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In this lesson, we dive deep into graphing exponential functions—a key concept in algebra. Whether you're learning for the first time or need a refresher, this video breaks it all down in a simple, relatable way.
What you'll learn:
How to recognize and graph exponential functions
Understanding the domain, range, and the role of asymptotes
Real-life examples of exponential growth (email chains, viral sharing)
Step-by-step graphing using key points
How transformations affect the graph: shifts, vertical stretches/shrinks, and reflections
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0:00
so what we're going to talk about you
0:01
weren't even here yesterday so skip both
0:04
we're going to talk about uh
0:07
graphing
0:09
exponential functions exponential
0:12
functions are functions that grow really
0:13
fast example let's say you know you ever
0:16
receive those emails where somebody goes
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hey i'm going to send you this email to
0:20
send you to five more people no no and i
0:23
mean it used to be something that was
0:24
used to happen forward you forward to
0:26
five more people right
0:28
so to figure out let's say you said you
0:31
send it to
0:33
like 10 people right 10 people do this
0:37
so how would you figure out how many
0:38
numbers of email was sent out it would
0:41
be what 5 to the^ of what 10 right so
0:45
the function that would best represent
0:47
that would be f ofx = 5 to the^ x
0:52
depending on the number of people that
0:53
are forwarding this email okay so
0:56
because you have if you have 10 people
0:58
doing this so you that means you send
1:00
about five to the power of 10 emails it
1:02
used to be something that people used to
1:03
do before or send this to five people or
1:05
something's going to happen to you today
1:06
i never bothered with it cuz it yeah
1:09
just chain stuff that didn't work for me
1:11
i'm not that one that never scares me
1:15
i could care less so that doesn't i'm
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not superstitious like that so i don't
1:20
really care i i just send it back to
1:22
them what they did that don't bother me
1:24
i don't need to send it okay so that's
1:26
that's the kind of step that we're going
1:28
to do today we're going to learn how to
1:29
graph these type of functions right now
1:32
uh an exponential function is always
1:34
going to be in the form f ofx= b to the
1:38
x b to the x okay now this function has
1:42
some specific things that we need to
1:43
know about the first thing that we need
1:46
to know is what is the domain of this
1:48
function and when i say domain of the
1:50
function i'm talking about for what
1:53
values does this function exist right so
1:58
if you have a function in the form b to
2:00
the x what are the values for which this
2:03
will exist i mean sort of what what is
2:07
it -21
2:10
i mean not here i mean right here the
2:12
domain will be what all real numbers
2:14
right every single number you put in
2:16
there you'll be able to evaluate this
2:18
function if you put 3 to the 100 you're
2:21
going to find it so any exponential
2:23
function in the form b to the x has for
2:26
domain all real
2:30
numbers now the good thing about this
2:32
exponential function is this what's the
2:34
range when the range of this function is
2:37
this when b is positive right when
2:39
there's no negative sign before b the
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range will be where all positive values
2:44
because we're going to have something
2:46
called asmtote right asmtote asmtote
2:48
yeah asintote is the point for which the
2:52
function does not cross over right so as
2:55
it's called an asmtote so
2:57
basically so you're going to see that a
2:59
lot in calculus right in calculus you're
3:01
going to see what we call an asmtote
3:03
okay so it's spelled a s
3:05
y b t o t e asintote right so basically
3:11
at this specific point the function is
3:14
like a limiting value a limiting it's
3:17
like a border you don't cross over that
3:19
right
3:21
so now how do we graph this function how
3:23
do you graph the exponential function so
3:26
let's take an example here if i graph i
3:29
want to graph y = 3 ^ x right 3 ^ x we
3:34
want to graph this so to graph this we
3:36
need to first find the domain of this
3:38
function the domain will be all real
3:39
numbers and then to graph it we're going
3:41
to choose some random points right now
3:46
for to keep to keep this easy to make it
3:50
easy i always want to choose like this
3:52
one here like -2 1 0 1 and two right
3:55
we're going to graph this and then we're
3:57
going to use those as your guiding line
3:58
to draw the function so for example when
4:01
x= -2 i'm going to plug it in here
4:05
y = 3 to the -2 we've already worked
4:08
with uh exponential um numbers right so
4:12
33 is basically what 1 over what 3 to
4:16
the 2 right which is 1 over 9 right
4:19
that's 1 n so this is 1 n so when x= -2
4:24
y is 1 9 and then when
4:27
x=1 it be 31 which is 1/3 right 1/3 here
4:33
and then when x is zero that's pretty
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one so because 3 to the 0 any number
4:38
raised to the 0 is one right and when x
4:40
is one pretty easy 3 to 1 is three and
4:44
when x is two is nine so with this four
4:47
point i can graph this exponential
4:49
function right i'm going to start by
4:50
okay when x is zero y is
4:54
1 when x is uh
4:57
-2 y is 1 n 1 9 is somewhere right here
5:01
somewhere close to that right when x is
5:04
uh
5:05
-1 y is 1/3 1/3 is going to be somewhere
5:08
here and when x is uh 1 y is three uh
5:14
one two and three somewhere here when x
5:17
is two y is nine so
5:20
two 3 3 4 5 6 7 8 9 somewhere here so
5:25
the function is going to look something
5:26
like this and like i say we have an
5:28
asmtote right so the function is always
5:32
going to get closer and closer to the
5:34
line this line y = x but it's never
5:37
going to cross it because that's your
5:38
borderline right it's never going to
5:41
cross this line but the function is
5:42
going to look like this this is called
5:43
an exponential function because it's
5:45
growing very fast you want exponential
5:48
meaning something that is exponential
5:50
growth something that grows bigger right
5:53
grows big so this is how the function is
5:55
going to look like so this is 3 to the x
5:58
okay so basically if you want to figure
6:00
out how many emails were sent by two
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people would be nine emails right man
6:07
three people that would be 27 if you use
6:10
that example of forwarding the emails
6:12
right so this function actually give you
6:15
um that type of like uh interpretation
6:17
okay now as example how would you graph
6:22
uh y = 4 to the x it would be pretty
6:25
much the same steps right y =
6:29
4x the domain of this function will be
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what all the numbers and then all you
6:35
have to do is just go through the same
6:36
step i will choose the same numbers -2 1
6:39
0 1 and two and then draw the function
6:42
okay now the problem comes when you have
6:45
like transformations okay
6:47
transformations and today i debated how
6:49
to teach this and i think i figure out
6:51
the best way to teach the
6:52
transformations for example if i have
6:55
this function here 2 to the x + one and
6:58
i want to graph it right you guys
7:00
remember what we did in a very like in
7:03
the second quarter no no well some of
7:07
the things that we did when we transform
7:08
a function right what do you see here 2
7:10
to the x + one so you're moving to the
7:12
left well this is the function you're
7:16
adding it do you are you adding this to
7:18
the x value or the whole function what
7:21
are we adding the one to the x value
7:23
only or the whole function here the
7:24
whole function so we're going to move on
7:26
what axis
7:29
up right so we're going to move up so
7:33
basically we're going to go one unit up
7:35
right so that's where we do the
7:36
transformation so let's say i wanted to
7:38
graph this function 2 to the x + one
7:41
right i'm going to go through the same
7:43
step i'm going to erase these here and
7:45
i'm going to do the same thing right so
7:47
i have 2 to the x and i want to graph
7:49
this 2 to the x + one right this is what
7:52
i want to graph so what i want to do is
7:54
this there's two ways to solve this we
7:56
can go the transformation way or we can
7:58
go the traditional way of just plugging
8:01
in stuff right so let me let me use the
8:04
same values here right so let's say i
8:06
want to use a transformation so i'm
8:07
going to start just graphing 2 to the x
8:10
okay i'm going to go when x is -2 right
8:13
this will be 1 2 -2 which is 1/4
8:18
right and then when x is -1 it' be what
8:24
12 i don't need to do all of that right
8:25
when x is zero is going to be one when x
8:28
is one will be two and when x is two be
8:30
what four right so i can put those on
8:34
here
8:39
i'm going to put those here and then
8:41
we're going to do something so when x is
8:43
-2 1/4 is somewhere here right when x is
8:49
-1 1/2 is in the middle right when x is
8:53
0
8:54
one x is one we got two
8:58
here uh and x is two we got four
9:01
somewhere here right so the function is
9:03
going to look like this but the the
9:06
question is this this is just 2 to the x
9:09
right but what am i trying to graph 2 to
9:13
the x + one so that means i'm going to
9:14
choose those key units and i'm going to
9:16
move them up how many units one unit
9:18
right so this guy is going to go up one
9:21
unit it's going to be right here this
9:23
one's going to go up one it's going to
9:24
be right here this one is going to go up
9:28
one up one and it's going to now the the
9:32
new graph is going to look like this
9:33
right so this is called a transformation
9:36
so you are basically transforming from 2
9:38
to the x you can transform it right by
9:41
just adding one unit okay i can do that
9:44
now what happens is it gets a little bit
9:47
more technical when we have more than
9:49
just one transformation
9:51
example if i have if i want to graph
9:54
this
9:55
graph right y = 2 to the x
10:00
-1 +
10:02
1 so how would you graph this 2x - 1 + 1
10:07
so how would you graph that function
10:11
move that function over right right did
10:17
you see what he says here and move that
10:19
function to the right or you can start
10:21
by doing what first what would you do
10:23
here once you graph 2 to the x what
10:26
would you do first move to the right one
10:28
unit up one and up one right so you
10:32
could do that with this you can move to
10:34
the right one unit and then move up one
10:36
because they now the transformation is
10:38
occurring at the x right so if he's up
10:41
here when he's negative we move to the
10:45
right and when he's positive we move to
10:47
the left yes what do you mean move to
10:50
the right all right so if i have 2 to
10:52
the x - one right this is the new
10:54
function if i give you this function you
10:56
have two transformations that are
10:58
happening okay so the first
11:01
transformation is this you start by
11:03
graphing 2 to the x right which we did
11:07
here we start by graphing 2 to the
11:10
x all right we start by graphing this
11:13
function and then now the next
11:15
transformation is this all the x values
11:18
what is happening to all the x values
11:20
subtract one right but we say when we
11:22
subtract we go to the right right so
11:26
this guy here is going to go one unit
11:28
here this one is going to go one here
11:30
this one's going to go here and this is
11:32
going to go here so your first
11:34
transformation is going to look like
11:35
this okay and then the next one because
11:38
we are adding one every single one of
11:41
those that moves is going to go up one
11:44
that's the last one right so you're
11:45
going to go up one up one
11:49
uh not here up one and then up one so
11:52
now the new graph is going to look like
11:54
this okay i know i'm using the same
11:56
marker that's why you can't see the
11:57
distinction but this is what we're doing
11:59
so these you have two transformations
12:01
here the first transformation is the x
12:03
values are going to change because this
12:05
is minus one you go to the right if it's
12:07
plus one you go to the left okay and
12:10
then the next transformation is the y
12:12
values on the horizontal is called a
12:14
vertical shift you're going to shift it
12:16
up one unit okay all right now there's
12:21
also something else that will occur and
12:24
i do want to talk about it but i'm not
12:25
sure if i should mention it now or save
12:27
it for tomorrow probably today i think
12:30
it's a good idea to state it no i'm not
12:33
listening
12:35
oh stop stop stop
12:38
stop stop
12:40
all right you know what i want to do is
12:42
i want to save this for
12:45
um
12:46
tomorrow
12:53
all right now let me mention this one
12:54
thing here
12:56
all right so now let's say i wanted to
12:58
graph um
13:05
12 right 2 to the
13:10
x minus 2 like
13:13
this
13:15
right so here's what i suggest we do
13:18
here right now you can see that there's
13:21
something that's happening with
13:23
our y values here right
13:27
there's no y well i'm saying whatever
13:30
there's a y there y is equal to this
13:33
right so i have 12 and the entire
13:35
function is being multiplied by what
13:39
no the entire function is being multiply
13:42
by what 12 right so here's what's going
13:45
to happen here
13:48
so it's one now before we do that right
13:51
let's just go ahead and graph this
13:53
function again and we're going to we're
13:54
going to do the transformations together
13:56
and figure out what's happening why did
13:58
multiplication stop being in x
14:01
because when we add in
14:04
right so let's start by graphing we're
14:07
going to start by graphing the function
14:09
that we know right pay attention here
14:11
because this is critical okay so we're
14:14
going to start here like i said again i
14:16
put in two i have a 1/2 here and the 1/2
14:20
is multiplying the entire function right
14:22
this is the function that we have and
14:24
the 1/2 is multiplying the entire
14:26
function so we're going to start again
14:28
by graphing 2 to the x okay so i'm going
14:31
to start by graphing 2 to the x so when
14:33
x is -2 1 2 y is 1/4 so right here
14:39
okay when x is -1 1/2 is right
14:44
here i put it a little bit up and then
14:47
when x is zero one is right here right
14:50
and then when x is one uh we have two
14:53
right here and when x is two we have
14:56
four three four somewhere right here
14:57
okay so this is the function that we
14:59
have right here this right here is 2 to
15:03
the x right that's just 2 to the
15:06
x all right are we good on that now the
15:10
next transformation is i'm doing what
15:12
after that
15:14
right i'm going to move to the right how
15:15
many units two unit right to the right
15:18
to the right so that means this guy's
15:20
going to go one and two right here one
15:23
and two right here one and two right
15:27
here and this is going to go one and two
15:30
right here right so we have
15:34
this right this is a new uh curve now
15:40
pay attention here what is happening i
15:43
got 1/2 right that's
15:46
multiplying all the y
15:49
values the x's are not changing right
15:53
only thing that's changing is the y
15:55
values here do you does that make sense
15:57
right so now what's going to happen is
15:59
this when x is two right now what's
16:02
going to happen is all the y values are
16:05
going to shrink by what 1/2 okay so
16:08
instead of um i ended up here now this
16:12
guy is going to end up where two two in
16:15
the middle somewhere right here right
16:17
two uh so this is a let me see one two
16:20
three four so it's going to end up right
16:21
here right where it was before no no no
16:24
you divide by multiply by 12 so it's
16:27
going to end up right here um
16:30
right and then this
16:33
time the x value stay the same yes sir
16:38
um you're um you're not dividing the x
16:42
by two the y by two yeah y by two or the
16:46
y by two yeah the y by two so if you
16:48
here four
16:49
right so this way it's going to go where
16:52
right here right so it's gonna
16:56
be right here now right because we have
17:01
two and two this is gonna go here and
17:03
then this guy because this is here and
17:06
then this was uh right
17:07
here now it's going to go in the middle
17:10
here so we we're going to have it right
17:12
here somewhere right and then this guy
17:14
here was here now it's going to go down
17:18
here and down here so now we're going to
17:21
have this
17:23
curve this is what you're going to get
17:25
right now the only reason why the reason
17:28
why this is a little confusing is
17:29
because i'm not using different colors
17:31
can you give us color pencils i can use
17:33
i can give you color pencils but what
17:35
you all you need to remember is when you
17:38
do the
17:39
transformations two to 2x to the two the
17:43
first transformation is you move two
17:45
unit to the to the uh right and then
17:47
next all the y values are going to be
17:50
divided by two the x stay the same i'm
17:53
going to repeat that again tomorrow so
17:54
that we can understand
17:58
all right so we're going to redo this
17:59
again tomorrow because i want to re
18:01
reemphasizes that okay so we're going to
18:03
do that again tomorrow so we're done
18:06
we're not done what are we what are we
18:08
doing now looks half dead