V = 55 – (1 – 0.02Y)
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 the function:
[
V = 55 – (1 – 0.02Y)
]
A. Does this function represent exponential growth or exponential decay?
This function does not represent exponential growth or decay. Exponential functions typically have the form:
[
V = V_0 \cdot e^{kt}
]
where (V_0) is the initial value, (e) is the mathematical constant, (k) is the rate of growth (if (k > 0)) or decay (if (k < 0)), and (t) represents time (or another variable). The given equation does not contain an exponential term involving (e), and it doesn’t represent growth or decay where the rate of change depends on the current value. Instead, the equation represents a linear relationship, where (V) depends on (Y).
B. What is your initial value?
To determine the initial value, we substitute (Y = 0) into the equation. This is because the initial value is typically the value of the function when the independent variable (in this case, (Y)) is zero.
Substituting (Y = 0):
[
V = 55 – (1 – 0.02(0)) = 55 – (1 – 0) = 55 – 1 = 54
]
So, the initial value of (V) when (Y = 0) is 54.
C. What is the rate of growth or rate of decay?
The rate of change in this equation is constant because this is a linear function. Specifically, we are concerned with the term involving (Y), which is (-0.02Y). This term tells us how (V) changes as (Y) increases.
The rate of change is (-0.02), meaning that for every 1 unit increase in (Y), (V) decreases by 0.02 units. This suggests a negative rate of change, which indicates a decreasing value as (Y) increases.
Thus, the rate of change is -0.02, which implies a constant linear decay (not exponential decay). The decay is uniform, and the value of (V) decreases at a steady rate as (Y) increases.
Summary
- A: The function does not represent exponential growth or decay, it is a linear equation.
- B: The initial value is 54.
- C: The rate of decay is constant at -0.02.