y = 1200 – (1 + 0.3)
A. Does this function represent exponential growth or exponential decay?
B. What is your initial value?
C. What is the rate of growth or rate of decay?
The correct answer and explanation is :
Let’s break down your question step by step and examine the equation ( y = 1200 – (1 + 0.3) ).
A. Does this function represent exponential growth or exponential decay?
The equation you provided, ( y = 1200 – (1 + 0.3) ), does not appear to represent either exponential growth or decay.
For a function to represent exponential growth or decay, it typically has the form:
[
y = a \cdot e^{kt}
]
where:
- ( a ) is the initial value,
- ( e ) is the base of the natural logarithm,
- ( k ) is the rate of growth (positive for growth, negative for decay), and
- ( t ) is time.
In contrast, the function you provided seems more like a linear equation or a simple arithmetic operation involving constants. ( y = 1200 – (1 + 0.3) ) simplifies to ( y = 1200 – 1.3 = 1198.7 ). This is a constant value, not a variable-dependent equation, and thus does not exhibit exponential growth or decay behavior.
B. What is your initial value?
Since the equation simplifies to ( y = 1198.7 ), this represents a constant value, which suggests that this function does not change over time (as there is no ( t )-variable involved).
However, in the context of an exponential function, the initial value typically corresponds to the value when ( t = 0 ). If we were dealing with an exponential equation such as ( y = 1200 \cdot e^{-0.3t} ), then the initial value would be ( 1200 ), as this would represent the starting value at ( t = 0 ). In your equation, there is no time-dependent variable (such as ( t )) to provide a true initial value for an exponential function.
C. What is the rate of growth or rate of decay?
Since the function you provided is not an exponential function, it does not have a rate of growth or decay. The equation represents a constant value, so there is no growth or decay happening here.
To understand the rate of growth or decay, you’d need a function that involves ( e^{kt} ), where ( k ) is the rate at which the quantity changes over time. In an exponential growth function, ( k ) would be positive, while in an exponential decay function, ( k ) would be negative.
If you were instead given an exponential function like ( y = 1200 \cdot e^{-0.3t} ), then:
- The initial value would be 1200 (the value at ( t = 0 )).
- The rate of decay would be ( -0.3 ), indicating exponential decay. The quantity decreases by 30% per time unit, as represented by the negative rate.
Conclusion:
- The given function does not represent exponential growth or decay.
- It simplifies to a constant value of 1198.7.
- There is no rate of growth or decay here, as it is not an exponential function.