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4-15 AIM3 8.4 Properties of Exponential Functions

Added Apr 15, 2024 •

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so Lo so let's take a look at some properties exponential functions both things you've seen before and something that might surprise you A variation on it first I guess I would first like to remind you keep that hitting a little bit just of our basic laws when they're multiplying the add the exponents or dividing we subtract now the power you multiply so we should have that will be referring to that a bit today to get this little bit more all the way okay sup do you see the graph of f x equals 2D X and then we're going to look at two variations of it f of x + 1 which does w think of transformational properties how does the graph of f x compared to the graph of what's one word to say it another way f of x + 1 effects of Truvia except for Xbox One to beach 2 to the x + 1 so what happens when you replace x with x + 1 and I hope you remember that just moves the graph left one so whatever happened to the X's we move it left one so for example here 012 we should go to -1 you were at 12 so we should be 02 how about here at 2 for the b-14 how about this one - 1 Ashley - 2 a half until we get let's see if I can make a reasonable sketch that's not horrible I suppose that's what that looks like and what about 2 x f of x to change on the graph if we do to x f of x which of course we just need two times tutu the app well I hope you would call that I can get us a vertical stretch by a factor of 2 so whatever the original y-values work or something text value of Darnell twice as much so for example here let's take 01 doublet we did 02 well how about that how about over here at 1 to we now have 14 - 12 -11 why it appears that the graphs they are the same we have two different expressions name the f of x + 1 + 2 x FX that for the graph of 2 to the X give us the same result why would that be well actually kind of hinted that it here so why don't we take 2 * x to the 1 Let's rewrite using an exponent properties so we could call it to the x x to the 1 and that means it would be two x to the x and two times to the ax out to The Exorcist FX till we get there we go there's half of x + 1 equals 2 x f of x from this case turns out they are the same that of course is it true every once in awhile there something interesting happened with exponential functions let's look at another example all right again my friend to the X there it is how about f of x - 20 first let's just remember what changes would occur what were the draft happened here well of course this would now equal to to the x - 2 and replace axe with x -2 then this would move the graph right to encouraging now right super sensitive her game today right until you take every Point yourself example -2 a half or what would that be one a little bit last night but move it right to is it right to resist you 2 they're so it looks like we're getting something that looks no first what effect does cheer while we have 14 x f voice really been 50 today half of Acts and that tells us we would have a vertical French by a factor 14 and let's take a look here the Y value is 4 now its 1 here at the Y value is two now it's a half it was one now it's supposed to be. order so yeah once again how are they related they are the same so we can have different expression for things that really mean to say but let's build on this again Joe you to revisit we've seen the fundamental law of exponents many times before you should have these totally locked in your head while this song is kind of interesting it happens when Justin said just thinking it is in your new your Expressions agreed to the X + Y 33 the X X through the Y what if we're doing with function exponential function to hear it is f of x equals B the extra the expression be that you got the function f of x equals B to the ACT but what can be some of the results of that well it turns out she'll drink it this way it turns out that the asthma going to look at one of them if you have your exponential function here if you have half of x + y does just f x x f of Y and look back to this law when you're multiplying exponents to hear if we add input it's after the first input times after the second and put if we're subtracting there was a tracking look what happened you divided we divide when we multiply exponents it comes from a power to power so that's so we can see the something relationship between our laws of exponents of the laws of exponents in the function version and you can figure out that's why these are true but just to help you see why this is true why this make sense let's look at this example of x + y well then f of x + y must be be to the x + y and now that we're in an exponential form we know the video place where I can be written as B the x x b to the Y and B the extra called is f of x b to the Y Must Be yes that's why I'm so there's an art proof of y f x + y equals f of x x f of why and how we can use Su you'll see it in just a moment it would be probably good practice for you by the way well for all of them really but using the kind of Coach you just saw good practice see if you can prove why these are true okay well let's see it in action how we can use these properties help us solve certain problems so here we go so we're dealing with the exponential function for the X and he just a couple value can check these on your calculator for to the 1.16 one would be really close to 5 for the 1.5 which after all let's remember might as well just mention this 4 to the 1.5 L famous for to the 3/2 and rational exponents this is your your roots this is the √ 4 which is just two and two cubed and so we get ate so that one. OK Google what can we do with Tom's like this but I've got these I'd like to find what value do you have what would equal 25 other words for to what we 25 but the only known to have are these and I suppose I also I could take advantage of stuff like this is well okay so how can I rewrite 25 using the two that I have no hopefully you've thought well I can call 25:5 Square why is that better because now that gives us two things we have power situation here and we also have five and we know what five is five is least approximately 1.1 610 and here's the beauty now that I have my function to a power look it's exactly this and so I can now bring it down make it a product and so I guess is Tupac Downs of product multi-lane I get to .32 you can check on your calculator for to the 2.32 to be very close to 25 how about 1.6 can you see how to rename 1.6 using the information we have I hope it's a ground Tab and you discovered that 8/5 1 3/5 is 1.6 an hour quotient of 1/8 and 5 + course we have those values precisely + 8 is she is 1.5 5 G of 1.1 610 and here's this so all means all I have to do is subtract them I got this. You could check if you didn't force the point 399 you discover it's really close to 1.6 and last one but about 40 well I hope you received 40 is 5 * 8 as we discussed we have functional forms for five and eight in the in the function was dealing with for the extra there's five there's eight and we vote. just that FX x f y is f of x + y and there they are so we can not all these are excellent in the traditional sense we week is now discovery of these laws for the functional sense to help us certain situations take care

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4-15 AIM3 8.4 Properties of Exponential Functions

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