HELLO, Mr. Tarrou. Ok. In this video we are going to take a look at graphing exponential
functions. We are going to start off with this parent function of y equals two to the
x. Ok So I have a little t table here set up. I am just going to plug in some points
and graph it and talk about this when we have a picture to look at. Getting a little bit
of a cold so please excuse the throat there, and that will probably happen through the
video. Two to the negative one. Ok. So how do you plug in negative one as an exponent
and deal with that. Well, negative exponents... You know when you write two to the negative
one that is ok in your work, but as a final answer we don't like to have negative exponents.
Certain not when we are graphing on the x and y axis. So, two to the negative one is
going to be one over two. So when I take the value of negative one and plug it in for x
in this exponential function which is the form of y equals a to the x... Your book may
have a b there or something like that but there is just a base and a variable up in
the exponent. So two to the negative one is one half. Ok. Two to the zero power. Well,
two to the zero power is equal to one. Anything to the zero power is equal to one. Let me
go off on a tangent here today and explain why anything to the zero power is one. Lets
take a look at... I am going to change the color of chalk because I am going off on a
tangent. What is x to the fourth divided by lets say x to the first. Well, we know this
as four x's on the top and one x in the bottom. That is going to cancel out giving us x to
the third. So we remember that when we divide like bases we subtract the exponents. X to
the fourth divided by x to the first is x to the four minus one which is three. Ok..
How is that supposed to show that everything to the zero power is one. What if I change
that one to a four? What if I say what does this equal to? You know that anything divided
by itself is equal to one. But if I apply the exponent rules that you learned about
dividing like bases, you would subtract the exponents. Well, what is four minus four?
Zero. And while this is certainly not a proof it is an example of how x to the zero power
is one, and indeed anything to the zero power is equal to one. I like to show that through
the division, your exponent rule... when you divide like bases you subtract the exponents
to give kids a concrete example of why we always say that anything to the zero power
is equal to one. Sometimes that gets put off as "just memorize it" That is again not a
proof, but it is a good example. Ok, to the first power. Two to the first power is equal
to two. And two to the second power is four. Alright. We have negative one one half. I
intended on filling up these tick marks. Negative one, one half is there. Zero one is here.
One comma two is over one and up two. And then finally two, four is over two and up
four. You start to see this...uh... that would be five. You kind of see this curvature starting.
But if you are not there yet, lets try three. Two to the third power is eight. And we get
this exponential curve. And it looks something like that! I am going to call that our parent
function. Because I have about as easy of a base as I can make with this y equals two
to the x. Exponential....Is that exponential growth or exponential decay? Well this is
going to be when a is greater than one it is going to be growth. And when a is greater
than zero but less than one, it is going to be decay. Now notice that I did not say when
a is less than one. I did not say the base is less than one it is decay, because exponential
functions you cannot have a base which is negative. If your base is value is greater
than one like this two is, that is growth and when your base is between zero and one
that will be decay. That graph will come down to the right and we will do that as our last
example before we run out of time. Some other characteristics of this basic exponential
growth graph that we have here is that all exponential equations of y equals a to the
x goes through the point zero one. I am about to...I am hesitating a little bit because
I am about to do some transformations that we have learned earlier. Vertical and horizontal
shifts and reflections over the x and y axis. We are going to change a little bit of this
while we go. But, when you are looking at this basic, most basic form of exponential
growth or decay, all of those equations will pass....if there are no horizontal or vertical
transformations.... they will all pass through zero one. Also, lets talk about horizontal
asymptotes. All exponential growth and decay problems do have a horizontal asymptote. Now
again, in this most basic form of y equals a to the x...again growth and decay... They
will all have a horizontal asymptote of, again again again with this this very basic form,
they are all going to have a horizontal asymptote of y equals zero. Because this graph is coming
down and approaching the y value of zero or approaching the x axis, in this case as x
goes to negative infinity. So, a little bit about basic exponential, in this case growth,
function. Either growth or decay, they all pass through (0,1) if there has not been a
vertical or horizontal transformation...shift, or reflection. They all have a horizontal
asymptote of y equals zero until there has been a vertical shift. We are going to do
one of those in a second. Their domain is going to be all real numbers. It goes to negative
infinity to positive infinity. And the range of your basic exponential function y equals a to the x is going to be.... Where
is the lowest y value that this graph includes. Well it is going down and approaching zero.
It is not going to become zero, but it will approach it. I am using interval notation
because I use that in my precalculus class. As an inequality it will by y is greater than
zero for the range. Ok. So that is about as much as I can say about the simple graph of
y equals two to the x. I am going to leave this up here and erase some of these notes
that I talked about. Over here lets do a couple of quick sketches from what we know about
transformations. There is a video about transformations in my library of videos so if I move too fast
please look that up. How about y equals two to the x minus two. If you have a plus or
minus inside your function that is a shift left or right. And you can simply do another
t table to get new coordinates without knowing that this has a horizontal shift to the right
because there is a minus two inside the math function. But we now do know these transformations.
This point is going to get moved to the right places. This point is going to get moved to
the right two places again because of the minus two inside the function. To the right
two places. To the right two places.... Again the graph will look something like that for
y equals two to the x minus two. Of course now the graph is not going to pass through
(0,1) because I have pushed the parent function right two places with the minus two up in
the exponent. If I were to change that to y equals two to the x minus two. So, now I
am writing the minus two as a constant at the end of the function not up in the exponent,
not inside the math function that is in the equation. This is going to be a vertical shift
down. So again, the parent function of two to the x power with a minus two at the end,
not inside the math function not inside the exponent. This is going to move the graph,
the white parent function that I made with the table, down two places. This point is
going down two. here. This point is coming down two places. This point is going to come
down two places. And down two places. And I don't think I can do many more graphs on
grid here. We have got something that probably looks something like this. Now look at that
yellow line. Do you see that there? What is the horizontal asymptote of that graph? This
graph, this yellow graph is now going to come down and not approach zero like the white
graph did. This is approaching a y value of negative two, because I pushed the graph down
two places. So this graph is going to have a domain of all real numbers or negative infinity
to positive infinity. But unlike the parent function, the white graph, where the range
is from zero to infinity our new graph is going to have a range of negative two to infinity.
(y>-2) Vertical shift and Horizontal shift. What...lets just squeeze one more in here
before I move on to decay. What would y equal to two to the negative x look like? Well,
if you place a negative in front of your math function, again we did this in transformations
in uh...well in my math book it was 1.7... If you start off with a negative leading coefficient,
that is a reflection over the x axis. If you have a negative in front of your x that is
inside your math function itself...uh.. we are not leading with a negative but it is
two to the negative x, that is going to be a reflection about the y axis. So again the
white graph with this y equals two to the negative x, this going to be a reflection
around the y axis. So, my point here at (-1,0) is going to be at (1,0) This point is at (0,1)
and will still be at (0,1) This point is going to reflect over the y axis and be right here.
This point at (2,4) is going to be at (-2,4). And this point at (3,8) is going to be at
(-3,8). I was going to do a decay problem but I believe I am going to run out of time.
But that kind of is decay. Uh... Lets talk about this red line for a second since I will
not have time to make a decay graph before time runs out. What is happening here? Why
does this look like this graph is going down to the right? This reflection over the y axis
that we get. Well, if I rewrite this equation so that my exponent is no longer negative...
Don't you get rid of a negative exponent by flipping the base? So this can be rewritten
as y equals, the 2 has a negative exponent so I am going to take that two and move it
down to the bottom (denominator) or flip the base, and get one half to the x power. So
I inadvertently did a decay question. I have a base that is between zero and one. It is
decay. BAM!!! I am Mr. Tarrou. Go do your homework. I hope this helps you graph exponential functions