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# Exponential function graph

Analyzing the features of exponential graphs through the example of y=5ˣ. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to graph y is
equal to 5 to the x-th power. And we'll just do this
the most basic way. We'll just try out
some values for x and see what we get for y. And then we'll plot
those coordinates. So let's try some negative
and some positive values. And I'll try to
center them around 0. So this will be my x values. This will be my y values. Let's start first with something
reasonably negative but not too negative. So let's say we start with
x is equal to negative 2. Then y is equal to
5 to the x power, or 5 to the negative
2 power, which we know is the same thing as 1 over 5
to the positive 2 power, which is just 1/25. Now let's try another value. What happens when x is
equal to negative 1? Then y is 5 to the
negative 1 power, which is the same thing as 1 over 5
to the first power, or just 1/5. Now let's think about
when x is equal to 0. Then y is going to be equal
to 5 to the 0-th power, which we know anything
to the 0-th power is going to be equal to 1. So this is going
to be equal to 1. And then finally,
we have-- well, actually, let's try a
couple of more points here. Let me extend this table
a little bit further. Let's try out x is equal to 1. Then y is 5 to the first power,
which is just equal to 5. And let's do one
last value over here. Let's see what happens
when x is equal to 2. Then y is 5 squared,
5 to the second power, which is just equal to 25. And now we can plot it to
see how this actually looks. So let me get some
graph paper going here. My x's go as low as negative
2, as high as positive 2. And then my y's go all the way
from 1/25 all the way to 25. So I have positive
values over here. So let me draw it like this. So this could be my x-axis. That could be my x-axis. And then let's make
this my y-axis. I'll draw it as neatly as I can. So let's make that my y-axis. And my x values, this
could be negative 2. Actually, make my
y-axis keep going. So that's y. This is x. That's a negative 2. That's negative 1. That's 0. That is 1. And that is positive 2. And let's plot the points. x is negative 2. y is 1/25. Actually, let me make
the scale on the y-axis. So let's make this. So we're going to go
all the way to 25. So let's say that this is 5. Actually, I have to do it a
little bit smaller than that, too. So this is going to
be 5, 10, 15, 20. And then 25 would be right where
I wrote the y, give or take. So now let's plot them. Negative 2, 1/25. 1 is going to be like there. So 1/25 is going to be really,
really close to the x-axis. That's about 1/25. So that is negative 2, 1/25. It's not going to
be on the x-axis. 1/25 is obviously
greater than 0. It's going to be really,
really, really, really, close. Now let's do this point here
in orange, negative 1, 1/5. Negative 1/5-- 1/5 on this
scale is still pretty close. It's pretty close. So that right over there
is negative 1, 1/5. And now in blue,
we have 0 comma 1. 0 comma 1 is going to
be right about there. If this is 2 and 1/2, that
looks about right for 1. And then we have 1 comma 5. 1 comma 5 puts us
right over there. And then finally,
we have 2 comma 25. When x is 2, y is 25. 2 comma 25 puts us
right about there. And so I think you see what
happens with this function, with this graph. The further in the
negative direction we go, 5 to ever-increasing
negative powers gets closer and closer
to 0, but never quite. So we're leaving 0, getting
slightly further, further, further from 0. Right at the y-axis,
we have y equal 1. Right at x is equal to 0,
we have y is equal to 1. And then once x starts
increasing beyond 0, then we start seeing what
the exponential is good at, which is just this
very rapid increase. Some people would call it
an exponential increase, which is obviously the
case right over here. So then if I just
keep this curve going, you see it's just going
on this sometimes called a hockey stick. It just keeps on
going up like this at a super fast rate,
ever-increasing rate. So you could keep going
forever to the left, and you'd get closer and
closer and closer to 0 without quite getting to 0. So 5 to the negative
billionth power is still not going
to get you to 0, but it's going to get you
pretty darn close to 0. But obviously, if you go to 5
to the positive billionth power, you're going to get
to a super huge number because this thing is just
going to keep skyrocketing up like that. So let me just draw
the whole curve, just to make sure you see it. Over here, I'm not actually on
0, although the way I drew it, it might look like that. I'm slightly above 0. I'm increasing above that,
increasing above that. And once I get into the
positive x's, then I start really,
really shooting up.