The given exponential function is:

$y = 51(0.91)^x$

Identifying Growth or Decay

An exponential function has the general form:

$y = a(b)^x$

• If $b > 1$, the function represents exponential growth.
• If $0 < b < 1$, the function represents exponential decay.

In this case, $b = 0.91$, which is less than 1. Therefore, the function represents exponential decay.

Determining the Percentage Rate of Decrease

To find the percentage rate of decrease, subtract $b$ from 1 and then multiply by 100:

$\text{Rate of decrease} = (1 - b) \times 100 = (1 - 0.91) \times 100 = 0.09 \times 100 = 9\%$

Answer

The function represents exponential decay with a 9% rate of decrease.

Would you like more details or have any questions?

Related Questions

1. How would the function change if the base were greater than 1, say 1.05?
2. What is the significance of the coefficient 51 in the given function?
3. How does the rate of change affect the shape of the graph?
4. What is the half-life of a function that decays at a 9% rate per unit of $x$?
5. How do you convert exponential decay into a logarithmic form?

Tip

In exponential functions, always examine the base to quickly determine whether the function represents growth or decay.