Video Transcript
Is the exponential function ๐ฆ equals four multiplied by 1.21 to the power of ๐ฅ growing or decaying?
To solve this problem, what weโre gonna first have a look at is what growth and decay actually are. And how you determine whether a function was in fact growing or decaying. First of all, weโre gonna start with growth. And if we actually have exponential growth, as you can see, the graph will show that we are actually increasing. And if we have our function in the form ๐ฆ equals ๐ multiplied by ๐ to the power of ๐ฅ, then our ๐ value would need to be greater than zero. And our ๐ value would have to be greater than one. So weโd actually need both of these conditions to be fulfilled for us to say yes, itโs actually exponential growth.
Okay, now weโre gonna move on to exponential decay. Well, if we take a look at decay, this time, our graph, you can see, is decreasing. What we also have is if we had the function in the form ๐ฆ equals a multiplied by ๐ to the power of ๐ฅ, then for us to identify it as decay, first of all, our ๐ would have to be greater than zero. And thatโs the same as in growth. And thatโs because if ๐ is actually not greater than zero, then itโs neither growth nor decay. But this time, itโs our ๐ value which is gonna be the difference. Because for decay, the ๐ value has to be greater than zero but less than one. And this is how weโre actually gonna determine whether it is a growing or decaying function.
So now, letโs take a look at our function. We have ๐ฆ is equal to four multiplied by 1.21 to the power of ๐ฅ. Well, actually great. This is already in the form ๐ฆ equals ๐ multiplied by ๐ to power of ๐ฅ. So therefore, we can identify that our ๐ is equal to four. And this means that actually yes, itโs gonna be growth or decay because itโs greater than zero. And our ๐ is equal to 1.21 because actually we have 1.21 to the power of ๐ฅ.
Okay, great. So what does this ๐ tell us? Well, our ๐ value is greater than one. So therefore, we can say that the exponential function ๐ฆ is equal to four multiplied by 1.21 to the power of ๐ฅ is a growing function. And itโs a growing function because when we have it in the form ๐ฆ is equal to ๐ multiplied ๐ to the power of ๐ฅ, our ๐ value is 1.21. And that is greater than one.