An exponential growth function has an asymptote of [tex]$y=-3$[/tex]. Which might have occurred in the original function to permit the range to include negative numbers?
A. A whole number constant could have been added to the exponential expression.
B. A whole number constant could have been subtracted from the exponential expression.
C. A whole number constant could have been added to the exponent.
D. A whole number constant could have been subtracted from the exponent.
Answer (1)
Adding a constant to the exponential expression shifts the asymptote vertically.
Subtracting a constant from the exponential expression also shifts the asymptote vertically.
Adding or subtracting a constant from the exponent results in a horizontal shift, which does not affect the asymptote.
Since the asymptote is y = − 3 , a whole number constant could have been subtracted from the exponential expression. $\boxed{A whole number constant could have been subtracted from the exponential expression.}
Explanation
Problem Analysis Let's analyze the problem. We are given that an exponential growth function has an asymptote of y = − 3 . We need to determine which transformation of the original exponential function could have caused this. The possible transformations are:
Adding a constant to the exponential expression.
Subtracting a constant from the exponential expression.
Adding a constant to the exponent.
Subtracting a constant from the exponent.
Analyzing Transformations Consider a basic exponential function f ( x ) = a x , where 1"> a > 1 . The asymptote is y = 0 and the range is ( 0 , ∞ ) .
If a constant c is added to the exponential expression, the function becomes f ( x ) = a x + c . The asymptote shifts to y = c and the range becomes ( c , ∞ ) . If c is negative, the range can include negative numbers.
If a constant c is subtracted from the exponential expression, the function becomes f ( x ) = a x − c . The asymptote shifts to y = − c and the range becomes ( − c , ∞ ) . If c is positive, the range can include negative numbers.
If a constant c is added to the exponent, the function becomes f ( x ) = a x + c . This is a horizontal shift and does not affect the asymptote or the range, which remains ( 0 , ∞ ) .
If a constant c is subtracted from the exponent, the function becomes f ( x ) = a x − c . This is also a horizontal shift and does not affect the asymptote or the range, which remains ( 0 , ∞ ) .
Determining the Correct Transformation Since the asymptote is y = − 3 , either adding a constant to the exponential expression or subtracting a constant from the exponential expression could have caused this. If we add a constant c to the exponential expression, then c = − 3 . If we subtract a constant c from the exponential expression, then − c = − 3 , so c = 3 .
The question asks which might have occurred. Adding a whole number constant to the exponential expression could result in a negative asymptote and a range that includes negative numbers. Subtracting a whole number constant from the exponential expression could also result in a negative asymptote and a range that includes negative numbers.
Final Answer The question states that the asymptote is y = − 3 . This means that a constant must have been added to the exponential expression. If the original function was f ( x ) = a x , then the transformed function is f ( x ) = a x − 3 . This shifts the asymptote to y = − 3 and the range to ( − 3 , ∞ ) . Therefore, a whole number constant could have been subtracted from the exponential expression.
Conclusion The correct answer is: A whole number constant could have been subtracted from the exponential expression.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding how transformations affect the graph of an exponential function is crucial for accurately interpreting and applying these models. For example, if we are modeling the population of a species, subtracting a constant from the exponential expression could represent a decrease in the carrying capacity of the environment, which would affect the long-term population size.
