The function given is y = 0.96x. To determine the percentage rate of change and whether this function represents exponential growth or decay, we need to analyze the base of the exponent.
In the case of functions of the form y = ax, the behavior of the function is largely dependent on the value of a:
- If a > 1, the function represents exponential growth.
- If 0 < a < 1, the function represents exponential decay.
Here, the base 0.96 is less than 1. Therefore, this means that the function y = 0.96x represents exponential decay.
Next, to find the percentage rate of change, we can calculate it using the formula for the percentage rate of change in an exponential function, which is given by:
Percentage Rate of Change = (f(x + 1) - f(x)) / f(x) * 100%
For the function we are analyzing:
- f(x) = 0.96x
- f(x + 1) = 0.96x + 1 = 0.96x * 0.96
Now substituting these into our formula:
Percentage Rate of Change = (0.96x * 0.96 - 0.96x) / 0.96x * 100%
This simplifies to:
Percentage Rate of Change = (0.96 - 1) * 100%
Calculating the above gives:
Percentage Rate of Change = (-0.04) * 100% = -4%
This indicates that the function experiences a 4% decrease for each unit increase in x. In summary, the function y = 0.96x shows an exponential decay with a percentage rate of change of -4%.