07 - What is an Exponential Function? (Exponential Growth, Decay & Graphing). - Free Educational videos for Students in K-12 | Lumos Learning

07 - What is an Exponential Function? (Exponential Growth, Decay & Graphing). - Free Educational videos for Students in k-12


07 - What is an Exponential Function? (Exponential Growth, Decay & Graphing). - By Math and Science



Transcript
00:00 Hello . Welcome back today . We're going to cover
00:02 the concept of the exponential function in math . It
00:06 has applications in algebra and trigonometry and pre calculus calculus
00:10 , all branches of advanced math and so on .
00:13 And it's not every day that I have the opportunity
00:16 to explain to you one of the most important functions
00:19 in all of math , all of science and all
00:21 of engineering . So I'm very excited to teach this
00:24 lesson here on the exponential function At the end of
00:27 this lesson you should understand what an exponential function is
00:30 . Have some idea of its importance . Although you
00:33 will continue learning its importance as we go on throughout
00:36 your your education in math and physics and chemistry and
00:39 other areas . Uh And also be able to understand
00:42 intuitive intuitively what the graph of the exponential function looks
00:46 like . And I have a computer demo that we're
00:48 gonna integrate into this lesson as well . So you
00:50 can see interactively what's really going on with this function
00:53 . I cannot overstate how important the exponential function is
00:57 in math . It's really honestly one of the most
00:59 important functions ever . Right , let me give you
01:02 a little motivation why . It's a crucially important function
01:05 for lots of reasons . But just off the top
01:08 of my head , uh if you like to make
01:09 money and you like to invest money in the stock
01:12 market or you like to put money in a savings
01:14 account and have it grow in the bank , the
01:16 interest that you earn on money ends up following an
01:19 exponential curve . So we're gonna learn a lot uh
01:23 after we understand what an exponential function is , we're
01:25 going to study the growth of money and that it
01:28 follows an exponential curve . So if you like making
01:31 money , exponential functions will do that for you .
01:33 Also when you take a loan out and you owe
01:35 money in form of a debt , the debt grows
01:38 monthly as well . And the growth of that debt
01:40 is also an exponential function . So if you want
01:42 to stay uh and keep your money and not have
01:45 as much debt . Exponential functions really are important for
01:48 you to understand . And going back more to the
01:51 pure science is when you have a bacteria that is
01:54 multiplying in a human body or in a laboratory ,
01:57 it's following a growth uh what we call exponential growth
02:01 . So the population of the bacteria is not growing
02:04 linearly , is growing exponentially . Right ? Also cancer
02:07 when cancer grows in your body , same kind of
02:09 thing is growing exponentially . Also populations , population of
02:13 Australia , Population of UK , population of the world
02:16 , population of some city somewhere . When you have
02:19 generation after generation uh producing offspring , the population follows
02:25 an exponential curve like this . So it's it has
02:28 practicality there . Getting more back into the sciences .
02:31 We have radio activity . Right ? When you have
02:33 uranium and it decays into some daughter product , you
02:36 might have heard the concept of half life when half
02:39 of the thing decays into something else . That also
02:41 follows an exponential type of decay . And here's the
02:44 granddaddy of them all . You will learn later in
02:47 your math and physics classes down the road engineering classes
02:51 that all waves , sine sine waves , which we
02:54 haven't even talked about sine waves yet . But these
02:57 up and down wave motions , right , They can
02:59 always be written in terms of exponential functions . So
03:03 that means literally every wave that propagates radio waves ,
03:06 microwaves , x rays , gamma rays , also ,
03:09 all of the waves that we use in quantum mechanics
03:12 to predict how matters going to behave . Those are
03:14 all waves too . And all of those waves can
03:17 always be written as exponential . All waves can be
03:20 written as exponential . So this exponential function that we're
03:23 learning literally has applications , every branch of math and
03:27 science and engineering . So that's why it is so
03:29 incredibly important . Now , what I want to do
03:31 is get on the board to show you what it
03:33 is . We're going to do the computer demo to
03:35 give you an idea what the graph looks like .
03:37 But then the rest of this lesson , we are
03:38 going to dissect this function so that you understand exactly
03:41 why the curve of this thing looks the way that
03:44 it does . And there's a couple little kind of
03:47 curves along the way . We'll be taking left turns
03:49 and right turns to make sure you understand every little
03:51 part of it . But ultimately , when you get
03:52 to the end , I want you to know what
03:54 this function is , what it looks like and why
03:56 it behaves the way it does . So what is
03:58 an exponential function ? Uh Let me just give a
04:01 couple of quick examples of exponential functions and I'm just
04:04 gonna write these down in no particular order . If
04:06 you have a function F of X equal to two
04:09 to the power of X , notice that the X
04:12 variable this is different than X squared . We've been
04:16 doing polynomial F of X is x squared . That's
04:19 a parabola . But in that case the X is
04:22 down low and the exponent X squared . The X
04:25 . The exponent is a number , just the number
04:27 two . In this case the exponent is not just
04:29 a single number . The exponent is the variable .
04:32 So when you see the variable up in the exponent
04:34 like that , that's an exponential function . Right .
04:37 Another example F of X is equal to 10 to
04:41 the power of X . This is a different exponential
04:43 function than this one , but it is still an
04:46 exponential function . Ffx Equals instead of a number like
04:51 two or 10 . We can have fractions as well
04:53 , we can have one half to the power of
04:55 X . Right ? So I have three examples here
04:59 and I could keep going on , I could say
05:00 , you know , 16 to the power of X
05:02 , 1/10 to the power of X , whatever .
05:04 But notice that they're all positive numbers down here at
05:07 the bottom and the exponent and every one of these
05:10 cases is up in the variable the variable is the
05:12 explosion . So what we can say about this exponential
05:16 function is the number down in the bottom is what
05:19 we call the base . So this base In this
05:23 case is equal to to the base is equal to
05:25 two . In this case , In this case you
05:27 might guess the base is equal to 10 . And
05:31 in this case you might guess that the base is
05:33 equal to one half . So you can see that
05:35 the base can be the base has to be positive
05:39 . And I'm gonna write all of these rules down
05:40 in a minute . It has to be positive and
05:43 it has to be uh well , I'm just gonna
05:45 leave it at that has to be positive . You
05:46 want to stay away from zero being a base and
05:48 you want to stay away from the number one being
05:50 the base . And I'll tell you why as we
05:52 get a little farther in why zero and one caused
05:55 problems for this function . But any other number than
05:58 that that's positive . We'll give you an exponential curve
06:01 just with different bases . All right . So what
06:03 I want to do now is go off to the
06:05 computer before we get too far into this and just
06:07 show you in general what this thing looks like .
06:10 And then once we finish that will come back and
06:12 we'll draw lots and lots of pictures . So you'll
06:13 understand exactly what this function looks like and how it's
06:16 calculated . Okay , welcome back here . We have
06:22 our computer demo this is what we call an exponential
06:24 function . This be right here is what we call
06:27 the base . So let's first take a look at
06:29 when the base is bigger than one , like two
06:31 to the power of X or three to the power
06:34 of X or 10 to the power of X .
06:35 Or 100 to the power of X and so on
06:38 . So what happens as I increase this guy ?
06:40 You see what happens uh as the base here is
06:43 the basis of 1.6 and then we have the power
06:45 of X . As the base gets bigger and bigger
06:47 and bigger . This curve starts out very very close
06:51 to zero . It rapidly ramps up like a Kind
06:54 of like a rocket launch and it kind of starts
06:55 going up much more and more vertical . And as
06:58 the base gets bigger and bigger and bigger all that
07:01 happens is the steepness of this part on the right
07:04 hand side gets more and more and more vertical .
07:06 So if I crank this thing up all the way
07:09 I think the maximum I have this thing set was
07:11 to 20 . But if I go to 100 or
07:13 300 all that's gonna happen is it's going to ramp
07:15 up more and more and more vertically . Right so
07:18 you might think of this curve representing the population growth
07:21 of bacteria starts off really really really small . But
07:24 then as it builds momentum in each generation starts generating
07:28 you know daughters and little daughter bacteria and little son
07:31 bacteria . And they continue to reproduce the Whammo ,
07:34 the thing goes straight up vertical like this 20 to
07:36 the power of X . Is a very steep exponential
07:38 curve . As I back this thing down you can
07:41 see we start to bend the thing down closer and
07:44 closer and closer and closer notice what happens when we
07:47 get down to around 1.5 to the power of X
07:49 . With a very gentle curve that goes up .
07:51 One thing I want to point out to you the
07:52 most important thing is that this exponential function . Look
07:55 at the scale of the Y axis , here's five
07:58 and here's 10 . So the tick marks are 12345
08:01 So this tick mark right where the curve crosses right
08:04 here is actually at the Y intercept of one .
08:07 Notice what happens when I change this curve , no
08:10 matter what it looks like , it always crosses the
08:13 y axis at the same spot the curve to the
08:16 left and the curve to the right changes shape but
08:19 it always goes through the same location . Why is
08:22 that ? Because if I take a value of X
08:24 is equal to zero and stick it in this ,
08:26 to the equation 4.9 to the zero power is just
08:29 one . So that's why it crosses at one .
08:32 If I change this curve to 20 to the power
08:34 of X , I put an X . Value of
08:36 0 , 20 to the power of zero is still
08:38 one . So it still crosses here . If I
08:40 go way way way down here to 3.33 point two
08:43 to the power of X , sticking a value of
08:45 X is equal to 3.2 to the zero power is
08:48 still one . So no matter how I play with
08:50 this curve , it always crosses there . Now notice
08:52 when I get down all the way I told you
08:54 get into problems when you have one to the power
08:57 of X . C 1.1 you have a very gentle
09:00 slope up . When you get down to one to
09:02 the power of X . Um It just flattens out
09:05 completely . And that's because one to the power of
09:08 ex uh no matter what I put in for X
09:11 , whether it's 21 to the power of two is
09:14 is 11 to the power of four is 11 to
09:18 the power of negative two is still one . No
09:19 matter what I put in for the X value .
09:21 When I say one to the power of anything ,
09:24 I always get one . So basically this exponential function
09:27 turns into a flat line when you put one as
09:30 the base . And that's why I told you need
09:32 to stay away from one being the base . Uh
09:35 And also the same thing , similar thing happens if
09:37 you put zero as a base , if you put
09:39 zero as a base zero to the anything power is
09:42 still zero . So that's not an exponential function either
09:45 . So you cannot have a base of zero and
09:47 you cannot have a base of one . But any
09:49 positive value notice everything bigger than one . Just yields
09:51 a steeper and steeper curve . Now let's go down
09:54 and take a look at what happens when the base
09:57 is not zero uh over here and it's not one
10:01 , but we just constrain it to be between 01
10:03 and one because the curve up here only looked at
10:06 what happens when the base is bigger than one .
10:07 Of course it gets steeper and steeper . But down
10:10 here we're just saying what happens when the base is
10:12 one half , what happens when the base is 3/4
10:14 ? What happens when the base is one ? 10
10:15 ? Things like that . So what I can do
10:17 is crank this thing this way and you can see
10:19 what starts to happen is the exponential function starts to
10:23 look like the exact mirror image of what's above .
10:26 Look at this guy here at .2 to the power
10:28 of X . And then I can crank this guy
10:30 up so that it looks something like a mirror image
10:33 . I mean I can adjust everything to make it
10:35 exactly the same . I'm not interested in making it
10:37 look the same . But you can see that the
10:39 shape of the curve looks the same as it does
10:41 in the previous case , it's just that for the
10:43 negative values of X , that's where the curve starts
10:46 to go into the stratosphere and then it just goes
10:48 down to zero as we go on to the positive
10:52 X direction . We're gonna get into exactly why it
10:54 does this in a minute . So here we have
10:56 the positive values of X . It gets steeper and
10:58 steeper still goes through the value the y intercept of
11:01 one . Here , if we have the base be
11:05 a fraction like one half or one third or whatever
11:07 to the power of X . We have the same
11:09 shape but it's just a mirror image on the other
11:12 side . Again , it goes through the y intercept
11:14 of one . Anything to the one to the anything
11:17 . Um I'm sorry anything zero any base to the
11:22 power of zero is just going to give you a
11:24 one . So it goes through the same spot here
11:26 . Now what we're gonna do is combine those two
11:29 things together , we're gonna say the base is gonna
11:30 be bigger than zero , but um uh can also
11:35 be a fraction as well . So when we go
11:37 larger and larger values of the base , we have
11:40 that nice skateboard type shape to the right , when
11:43 we back it down , uh here what ? We
11:46 have one to the power of X . We get
11:48 to where it's flat . And then when I go
11:50 just on the other side of one , then you
11:52 can see this thing curving up the other direction .
11:54 So you see the nice symmetry here , What's going
11:56 on when the basis of fraction is , you can
11:58 make it steeper and shallow over here . And then
12:01 when you go through one it gets flat exactly flat
12:05 . And when you go through one you get you
12:06 have that nice shape happening on the other side .
12:08 So this is what the exponential function does notice it
12:12 always crosses through the y intercept of one . Okay
12:16 , Like this and that is all I'm going to
12:20 show you right now . I do want to show
12:21 you what happens when the base gets less than zero
12:24 , but I'm not gonna do that until the end
12:25 of the lesson because I think it's gonna make a
12:26 lot more sense then for now , what I want
12:29 you to realize is when the base is bigger than
12:31 one , you have this very steep curve on the
12:34 positive part of the X values . When the base
12:37 is a fraction like one half or one third or
12:39 whatever , you have the steep part of the curve
12:41 on the negative X . Side . And when you
12:44 put those two things together , you can just see
12:45 how it behaves . Going through a flat curve ,
12:48 a flat line and being exponential on both sides of
12:52 this guy . So what I want to do now
12:53 is stop the computer demo go back to the board
12:55 and really understand why this function behaves the way that
12:59 it does . All right , welcome back . I
13:02 hope you've enjoyed the computer demo to really get an
13:05 intuitive feel for what these exponential functions really look like
13:09 . Now . What we learned from the computer plotting
13:12 that we did was that the exact shape of the
13:14 exponential function really is going to depend upon the base
13:17 , the number that we have down in the bottom
13:19 below the exponent is going to determine the exact shape
13:22 of the curve . However , the general shape of
13:24 the thing , as far as its sloping upward like
13:27 that generally looks the same among all of those .
13:29 But the exact steepness of it of course , depends
13:32 on the base . So , just a couple of
13:34 rules , I want to make sure and get down
13:36 here on the board to make sure they're not lost
13:39 is . So I'll just kind of put this under
13:41 some miscellaneous note , right ? The base must be
13:48 positive . Notice that when we did the computer demo
13:53 , we did uh fractions like one half , one
13:56 third . We did fractions all the way , just
13:57 above zero all the way to 22 positive 20 .
14:00 So any positive numbers . Okay . Uh except there's
14:04 a couple of gotchas that you have to be careful
14:06 about . Base must be positive . However , the
14:09 base The base must not equal one . Why ?
14:16 Because if I put a base of one here ,
14:17 one to the power of X No matter what I
14:20 pick for X . 1 to the power of anything
14:22 affects his one affects us to affect his negative 11
14:25 to the power anything is just one . So we
14:27 saw that when the base was one in the computer
14:30 demo , it was just a flat line that's not
14:32 really considered exponential . In order to be an exponential
14:34 curve , it needs to slope upward one direction or
14:37 another . So the base cannot be won . If
14:39 you put a one in there , it's just not
14:40 gonna really be an exponential curve anymore . Also the
14:44 base must not equal zero for similar reasons . If
14:49 I put a zero in here , zero to the
14:51 power of anything is just going to give me zero
14:53 . So if you put a number one in for
14:55 the base , you're not gonna get an exponential function
14:57 . If you put a zero in for the base
14:58 you're still not going to get an exponential function .
15:00 Any other positive number is going to give you an
15:03 exponential function with varying shapes . Now You might probably
15:07 be wondering what happens if the base is negative ?
15:09 What if I put a negative two for the base
15:11 ? What if I put a negative 10 for the
15:12 base ? I'm gonna save that for the end of
15:14 this lesson . I would like you to know what
15:16 happens , but I don't want to talk about it
15:17 now because it doesn't give you an exponential function either
15:20 . For now , just know that the base has
15:22 to be positive , but it can't be one and
15:24 it cannot be zero . All right now , some
15:27 of you might be looking at this and saying ,
15:28 well , we've kind of studied functions similar to this
15:31 in the past . We've studied things like for instance
15:33 , ffx is equal to x squared . This kind
15:37 of has an exponent there , we've studied ffx uh
15:40 x cubed X to the fourth , but this is
15:43 different . This is f of x is something like
15:46 , you know , whatever , I can pick the
15:47 base that I want . Six to the power of
15:48 X . You see , I really want you to
15:50 understand the difference here . These things are just polynomial
15:54 , is they are not exponential . These are not
15:57 exponential . Yeah , But this one is exponential .
16:05 It's an exponential function . So the only thing you
16:07 have to look for is if the variable is in
16:09 the exponent , not a solid number here , but
16:12 the variable itself is up top in the exponent .
16:14 Then you have an exponential function . Then the exact
16:17 shape of the curve is going to be governed by
16:20 the base , right ? So we have to split
16:22 it up into two pieces first . Like we did
16:24 in the computer demo , I want to talk to
16:25 talk to you in more detail what happens when the
16:27 base Is bigger than the # 1 ? Right .
16:30 And then we're gonna talk about what happens when the
16:32 basis of fraction somewhere between zero and one . Because
16:34 we saw in the computer demo it flipped everything around
16:37 and made the curve kind of mirror image and that
16:40 we need to explore a lot more in a lot
16:42 more detail . But for right now let's go and
16:45 talk about something very very important . Let's take a
16:47 look at an example . One of these ffx is
16:51 equal to two to the power of X . That's
16:53 an exponential function with the base of two . And
16:56 what I want to do is just spend a minute
16:58 trying to draw a pretty good graph of this .
17:02 Now is this graph gonna be as good as a
17:03 computer ? No that's actually why I showed you the
17:06 computer first . Um It's not gonna be as good
17:08 but we can still learn a lot . Why does
17:11 it start out low and why does it get steep
17:13 like this ? And why does the steepness change depending
17:16 on what the base is ? I want to get
17:18 very very very comfortable with that . Now we saw
17:22 in the computer demo that when X . is equal
17:24 to zero all of these curves cross at an intercept
17:29 point of one . I'm gonna put the number one
17:32 up above here . Okay , so we saw that
17:35 when the base was a positive number bigger than one
17:39 like this , it starts way over here , close
17:42 to zero . I'm probably gonna mess this curve up
17:44 but it goes really really gently slowly increasing right through
17:47 this point and then kind of going up something like
17:51 this . Is that perfect ? No it's a little
17:52 bit lumpy . It should be nice and smooth now
17:55 the exact shape of it of course , depends on
17:57 the base . As the base gets bigger , bigger
17:59 , bigger . This thing gets steeper , steeper steeper
18:02 as the base gets smaller smaller smaller it lays more
18:04 flat till eventually it's just a flat line and then
18:07 it flips over and goes the other direction . We
18:09 saw that in the computer . But why is the
18:11 thing curve this way ? Because when you take and
18:15 put an X . Value of zero into this thing
18:17 , what do you get ? I'm going to do
18:18 a lot of little substitution is down here below you
18:21 get two to the power of zero but two to
18:24 the power of zero is just one . So you
18:26 see the number one it crosses at the uh the
18:29 uh Why Value of one . Now let's go and
18:32 put a few values in there . 12345 negative one
18:37 negative two negative three negative four negative five . So
18:40 this is negative five negative four , negative three negative
18:43 two negative 11234 and five . And what I'd like
18:50 to do is just spend a second to understand how
18:55 this thing uh increases . If we put the value
18:58 of one in here then what we're going to get
19:00 is two to the power of one . That just
19:03 means to because that's all it is . Two to
19:05 the power of one . So we get a value
19:06 of two . So I'm not going to plot it
19:07 . This graph this why access I have is not
19:10 exactly right . My curve is not exactly right .
19:12 But you can still see that as I go up
19:14 here . When I put a two in there ,
19:16 it'll be two to the power of to this goes
19:18 into the exponent location which gives you a four .
19:22 Right ? And then you can see that as I
19:24 put three in there , it's two to the power
19:25 of 32 times two times two . This gives you
19:28 an eight . So you see what's happening is it's
19:30 increasing faster and faster . When I put a value
19:33 of foreign to to the power of four , I'm
19:36 going to get 16 two times two times two times
19:39 2 16 . And then when I put a value
19:42 of five and I'll get to to the five uh
19:45 and I'm gonna get 32 . So you can see
19:47 these numbers 248 16 32 . It's doubling every time
19:53 . And that's kind of the genesis of the exponential
19:56 growth part . When when you put money in the
19:58 bank and it earns interest , we're going to study
20:00 it more later . But you have an exponential effect
20:03 because when you earn money one year that money is
20:06 still in the bank for the following year to help
20:09 earn more money . So the more money you earn
20:11 , the more interest you earn the following year .
20:13 Because the money you earned in year one starts to
20:17 earn more money in year two . But then in
20:19 year two you have even more money which then generates
20:22 more money . And so it goes up and up
20:23 and up in an exponential fashion . So that's why
20:26 the curved gets so steep . So quickly look at
20:28 how fast these numbers are increasing . Now when we
20:30 go to the other side here switch colours for this
20:33 , I want to go to the negative one .
20:35 A lot of students don't really quite understand why this
20:37 works the way it does . If you put a
20:39 negative one in for the experiment it'll be two to
20:41 the power of negative one . But this is just
20:44 the same thing as 1/2 to the first power ,
20:48 Which is the same thing as 1/2 . Okay ,
20:52 so we have one half here . Um And that's
20:56 why it gets smaller than one , notice it gets
20:57 smaller than one . Now we have a similar thing
20:59 right here . If we put two and this exponent
21:01 of negative to here , then this becomes 1/2 squared
21:06 which becomes 1/4 . So it gets even smaller and
21:10 you can see the pattern here . This I'm not
21:12 gonna substitute every little thing , it's just gonna be
21:14 1/8 . This guy is gonna be 1/16 and this
21:18 guy is gonna be 1/32 . So you see the
21:22 numbers that you have on the positive side , Just
21:25 become one over those numbers on the negative side .
21:27 So as fast as this thing is increasing this direction
21:30 , it is chopping it down just that fast going
21:33 the other direction . This is why the shape of
21:35 the curve for the exponential function in this case we
21:38 just pick two to the power of X starts out
21:40 very small because 1/32 is a really small number .
21:44 But as we get It builds very very , very
21:46 slowly . It always goes through the .1 , but
21:49 then it just rockets up because when you get into
21:52 positive territory , everything starts stacking as we just described
21:56 here and that's called exponential growth . Okay , if
22:00 we change the base , we're still going to be
22:03 going through this crossing point of why is equal to
22:06 one . That's really , really important for you to
22:08 know . Um In fact , um I do want
22:11 to take just a second before we talk about the
22:13 negative side just to kind of draw some comparisons .
22:15 Right ? So we might , you might say ,
22:17 well we've studied functions like this before and we just
22:19 talked about the fact that these are not exponential functions
22:22 , but a curve that looks kind of like this
22:24 is the X square curve . I'm not going to
22:26 clutter up this curve , but you remember the X
22:27 square parabola goes down and then up . So it
22:30 kind of has an upward slope like this . But
22:32 one thing that's not , I think made clear enough
22:35 in most books , is that an exponential function like
22:39 this ? If you give it enough time , if
22:41 you go out far enough will always go up steeper
22:45 than a polynomial of the of a similar degree .
22:47 Right ? So in other words , if I were
22:49 to plot X squared , what's going to happen is
22:52 if you square this guy , you're gonna get one
22:55 . If you square this guy , you're going to
22:57 get four . If you swear this guy you're gonna
22:59 get three times three is nine . So you see
23:01 the X Squared functions actually beating this function here Because
23:04 nine for X squared if I did three square being
23:08 nine is bigger than this guy . Four square to
23:10 16 . So right at this point the X squared
23:12 function is going to kind of intersect here and it's
23:15 going to start to not win anymore because if you
23:17 square this guy five square you're gonna get 25 that's
23:20 not as big as 32 . So what happens is
23:23 the the the parabola function X squared , which is
23:26 kind of similar because the basis to an X .
23:28 Squared has a two in the exponents . It's not
23:30 gonna it's gonna actually beat this function in the beginning
23:33 . But then eventually if you give it enough time
23:35 , the exponential function will always overtake a polynomial and
23:39 rocket higher . So that's why if you ever want
23:41 to you know , you know , earn money or
23:44 interest or study bacteria or anything that has an exponential
23:46 kind of growth , that's why it gets out of
23:48 control so fast . Especially in terms of debt when
23:51 you can owe so much money so fast because it
23:54 just builds on each other on itself year after year
23:57 after year and it kind of just rockets up and
23:59 starts to go vertical like that . Okay . Um
24:02 One more thing I want to bring very uh importantly
24:06 home is that uh zero comma . One is uh
24:13 or I should say lies on all exponential functions .
24:22 I've said this several times but it's important enough to
24:24 make sure you understand what I mean by this is
24:27 the Y intercept is one . Right ? If I
24:29 take for instance ffx let's take a crazy exponential function
24:33 . Let's say it's 100 to the power of X
24:35 . That's an enormously steep exponential function . Right ?
24:39 But if I put the value of X is equal
24:41 to zero in here , I'm gonna get 100 to
24:44 the power of zero , which is just equal to
24:45 one . So this means that the 10.0 comma one
24:48 is on this curve . So just like we saw
24:51 in the computer demo even if I crank the bass
24:53 up to 100 all that's gonna happen is this is
24:55 going to get very , very steep , but it's
24:56 still going to come down through this point and then
24:58 it's gonna get really , really small on this side
25:01 . Okay , now so far , we've talked about
25:05 positive values of the base . I want to draw
25:07 a few more pictures about the positive values of the
25:09 base and then we're gonna move on and talk about
25:11 what happens . I should say we've talked about positive
25:13 basis that are bigger than the number one base is
25:16 equal to two . Base is equal to three .
25:18 Base is equal to four in just a second .
25:20 We're gonna talk about fractional basis where we see the
25:23 exponential function flip over to the other side . Okay
25:26 . But before I do that right underneath this ,
25:29 I want to draw a few different cases which we
25:32 saw in the in the demo , but I want
25:35 to draw a couple of cases to make sure you
25:36 understand why this thing gets steeper and steeper and steeper
25:40 as you increase the base . So let's take the
25:42 case that we just have right here . Ffx is
25:45 equal to two to the power of X . And
25:48 we're gonna graph this guy right next door to a
25:51 couple of alternatives just so we can kind of see
25:53 what happens . So it's gonna go through the value
25:56 why is equal to one . We know that it's
25:58 going to start something like this and something like this
26:03 . And we know we can just pick whatever number
26:05 we want . Let's pick the number three just as
26:08 a point of comparison X . Is equal to three
26:10 when we put a value . Uh And and of
26:13 course my scale is not right at all here ,
26:15 but if I go up here and I go right
26:17 here , what's going to happen whenever I do this
26:20 ? Is that um this value right here , it's
26:24 gonna be two to the power of three , which
26:26 is eight . So my scale is not right okay
26:28 because there's one here and there's an eight here ,
26:30 but you get the idea we're gonna cross up and
26:32 we're gonna come up to a value of eight .
26:34 Okay , now let's take another example and let's see
26:38 what happens . If we increase the base to F
26:41 . Of X . Uh is equal to 3 to
26:44 the power of X . What will something like this
26:47 look like ? Well we're gonna have a curve that's
26:50 gonna go like this is gonna be X . And
26:52 ffx like this . We know it's going to go
26:54 through the number one . Yes , I know that
26:56 my scale is not perfect . So don't don't you
26:58 know , don't get too mad at me . But
27:00 basically what I'm telling you is just what we saw
27:02 in the computer program . It's gonna start out very
27:04 very slow and then it's gonna go a little bit
27:07 steeper . Why is it going to go a little
27:09 bit steeper ? Because again , if I take The
27:12 same number three here and put it in here ,
27:15 It's going to go , it's going to be much
27:17 much higher . Why ? Because this point is going
27:19 to be three to the power of three , and
27:22 three to the power of three is 27 . So
27:25 the crossing point here is actually at 27 . So
27:28 that's why increasing the base makes the thing gets steeper
27:32 because here we're just as we go along and X
27:34 . We're just going to times two times two .
27:36 However many times you're doing it here , we're going
27:38 three times three times three . However many times we
27:40 do it . So if we pick the same value
27:42 of X , of course the curve is going to
27:43 be steeper here and we can make a similar argument
27:46 on the other side is going to get closer to
27:48 the X axis on the other side as well .
27:50 Okay , so as the base gets bigger and bigger
27:53 , larger than one , the exponential function gets steeper
27:56 and steeper , but it always goes through the y
27:58 intercept of one . Okay , now , what we
28:02 want to do is we want to take a look
28:05 at what happens when this base here , in this
28:08 case it's the number two . When this base is
28:09 allowed to be below the value of one , because
28:13 remember when the base is one , it's exactly equal
28:15 to one . We don't have an exponential function at
28:17 all . But when it's less than one but bigger
28:19 than zero , we have something really interesting happened .
28:22 We saw in the demo that this curve flips backwards
28:25 and starts high on this side and gets very low
28:27 on this side . So we want to describe um
28:31 exactly why that happens . So , let's take a
28:34 look at that . Let's take another exponential function .
28:38 Let's say f of X is going to be equal
28:41 to uh instead of two to the power of X
28:44 . Let's take a look at one half to the
28:46 power of X . Same kind of thing . We're
28:49 just gonna draw crude graph . It's not going to
28:51 be exact at all , but we're gonna do our
28:54 best . All right ? So let's do the same
28:58 kind of numbers 12345 So 543210 negative one , negative
29:06 two negative three negative four negative five -4 -3 -2
29:11 -1 . Alright , What I'm claiming is that this
29:14 function is going to go through the exact same point
29:17 . It's still going to go through one . It's
29:19 still going to go through one . But instead of
29:21 being high on this side , it's going to be
29:22 high on this side . So I'm probably not gonna
29:25 be able to draw this right . Apologize let me
29:26 start this way and see if I can get it
29:29 go like this right through this point like this .
29:33 That's not actually too bad . You see it's a
29:35 mirror image of what we have over here . Why
29:38 is it a mirror image ? Let's take a look
29:40 at it . If we take the number zero and
29:43 put it in there , we're gonna have one half
29:45 to the power of zero , that equals one .
29:47 So we have to go through the 10.1 . Just
29:49 like always this guy is going to be one half
29:53 to the power of one . So you can see
29:55 right away . This is going to give you a
29:56 value of one half . So it has to be
29:58 lower . And then this guy , I'm running out
30:00 of space here , it's gonna be one half to
30:03 the power of to which is gonna be 1/4 .
30:07 So it's gonna be 14 so it's going to get
30:09 smaller and smaller . And then this guy you can
30:11 see it's gonna be uh one half to the third
30:16 power . So two times two times two is 1/8
30:19 whoops , 1/8 . So it's gonna get smaller .
30:22 And then over here you're gonna have 1/16 . And
30:26 then over here you're gonna have 1/32 . Okay ,
30:30 now let's go the other direction , we'll pick another
30:32 color for this and go this way . Now take
30:34 a look at what happens right here . Um Right
30:36 here we have 1/2 To the -1 . Power to
30:42 the negative one . Power means it's the same thing
30:44 as 1/1 half to the one power . But I
30:48 can flip this thing over and then that means this
30:50 value is going to give me a two . So
30:52 this side actually pops up higher than the one .
30:54 Okay , here it's gonna be one half to the
30:58 power of negative two , and it's going to be
30:59 the same kind of thing , 1/1 half squared .
31:05 So this is gonna be 1/4 . And when you
31:07 flip this guy over , when you do the division
31:09 one divided by means you flip it over , you're
31:10 gonna get a four . Same thing is going to
31:13 happen here except you're gonna get an eight and right
31:16 here you're gonna get a 16 and right here you're
31:18 going to get that 32 . So you see the
31:21 numbers the 32 16 , 842 is are the same
31:24 numbers on the bottom here , which are the exact
31:27 same numbers we got here . So it literally is
31:30 the mirror image here . The positive values of X
31:32 gave me large values and here they are over here
31:35 when the base is one over the base we had
31:37 there the numbers are the same . It's just that
31:40 the larger side is on the left and the fractional
31:42 side is on the right . So it literally is
31:44 like taking that function and reflecting it to the other
31:47 side . Like this . It's an exact mirror image
31:50 because this base is just one over this base .
31:53 But of course , if I change this base so
31:56 that it's different than one half , then I'm going
31:59 to of course have a slightly different curve there .
32:01 And as I play around with that base , uh
32:04 making it closer to zero and closer to the number
32:07 one . Then I'm going to change and make it
32:09 steeper steeper or less steep . And I do want
32:11 to take just a second to show you what that
32:14 would look like . All right . So what I
32:16 want to do is just do a little bit of
32:18 a comparison to see how this curve changes as we
32:21 change the base . We saw that in the computer
32:23 demo also . So , let's start out by coming
32:26 over here and taking a look at what happens if
32:28 we change . Well , let's go in and draw
32:30 what we have actually , let's say we have one
32:32 half to the power of X . This is what
32:34 we just drew . I'm just going to kind of
32:35 re sketch it over here as a point of comparison
32:38 like this . We know this function is going to
32:40 go through the number one comma zero , uh sorry
32:43 , zero comma one . And we know it's gonna
32:45 be pretty gradual in general . It's gonna go start
32:48 out something like this . It's gonna go through this
32:51 point as close as I can get it . Something
32:53 like this . Now , let's pick a point of
32:55 comparison . Let's pick the point negative three out here
32:58 . Which is going to intersect something like this .
33:01 What is this point gonna look like It's going to
33:03 be when we do uh one half to the negative
33:07 three power , Right ? It's one half to the
33:09 negative three power . We go over here and we
33:11 see we're going to get an eight . How do
33:13 we know that ? One half to the negative three
33:15 power ? It's gonna be 1/1 half cube . So
33:18 the thing is really big and what we're gonna get
33:21 is eight . We just saw that here . So
33:23 we see that it's higher . Now let's do the
33:25 exact same curve . But let's change it so that
33:29 we say F of X Is equal to change the
33:33 base to make it even smaller . 1/10 to the
33:36 power of X . Kind of draw a little dividing
33:38 line right here . So I'm comparing as to what
33:40 happens over here . What's going to happen whenever we
33:43 make the base really , really tiny . Close to
33:44 zero ? Well , we're gonna show in just a
33:48 second what's going to happen is this function is going
33:51 to be really , really steep , but still going
33:54 to go through this point and it's going to get
33:55 like this . You see how this one's a lot
33:57 steeper . I may not have drawn it perfectly ,
33:59 but this one's supposed to be a lot steeper .
34:01 Why is it gonna be steeper ? Because if I
34:03 go to the same place , negative three , what's
34:07 this point gonna look like ? It's gonna be 1/10
34:10 to the negative three , Which is one over 1/10
34:15 to the power of three . Which this is going
34:17 to be won over 10 to the power of three
34:20 is gonna be 1000 and then one over . Uh
34:24 Or I should say , let's just do it like
34:26 this 1/1 over 1000 And that's going to equal 1000
34:32 . You see it's one over the fraction one over
34:35 1000 . You flip this guy over , you're gonna
34:36 get 1000 . So this value right here is actually
34:39 way up here at 1000 . Now of course I
34:42 didn't draw it right . This thing is actually much
34:45 , much steeper . It should be going really ,
34:46 really low and getting really really really steep when you
34:49 make the thing uh so small like this . So
34:52 the bottom line is that as you start off with
34:56 a base very very close to zero , you have
34:58 a really steep curve . Really steep as you make
35:02 the base bigger and bigger and bigger closer to one
35:04 . This bends down so that it's getting closer and
35:07 closer . When you make the base equal to one
35:10 . Exactly , you have a flat curve that's not
35:13 even really an exponential function . But then as you
35:16 increase the base past one , so you get to
35:18 to it gets a little steeper . And then when
35:20 you get even higher gets a little steeper . This
35:22 behavior of starting high getting flat and then going up
35:26 again is what we saw in the computer demo .
35:28 And that's why it's because when you have these negative
35:31 values here for X and you're putting them in here
35:34 , then you have to do the fraction arithmetic to
35:36 flip it upside down . And that's why it gets
35:37 getting bigger on this one side here . It's extremely
35:41 confusing when you first learned an exponential function , why
35:44 is it getting steeper when the base is a fraction
35:47 on the on the opposite side of when it's getting
35:50 steep when the base is bigger than one ? Okay
35:53 , so now that we have all of that background
35:55 in place , Mhm . We can finally write down
35:59 what you'll probably see in your textbook this is an
36:03 exponential function , exponential function . And in general ,
36:10 you know , we've already talked about this but I
36:13 want to put it all in one place . The
36:14 exponential function is the following . It is a function
36:18 F of X is equal to some base to the
36:22 power of X , right ? With be greater than
36:29 zero . The base has to be greater than zero
36:31 and B is not equal to one . This is
36:35 exactly what I told you in the beginning , I
36:36 told you the base can't be zero and the base
36:39 can't be one , but the base must be positive
36:42 . Okay , so that's all I'm saying is how
36:45 you would typically see it in a book . It'll
36:47 be exponential function with B to the power of X
36:49 . B has to be greater than zero . That
36:51 means be can't be zero , but it cannot be
36:53 equal to one . Any other number is fair game
36:55 . And then you have different cases , right ?
36:58 And we've already written all this down , but I'm
36:59 putting it all in one place if B is greater
37:04 than zero , but less than one . This means
37:06 it's a fraction somewhere between zero and one than in
37:09 general . The exponential function is going to look something
37:12 like this , It's going to go through the .001
37:17 on the left hand side , right through this point
37:19 , something like this . Okay . If uh B
37:26 is exactly equal to one , which you can't really
37:29 have , but I'm just putting it here for completeness
37:31 . So you can see what it actually in your
37:33 mind , you can see what would happen . Is
37:35 the line is not an exponential function at all ,
37:37 it's just flat . So you can't really have this
37:39 is not an exponential function . I'm just putting it
37:41 here to show you what would happen if it is
37:42 equal to one . And then finally , if B
37:47 is greater than one , what you get is that
37:50 standard exponential function that we have already talked about ,
37:53 where it goes through zero comma , one starts off
37:56 really , really small , goes through this point and
37:59 rapidly goes up high like this bigger than one .
38:03 If you have a base of one , it's not
38:07 really an exponential function . If you have a base
38:08 of zero , it's not an exponential function . So
38:11 that's why those constraints are there . Now , The
38:13 one point of curiosity that most people have is what
38:16 happens if the base is negative ? We said the
38:19 base has to be bigger than zero . We understand
38:21 zero to the power of whatever is not gonna be
38:23 an exponential function . But we said it has to
38:24 be bigger than zero and it has to not be
38:26 one that all makes total sense . What happens if
38:29 we put a value a negative value for the base
38:32 into this equation ? The answer is it's not an
38:35 exponential function and I want to tell you just a
38:38 second why it's not an exponential function . Let me
38:41 go find some room . I think over here let's
38:45 say the base is negative . Let's just pretend you
38:49 cannot have . This is not going to be an
38:51 exponential function , but what's going to happen ? Let's
38:53 say F of X is equal to let's just say
38:57 negative two to the power of X . I mean
39:00 , at first glance it looks like it would be
39:01 an exponential function . But think about it for a
39:04 second . What's going to happen if you put a
39:07 value of one here , Let's just pretend let's say
39:11 f of one . Okay , that's gonna be negative
39:14 two to the first power . So F of one
39:18 is going to be equal to negative two . All
39:20 right , so so far so good . Let's say
39:22 we go up to the next guy and say let's
39:24 put a value of X . Is equal to two
39:26 in here . Two . Okay . And that means
39:30 that we have negative two to the second power .
39:34 Now , you see because negative times negative is positive
39:37 . This is actually going to be a positive for
39:41 All right . Now let's do one more . What
39:42 if we go up one more and say F .
39:44 Of three ? It's gonna be negative two to the
39:47 three power . So F of three is gonna be
39:50 negative two times negative two times negative two . So
39:53 negative two times negative is positive for positive . Four
39:56 times negative two is actually negative . Eight . You
39:59 see what's going on here as we as we put
40:02 different values of action for our regular exponential function ,
40:05 it's a very smooth increasing value that just goes up
40:08 and up and up . But here when the base
40:10 is negative you have this oscillation happening here . We
40:14 just had when we put a value of of one
40:17 in we got a negative value out but then we
40:20 put a value of two in and we got a
40:21 positive value of out . But then we put a
40:23 value of three in and we got another negative value
40:25 out . So it's like it's going positive negative positive
40:28 negative . So what's gonna happen is it's gonna get
40:30 bigger . 248 16 32 but positive negative positive negative
40:36 positive negative , it's going to oscillate and that is
40:38 not an exponential function . So that's why when we
40:41 say an exponential function is when the base is greater
40:44 than zero but it can't be equal to one .
40:47 The reason we're saying that is because you put negative
40:49 bases in here , you can calculate the answers but
40:51 it's going to have this oscillation effect up and down
40:54 bouncing and it's not gonna be a smooth exponential function
40:57 . So let's take just a second to go .
40:59 Take a look at that in a computer just to
41:00 satisfy your curiosity to see what that actually looks like
41:04 . Hello , Welcome back . So here's what we're
41:06 going to do is see what happens when the base
41:08 is less than zero . Now , first , let's
41:10 remind ourselves what happens when we increase the base .
41:13 This is a positive base getting bigger , bigger ,
41:15 bigger , bigger , bigger . We have this nice
41:17 exponential function going really , really , really steep as
41:20 we get down here to where the base is now
41:22 getting close to the number one , it gets very
41:24 flat . The scale is a little different here ,
41:26 so it gets very flat . And then when we
41:28 get to fractions , we see that the exponential function
41:31 gets larger on the other side , mirror image like
41:34 we just described . Now , this is .2 to
41:36 the power of X . Let's still upside gave the
41:39 punchline away , let's go and let it get to
41:42 where we have a negative value in here for this
41:44 base . It gets very , very high . And
41:47 then if I can tweak it and get a negative
41:49 value here , we have this oscillation kind of effect
41:52 . You see how these oscillations are happening , We're
41:54 bouncing positive negative , it's still getting bigger . It's
41:57 just the absolute value is getting positive , negative ,
41:59 positive negative . And if we crank it up even
42:02 more than that , we're going to see this thing
42:03 go out of control and they're gonna see an oscillation
42:05 effect on the other side . So you see how
42:08 we still have a mirror image kind of thing going
42:09 on , where it gets bigger on one side ,
42:11 then bigger on the other side when the base is
42:13 negative . But this oscillation effect that I just described
42:16 on the board is what prevents this thing from becoming
42:18 an exponential function . This is a perfectly valid function
42:21 . It's just not an exponential function . So let's
42:24 go ahead and wrap up the computer demo and go
42:26 off and wrap up the lesson . All right ,
42:29 welcome back . I hope you understand now what an
42:31 exponential function is and why it is so important .
42:34 The bottom line is an exponential function is when we
42:36 have a number raised to a power raised to an
42:39 X variable like this , the number on the bottom
42:42 is what we call the base . The shape of
42:44 the exponential curve depends on the base and went through
42:46 all of this kind of background information . Ultimately arriving
42:50 at something like this . We can see that we
42:52 put bigger and bigger values in for a positive base
42:54 bigger than one , it gets steeper and steeper .
42:57 If we change the base , the curve still goes
42:59 through one , but just gets steeper . If we
43:02 increase the base , we can actually see it getting
43:04 steeper right here . Then if we flip it around
43:07 and say , well , what happens when the basis
43:09 of fraction between zero and one ? We have the
43:11 exact mirror image we have of getting steep on the
43:13 left and getting very small on the right . The
43:16 reason it does that is because of the way fractions
43:18 work with powers . When we have negative power is
43:21 applied to a fraction and ends up becoming one over
43:23 that fraction , which flips it over and makes it
43:25 very steep . Similar reasons is why this becomes very
43:29 shallow right here . But still , when we adjust
43:31 this base , even if it's in between zero and
43:33 one , we can change the shape of this exponential
43:35 curve as we make the base closer to zero .
43:38 This is much smaller than that one . It gets
43:40 much much much very very very steep . As we
43:43 get it closer to uh closer and closer to the
43:46 number one . Then of course it bends down as
43:48 we have seen it do in the the computer demo
43:53 and then finally we have the exponential function , it's
43:55 B to the power of X . The base has
43:57 to be bigger than zero , but it cannot be
43:59 equal to one . If the basis of fraction between
44:01 zero and one , you're gonna in general have a
44:03 sloping downward type of function that's big on the left
44:06 and small on the right . If you have a
44:09 base larger than one , it's the exact opposite ,
44:11 small on the left , big on the right .
44:13 Of course the exact shape depends on the value of
44:15 the base . If the base is exactly equal to
44:18 one , the function is not an exponential function anymore
44:20 , it's completely flat . And we talked about why
44:23 that's the case and that's why you cannot have a
44:25 base uh equal to one . And we just saw
44:27 on the computer demo if you actually got crazy and
44:30 put a negative value in for the base , then
44:32 you have this kind of like oscillating beast . That's
44:35 not an exponential function at all . That's why the
44:36 base can be negative . So I hope you understand
44:39 exponential functions . Burn these graphs into your mind please
44:44 because really and truly exponential functions is one of the
44:47 most important functions in all of math . We are
44:49 going to come back to it again and again and
44:51 again . And when you get into calculus and other
44:54 advanced subjects it's going to become even more critical .
44:56 So please make sure you understand this . Follow me
44:58 on to the next lesson . We're going to learn
45:00 how to solve equations that contain exponential functions .
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