For the following exercises, find the formula for an exponential function that passes through th

For the following exercises, find the formula for an...

Key Concepts

Exponential Decay

Exponential decay occurs when the base 'b' of an exponential function is between 0 and 1, meaning the value of the function decreases as 'x' increases. Identifying whether the scenario represents decay or growth is fundamental in understanding the behavior and application of the exponential function.

Determining Parameters from Points

When an exponential function passes through two given points, each point can be substituted into the function y = ab^x. One of these points, typically the one with x=0, directly gives the value of 'a'. The additional point then enables solving for 'b' by forming an equation that usually necessitates algebraic manipulation or the use of logarithms.

Exponential Function

An exponential function is generally represented in the form y = ab^x, where 'a' is the initial value and 'b' is the base that determines the rate of growth or decay. This function is useful for modeling situations where the rate of change in a quantity is proportional to the quantity itself.

Transcript

00:01 All right, so find the formula for an exponential function that passes through the two points, 0, 2000, and 220.

00:11 All right, so let's see.

00:13 We're trying to find an exponential graph, an exponential equation that satisfies these two points.

00:19 So that can be in the form y is equal to a times b to the x.

00:25 Now, this first corner is actually really interesting because it's at the point 0 ,000.

00:32 So it's actually, if we have the graph right here, it's at the beginning.

00:36 It's the starting value.

00:37 Assuming we're only doing the positive values for x and stuff, it's either going to go over here, over here.

00:44 But we know it's starting at 2000.

00:47 Now, why is important? well, because when we have an exponential equation, we know that the a value right over here is going to be the beginning number.

00:59 And from there, it's either going to go up or down.

01:00 So we know that this right here is 2000.

01:04 And then from there, we can use the other coordinate to find b, and then from there we can get the final equation.

01:10 But we know that so far it is 2000 times b to the x.

01:19 Now, in order to find b, we can use the other coordinate right over here because it gives us an x and a y value.

01:27 And from there, we can literally just plug it in, and we have one value left over, which is b, which is what we're trying to find.

01:35 So let's do that, actually.

01:42 Let me just draw out a equation...