Is the exponential function 𝑦 equals four multiplied by 1.21 to the power of 𝑥 growing or decaying?
To solve this problem, what we’re 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, we’re 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 𝑦 equals 𝑎 multiplied by 𝑏 to the power of 𝑥, then our 𝑎 value would need to be greater than zero. And our 𝑏 value would have to be greater than one. So we’d actually need both of these conditions to be fulfilled for us to say yes, it’s actually exponential growth.
Okay, now we’re 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 𝑦 equals a multiplied by 𝑏 to the power of 𝑥, then for us to identify it as decay, first of all, our 𝑎 would have to be greater than zero. And that’s the same as in growth. And that’s because if 𝑎 is actually not greater than zero, then it’s neither growth nor decay. But this time, it’s our 𝑏 value which is gonna be the difference. Because for decay, the 𝑏 value has to be greater than zero but less than one. And this is how we’re actually gonna determine whether it is a growing or decaying function.
So now, let’s take a look at our function. We have 𝑦 is equal to four multiplied by 1.21 to the power of 𝑥. Well, actually great. This is already in the form 𝑦 equals 𝑎 multiplied by 𝑏 to power of 𝑥. So therefore, we can identify that our 𝑎 is equal to four. And this means that actually yes, it’s gonna be growth or decay because it’s greater than zero. And our 𝑏 is equal to 1.21 because actually we have 1.21 to the power of 𝑥.
Okay, great. So what does this 𝑏 tell us? Well, our 𝑏 value is greater than one. So therefore, we can say that the exponential function 𝑦 is equal to four multiplied by 1.21 to the power of 𝑥 is a growing function. And it’s a growing function because when we have it in the form 𝑦 is equal to 𝑎 multiplied 𝑏 to the power of 𝑥, our 𝑏 value is 1.21. And that is greater than one.