of the exponential function y = 2 x is y = 3 c d o t 2 x .
Explanation
Understanding the Problem We are given three functions: y = 2 x , y = 3 c d o t 2 x , and y = 2 3 x . We need to identify which function represents a vertical stretch of the exponential function y = 2 x .
Recalling Vertical Stretch A vertical stretch of a function f ( x ) by a factor of k is given by k c d o t f ( x ) , where 1"> k > 1 . We need to compare the given functions to this form.
Identifying the Vertical Stretch The function y = 3 c d o t 2 x is a vertical stretch of y = 2 x by a factor of 3, since it is in the form k c d o t f ( x ) where k = 3 and f ( x ) = 2 x .
Analyzing the Other Functions The function y = 2 3 x can be rewritten as y = ( 2 3 ) x = 8 x . This represents a horizontal compression of y = 2 x , not a vertical stretch.
Final Answer Therefore, the function that represents a vertical stretch of the exponential function y = 2 x is y = 3 c d o t 2 x .
Examples
Vertical stretches of exponential functions are used in various real-world applications, such as modeling population growth or radioactive decay. For example, if a population of bacteria doubles every hour, the growth can be modeled by the function y = 2 x , where x is the number of hours. If we introduce a factor that triples the initial population, the new growth model becomes y = 3 c d o t 2 x , representing a vertical stretch of the original function. This concept is also applicable in finance, where interest rates can cause exponential growth of investments, and scaling factors can represent changes in investment strategies or market conditions.
The function that represents a vertical stretch of the exponential function y = 2 x is y = 3 ⋅ 2 x . This function indicates an increase in the output by a factor of 3 compared to the original exponential function. The other function, y = 2 3 x , represents a horizontal compression.
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