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# Graphs of exponential growth

CCSS.Math:

## Video transcript

all right we are asked to choose the graph of the function and the function is f of X is equal to 2 times 3 to the X and we have three choices here so pause this video and see if you can determine which of these three graphs actually is the graph of f of X all right let's work through this together so whenever I have a function like this which is an exponential function because I'm taking some number and I'm multiplying it by some mother number to some power so that tells me that I'm dealing with an exponential so I like to think about two things what happens when x equals zero what is what is the value of our function well when you just look at this function this would be 2 times 3 to the 0 which is equal to 3 to the 0 is 1 it's equal to 2 so one way to think about it in the graph of y is equal to f of X when x is equal to 0 Y is equal to 2 or another way to think about it is this value in exponential function sometimes called the initial value if you were thinking of the x-axis instead of the x-axis or think about the time axis or the T axis that's why it's sometimes called the initial value but the y-intercept is going to be described by that when you have a function of this form and you saw it right over there F of 0 3 to the 0 is 1 you're just left with the 2 so which of these have a y-intercept of 2 well here the y-intercept looks like one here the y-intercept looks like 3 here the y-intercept is 2 so just through elimination through that alone we can feel pretty good that the third graph is probably the choice but let's keep analyzing it to feel even better about it and so that we have the skills for really any exponential function that we might run into well the other thing to realize this number 3 is often referred to as a common ratio and that's because every time you increase X by 1 you're going to be taking 3 to a 1 higher power or you're essentially going to be multiplying by 3 again so for example f of 1 is going to be equal to 2 times 3 to the 1/2 times 3 to the 1 or 2 times 3 which is equal to 6 so from F of 0 to F of 1 you sense you have to multiply by 3 and you keep multiplying by 3 F of 2 F of 2 you're going to multiply by 3 again it's going to be 2 times 3 squared which is equal to 18 and so once again when I increased my X by 1 I'm multiplying the value of my function by 3 so let's just see which of these do this this one we said it has the wrong y-intercept but as we go from x equals 0 to x equals 1 we are going from 1 to 3 and then we are going from 3 till looks like we closed pretty close to 9 so it does look like this does have a common ratio of 3 just has a different y-intercept than the function we care about this looks like the graph f of X is equal to just 1 times 3 to the X here we're starting at 3 and then when x equals 1 it looks like we are doubling every time x increases by 1 so this looks like the graph of y is equal to I have my what we could call our initial value or y-intercept 3 and if we're doubling every time we increase by 1 3 times 2 to the X that's this graph here as I said this first graph looks like Y is equal to 1 times 3 to the X we are tripling every time 1 times 3 to the X or we could just say Y is equal to 3 to the X now this one here better work because we already picked it as our solution so let's see if that's actually the case so as we increase by 1 we should multiply by 3 so 2 times 3 is indeed 6 and then when you increase by another one we should go to 18 and that's kind of off the charts here but it does seem reasonable to see that we are multiplying by 3 every time and you could also go the other way if you're going down by one you should be dividing by 3 so 2 divided by 3 this does look pretty close to 2/3 so we should feel very good about our third choice