An exponential growth function has an asymptote of $y=-3$. 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)
Vertical shifts of exponential functions are determined by adding or subtracting a constant from the exponential expression.
Subtracting a constant c from the exponential expression, f ( x ) = a x − c , shifts the horizontal asymptote to y = − c .
Since the asymptote is y = − 3 , the transformation is f ( x ) = a x − 3 , which includes negative numbers in its range.
Therefore, 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
Understanding the Problem Let's analyze the problem. We're given that an exponential growth function has an asymptote of y = − 3 and that its range includes negative numbers. We need to determine which transformation of the original exponential function could have caused this. The general form of an exponential function is f ( x ) = a x , where 0"> a > 0 and a e q 1 . The basic exponential function f ( x ) = a x has a horizontal asymptote at y = 0 and a range of ( 0 , ∞ ) .
Analyzing Transformations Now, let's consider the possible transformations:
Adding a constant to the exponential expression: f ( x ) = a x + c . This shifts the graph vertically by c units. The horizontal asymptote becomes y = c , and the range becomes ( c , ∞ ) .
Subtracting a constant from the exponential expression: f ( x ) = a x − c . This shifts the graph vertically by − c units. The horizontal asymptote becomes y = − c , and the range becomes ( − c , ∞ ) .
Adding a constant to the exponent: f ( x ) = a x + c . This shifts the graph horizontally by − c units. The horizontal asymptote remains y = 0 , and the range remains ( 0 , ∞ ) .
Subtracting a constant from the exponent: f ( x ) = a x − c . This shifts the graph horizontally by c units. The horizontal asymptote remains y = 0 , and the range remains ( 0 , ∞ ) .
Determining the Correct Transformation Since the asymptote is given as y = − 3 , we need a vertical shift that results in an asymptote at y = − 3 . This can be achieved by subtracting a constant from the exponential expression. Specifically, if we subtract 3 from the exponential expression, we get f ( x ) = a x − 3 . The asymptote is y = − 3 , and the range is ( − 3 , ∞ ) , which includes negative numbers.
Conclusion Therefore, the correct transformation is subtracting a whole number constant 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 population growth, the asymptote represents the carrying capacity of the environment, which is the maximum population size that the environment can sustain. If the population starts above the carrying capacity, the population will decline towards the carrying capacity. Subtracting a constant from the exponential expression shifts the carrying capacity downwards, which can be used to model the impact of environmental factors on population growth.
