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 (2)
Subtracting a constant from an exponential expression shifts the horizontal asymptote vertically.
A basic exponential function f ( x ) = a x has a horizontal asymptote at y = 0 .
Subtracting a constant c from the exponential expression, f ( x ) = a x − c , shifts the horizontal asymptote to y = − c .
Therefore, to have an asymptote at y = − 3 , a constant must be subtracted from the exponential expression: A whole number constant could have been subtracted from the exponential expression.
Explanation
Understanding the Problem Let's analyze the problem. We are given that an exponential growth function has a horizontal asymptote at y = − 3 . We need to determine which transformation of the original exponential function could have caused this. The options are: adding a constant to the exponential expression, subtracting a constant from the exponential expression, adding a constant to the exponent, or subtracting a constant from the exponent.
Analyzing Transformations Consider a basic exponential function f ( x ) = a x , where 1"> a > 1 . This function has a horizontal asymptote at y = 0 and a range of ( 0 , ∞ ) . Adding or subtracting a constant from the exponential expression shifts the horizontal asymptote vertically. Adding a constant c gives f ( x ) = a x + c , shifting the asymptote to y = c . Subtracting a constant c gives f ( x ) = a x − c , shifting the asymptote to y = − c . Adding or subtracting a constant from the exponent shifts the graph horizontally. Adding a constant c to the exponent gives f ( x ) = a x + c , which is a horizontal shift to the left. Subtracting a constant c from the exponent gives f ( x ) = a x − c , which is a horizontal shift to the right. Horizontal shifts do not affect the horizontal asymptote or the range (which remains ( 0 , ∞ ) ).
Determining the Correct Transformation Since the asymptote is at y = − 3 , a constant must have been subtracted from the exponential expression. Specifically, subtracting 3 from the exponential expression, f ( x ) = a x − 3 , shifts the horizontal asymptote to y = − 3 and the range to ( − 3 , ∞ ) , which includes negative numbers.
Conclusion Therefore, the correct answer is that 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. In the context of compound interest, the amount of money you have after a certain time can be modeled by an exponential function. If you start with an initial investment and the interest is compounded continuously, the function might look like A ( t ) = P e r t + c , where P is the principal, r is the interest rate, t is the time, and c is a constant representing an initial debt. The asymptote y = c represents the limit of how much your investment can decrease due to the initial debt. Understanding how constants shift exponential functions helps in predicting financial outcomes.
To have a horizontal asymptote at y = − 3 , a whole number constant must be subtracted from the exponential expression, shifting the asymptote down. This allows the range of the function to include negative values, making the final answer B. A whole number constant could have been subtracted from the exponential expression.
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