Question

Given the following exponential function, identify whether the change

represents growth or decay, and determine the percentage rate of
increase or decrease.
y = 210(1.09)^x
Growth or decay?
% increase?

Answer 1

The given exponential function represents growth with a percentage rate of increase of approximately 9.48%.

Step-by-step explanation:

The given exponential function is y = 210(1.09)^x. To determine if the change represents growth or decay, we examine the base of the exponential function. In this case, the base is 1.09. Since the base is greater than 1, the function represents exponential growth.

To determine the percentage rate of increase, we can compare the value of the function at two different points. Let's take x = 0 and x = 1. Substituting the values into the function, we get y(0) = 210 and y(1) = 1.09 * 210 = 229.9.

The percentage rate of increase can be calculated using the formula: percentage rate of increase = [(new value - initial value) / initial value] * 100%. Substituting the values, we have [(229.9 - 210) / 210] * 100% = 9.47619%. Therefore, the percentage rate of increase is approximately 9.48%.

Answer 2

Growth, with a 9% increase.

Step-by-step explanation:

Exponential patterns are usually defined by some kind of repeated multiplication. In most exponential patterns we start with some initial value (let's call it a) and some multiplier (let's call it b), and a is multiplied by b some number x times. We capture this as a function y with the equation
`y=ab^x` There are three broad categories for this pattern:

• No growth/decay - The function has an initial value that never changes. In order for the initial value to stay constant, we need
  `b=1`, since multiplication by 1 doesn't change the number it's multiplied by.
• Growth - The function grows at some exponential rate; for every increase in x, our initial value will be at least as big as it was for smaller values of x, so
  `b>1` in this case. (Example: A bacteria population starts at size 3 and doubles in size every hour; after x hours, there will be
  `3\cdot2^x` bacteria)
• Decay - The function decays or decreases at some exponential rate; for every increase in x it can't be as large as it was for smaller values of x, so
  `b<1` here. (Example: An Instagram post gets an initial wave of 100 new likes, and every hour it gets half the number of likes as the previous hour, so the number of new likes per hour is
  `100\cdot(0.5)^x`.

In the function
`y=210(1.09)^x`,
`b>1`, so we have exponential growth, but by what percent? Well, for every step in x, we take 1.09 (109%) of the previous value, so that's a 9% increase.