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# Analyzing graphs of exponential functions: negative initial value

CCSS.Math:

## Video transcript

so we have a graph here of the function f of X and I'm telling you right now that f of X is going to be an exponential function it looks like one but it's even nicer when someone tells you that and our goal in this video is to figure out at what x value so when when does f of X so what at what x value is f of X going to be equal to negative 1 25th and you might be tempted to just eyeball it over here but when when f of X is negative 1 25th that's like right below the x axis so if I try to if I try to eyeball it it would be very difficult it's very difficult to tell what value that is it might be it 3 it might be at 4 I am Not sure so instead of actually it looks like now maybe 3 well I don't want to I just eyeball it guess it instead I'm going to actually find an expression that defines f of X because they've given us some information here and then I can just solve for X so let's do that well since we know that f of X is an exponential function we know it's going to take the form f of X is equal to our initial value a times our common ratio R to the X power well the initial value is straightforward enough that's going to be the the value that the function takes on when x is equal to zero and you can even see here if X is equal to 0 the art of the X would just be 1 and so f of 0 will just be equal to a and so what is f of 0 well when X is equal to 0 this is essentially we're saying where does it intersect where does it intersect the y axis we see f of 0 is negative 25 so a is going to be negative 25 negative 25 when x is 0 the art of the X is just 1 so f of 0 is going to be negative 25 we see that right over there now to figure out the common ratio there's a couple of ways you could think about it the common ratio is the ratio between the two successive two successive values that are separated by 1 what do I mean by that well you could view it as the ratio between F of 1 and F of 0 that's going to be the common ratio or the ratio between F of 2 and F of 1 that is going to be the common ratio well lucky for us we know F of 0 is negative 25 and we know that F of 1 F of 1 X is equal to 1 Y or f of X or F of 1 is equal to negative 5 negative 5 and so just like that we're able to figure out that our common ratio R is negative 5 over negative 25 which is the same thing as 1/5 divide a negative by negative you get a positive so you're going at 5 over 25 which is 1/5 which is 1/5 so now we can write we can write an expression that defines f of X f of X is going to be equal to negative 25 x times 1/5 to the X power and so let's go back to our question when is this going to be equal to negative 125th so when does this equal to negative 120 v motor so let there's a siren outside I know if you hear it so negative I'll power through alright negative so let's see what at what x value does this expression equal negative 125th let's see we can multiply well actually we want to solve for X so let's see let's divide both sides by negative 25 and so we are going to get 1/5 to the X power is equal to if we divide both sides by negative 25 this negative 25 is going to go away and on the right hand side we're going to have divide a negative by negative couldn't be positive it's going to be 1 over 625 it's going to be 1 over 625 and 1/5 to the X power this is the same thing as 1 to the X power over 5 to the X power is equal to is equal to 1 over 625 1 over 6:25 well one to the X power is just going to be equal to it's just going to be equal to one so we could really doesn't matter that we have this to the X power over here and so we can see how I thought I was erasing that with a black color there you go that's a black color right over there so we can see that 5 to the X power needs to be equal to 625 so let me write that over here 5 whoops didn't change my color 5 to the X power needs to be equal to 625 now the best way I could think of doing this is let's just think about our powers of 5 so 5 to the first power is 5 5 squared is 25 5 to the third is 125 5 to the fourth we'll multiply that by 5 you're going to get 625 so X is going to be 4 it is going to be 4 because 5 to the 4th power is 625 so we can now say that F of 4 F of 4 is equal to is equal to negative 1 25th is equal to negative 125th now once again you can verify that you can verify that right over here 1/5 to 1/5 to the fourth power is going to be 1 over 6 25 25 negative 25 over positive 625 is going to be negative 125th so hopefully that clears things up a little bit