Question

Given the following exponential function, identify whether the change

Answer 1

Answer:

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 $b$ 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.