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# Analyzing graphs of exponential functions

CCSS.Math:

## Video transcript

so we have the graph of an exponential function here and the function is M of X and what I want to do is figure out what is M of 6 going to be equal to and like always pause the video and see if you can work it out well as I mentioned this is an exponential function so M is going to take the form let me write it this way M of X is going to take the form a times R to the X power where a is our initial value and R is our common ratio well the initial value is pretty straightforward is just going to be what M of 0 is so a is going to be equal to M of 0 and we can just look at this graph when X is equal to 0 the function is equal to 9 so it's equal to 9 and now we need to figure out our common ratio so let me set up a little bit of a table here just to help us with this so let me draw some straight lines and so this is X and M of X we already know that when x is 0 M of X is equal to 9 we also know when X is let's see when X is 1 when X is 1 M of X is 3 M of X is 3 so when we when we increase our X by 1 what happened to our M of X what did we have to multiply it by well to go from 9 to 3 you multiplied by 1/3 so that's going to be our common ratio in fact if we wanted to care what what M of 2 is going to be we would multiply by 1/3 again and M of 2 should be equal to 1 and we see that right over here M of 2 is indeed equal to 1 so our common ratio our common ratio right over here is equal to 1/3 so f of X we can write it as M of X is going to be equal to our initial value a which we already figured out as a is a is equal to nine so it's going to be nine times our common ratio times our common ratio 1/3 to the X power so I was able to figure out the the formula for our definition for M of X but that's not what I wanted I just wanted to figure out what M of six is going to be so we can write down that mo6 M of six is going to be 9 times 1 over 3 to the sixth power let's see that is going to be equal to that's the same thing as 9 times well 1 to the sixth is just one it so it's going to be 1 to the 6 which is just 1 over 3 to the 6th power now what is 3 to the 6 power in fact I could even simplify this a little bit more I could recognize that 9 is 3 squared so I could say this is going to be 3 squared over 3 to the sixth 3 squared over 3 to the sixth and then I could tackle this a couple of ways I could just divide the numerator and the denominator by 3 squared in which case I would get 1 over 3 to the 4th power or another way to think about it this would be the same thing as 3 to the 2 minus 6 power which is the same thing as 3 to the negative 4 power which of course is the same thing as 1 over 3 to the 4th so what's 3 to the 4th so 3 squared is 9 3 to the third is 27 3 to the 4th is 81 so this is going to be equal to 1 over 81 M of 6 is equal to 1 over 81 we could also be done that if we kept going with our table M of 3 multiplied by 1 3 is going to be 1/3 mo 4 multiplied by one third again is going to be 1/9 and we could Emma 5 is going to be one 27th and M of 6 is going to be 180 first