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# Graphs of square-root functions

Sal graphs y=√x. Then, he shows a few examples of how we can shift and stretch the graphs of y=√x and y=x^2, and how the equations of those graphs look. Created by Sal Khan and CK-12 Foundation.

Video transcript

I think you're probably
reasonably familiar with the idea of a square root, but I
want to clarify some of the notation that at least, I always
found a little bit ambiguous at first. I
want to make it very clear in your head. If I write a 9 under a radical
sign, I think you know you'll read this as the square
root of 9. But I want to make one
clarification. When you just see a number
under a radical sign like this, this means the principal
square root of 9. And when I say the principal
square root, I'm really saying the positive square root of 9. So this statement right
here is equal to 3. And I'm being clear here because
you might already know that 9 has two actual
square roots. By definition, a square root is
something-- A square root of 9 is a number that, if
you square it, equals 9. 3 is a square root, but
so is negative 3. Negative 3 is also
a square root. But if you just write a radical
sign, you're actually referring to the positive square
root, or the principal square root. If you want to refer to the
negative square root, you'd actually put a negative in front
of the radical sign. That is equal to negative 3. Or if you wanted to refer to
both the positive and the negative, both the principal and
the negative square roots, you'll write a plus or
a minus sign in front of the radical sign. And of course, that's equal to
plus or minus 3 right there. So with that out of the way,
what I want to talk about is the graph of the function, y
is equal to the principal square root of x. And see how it relates to the
function y is equal to x-- Let me write it over here because
I'll work on it. See how it relates to y
is equal to x squared. And then, if we have some time,
we'll shift them around a little bit and get a better
understanding of what causes these functions to shift up
down or left and right. So let's make a little value
table before we get out our graphing calculator. So this is for y is equal
to x squared. So we have x and y values. This is y is equal to the
square root of x. Once again, we have x and
y values right there. So let me just pick some
arbitrary x values right here, and I'll stay in the
positive x domain. So let's say x is equal
to 0, 1-- Let me make it color coded. When x is equal to 0, what's
y going to be equal to? Well y is x squared. 0 squared is 0. When x is 1, y is 1 squared,
which is 1. When x is 2, y is 2 squared,
which is 4. When x is 3, y is 3 squared,
which is 9. We've seen this before. And I could keep going. Let me add 4 here. So when x is 4, y is
4 squared, or 16. We've seen all of this. We've graphed our parabolas. This is all a bit of review. Now let's see what happens
when y is equal to the principal square root of x. Let's see what happens. And I'm going to pick some x
values on purpose just to make it interesting. When x is equal to 0, what's
y going to be equal to? The principal square
root of 0? Well it's 0. 0 squared is 0. When x is equal to 1, the
principal square root of x of 1 is just positive 1. It has another square root
that's negative 1, but we don't have a positive or
negative written here. We just have the principal
square root. When x is a 4, what is y? Well, the principal square
root of 4 is positive 2. When x is equal to
9, what's y? When x is equal to 9,
the principal square root of 9 is 3. Finally, when x is equal to 16,
the principal square root of 16 is 4. So I think you already see how
these two are related. We've essentially just swapped
the x's and the y's. Well, these are the same x and
y's, but here you have x is 2, y is 4. Here x is 4, y is 2. 3 comma 9, 9 comma 3. 4 comma 16, 16 comma 4. And that makes complete sense. If you were to square both sides
of this equation, you would get y squared is equal
to x right there. And, of course, you would want
to restrict the domain of y to positive y's because this can
only take on positive values because this is a principal
square root. But the general idea, we just
swapped the x's and y's between this function and this
function right here, if you assume a domain of positive
x's and positive y's. Now, let's see what the
graphs look like. And I think you might already
have a guess of-- Let me just graph them here. Let me do them by hand because
I think that's instructive sometimes before you take out
the graphing calculator. So I'm just going to stay in
the positive, in the first quadrant here. So let me graph this first. So we have the point 0, 0, the
point 1 comma 1, the point 2 comma 2, which I'm going to have
to draw it a little bit smaller than that. Let me mark this is 1, 2, 3. Actually, let me do
it like this. Let me go 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. That's about how far
I have to go. And then I have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. That's about how far I have to
go in that direction as well. And now let's graph it. So we have 0, 0, 1, 1,
2 comma 1, 2, 3, 4. 2 comma 4 right there. 3 comma 9. 3 comma 5, 6, 7, 8, 9. 3 comma 9 is right
about there. And then we have 4 comma 16. 4 comma 16 is going to
be right above there. So the graph of y is equal
to x squared, and we've seen this before. It's going to look something
like this. We're just graphing it in the
positive quadrant, so we get this upward opening
u just like that. Now let's graph y is equal
to the principal square root of x. So here, once again,
we have 0, 0. We have 1 comma 1. We have 4 comma 2. 1, 2, 3, 4 comma 2. We have 9 comma 3. 5, 6, 7, 8, 9 comma 3
right about there. Then we have 16 comma 4. 16 comma 4 is right
about there. So this graph looks like that. So notice, they look like
they're kind of flipped around the axes. This one opens along the y-axis,
this one opens along the x-axis. And once again, it makes
complete sense because we've swapped the x's and the y's. Especially if you just consider
the first quadrant. And actually, these are
symmetric around the line, y is equal to x. And we'll talk about things like
inverses in the future that are symmetric around the
line, y is equal to x. And we can graph this better
on a regular graphing calculator. I found this on the web. I just did a quick web search. I want to give proper credit
to the people whose resource I'm using. So this is my.hrw.com/math06. You could pause this video. And hopefully, you should
be able read this. Especially if you're looking
at it in HD. But let's graph these
different things. Let's graph it a little
bit cleaner than what I can do by hand. And actually, let me have some
of what I wrote there. So that should give you-- OK. So let's first just graph
y is equal to x squared. And then in green, let me
graph y is equal to the square root of x. They have some buttons here on
the right, just so you know what I'm doing. I have some buttons here on
the right: squared and the radical sign and all of that. Let me just focus on this. So let me just graph those. So first it did x squared
and then it did the square root of x. Look, if you just focus on the
first quadrant right here, you see that you get the exact same
result that I got over there, although mine
is messier. Now, just for fun and, you know,
I really didn't do this yet with the regular
quadratics, let's see what happens. What we need to do to shift
the different graphs. So with x squared, I'm going
to do two things. I'm going to scale the graphs
and I'm going to shift them. So that's x squared. So let's just focus on the x
squared and see what happens when we scale it. And then I'll do it with the
radical sign as well. This will really work
for anything. Let's see what happens when you
get 2 times-- no, not 2 squared --2 times x squared. And let's do another one that is
1.5 times our 0.-- I could just do 0.4 actually. 0.5 times x squared. Let's graph these right there. So x squared. So notice, our regular x
squared is just in red. If we scale it by 2, it's still
a parabola with the vertex at the same place, but
we go up faster in both directions. And if we have 0.5 times x
squared, we still have a parabola, but we go up
a little bit slower. We have a wider opening u
because our scaling factor is lower than 1. So that's how you kind of decide
how wide or how narrow the opening of our
parabola is. And then if you want to shift
it to the left or the right, and I want you to think
about why this is. So that's x squared. Let's say I want to just take
the graph of x squared and I want to shift it four
to the right. What I do is I say, x minus 4. x minus 4 squared. And if I want to shift it two
to the-- Let's say I want to shift it two to the
left. x plus 2 squared, what do we get? Notice it did exactly what I
said. x minus 4 squared was shifted four to the right. x plus 2 squared, was shifted
two to the left. And it might be unintuitive at
first, this shifting that I'm talking about. But really think about
what's happening. Over here, the vertex is
where x is equal to 0. When you get 0 squared
up here. Now over here, the vertex
is when x is equal to 4. But when x is equal to 4,
you stick 4 in here, you get 4 minus 4. So you're still squaring 0. 4 minus 4 is 0 and that's
what you're squaring. Over here, when x is equal to
negative 2-- negative 2 plus 2 --you are squaring 0. So, in other words, whatever
you're squaring, that 0 is equivalent to 4 here. Or 4 is equivalent to 0. And negative 2 is equivalent
to 0 over there. So I want you to think about
it a little bit. Another way you could think
about it, when x is equal to 1, we're at this point
of the red parabola. But when x is equal to 5 on
the green parabola, you have 5 minus 4. Inside of the parentheses you
have a 1, just like x is equal to 1 over here, up here. So you're at the same point
in the parabola. So I want you to think about
that a little bit. It might be a little
non-intuitive that you say minus 4 to shift to the
right and plus 2 to shift to the left. But it actually makes
a lot of sense. Now, the other interesting
thing is to shift things up and down. And that's actually pretty
straightforward. You want to shift
this curve up. Let's say we want to shift the
red curve up a little bit. You do x squared plus 1. Notice it got shifted up. If you want this green curve to
be shifted down by 5, put a minus 5 right there. And then you graph it and it
got shifted down by 5. If you want it to open up a
little wider than that, maybe scale it down a little bit. Scale it down and let's
say 0.5 times that. So now the green curve will be
scaled down and it opens slower, it has a
wider opening. And the same idea can
be done with the principal square roots. So let me do that. Let me do the same idea. And the same idea actually, can
be done with any function. So let's do the square
root of x. And in green, let's do
the square root of x. Let's say, minus 5. So we're shifting it over
to the right by 5. And then let's have the square
root of x plus 4. So we're going to shift
it to the left by 4. Let's shift it down by 3. And so lets graph
all of these. The square root of x. Then have the square
root of x minus 5. Notice it's the exact same thing
as the square root of x, but I shifted it to
the right by 5. When x is equal to 5, I have
a 0 under the radical sign. Same thing as square
root of 0. So this point is equivalent
to that point. Now, when I have the square root
of x plus 4, I've shifted it over to the left by 4. When x is negative 4, I have
a 0 under the radical sign. So this point is equivalent
to that point. And then I subtracted 3, which
also shifted it down 3. So this is my starting point. If I want this blue square root
to open up slower, so it'll be a little
bit narrower, I would scale it down. So here, putting a low number
will scale it down and make it more narrow because we're
opening along the x-axis. So let me to do that. Let me make this green one--
Let me open up wider. So let me say it's 3 times the
square root of x minus 5. So let's graph all of these. So notice, this blue one now
opens up more narrow and this green one now opens up
a lot, I guess you could say, a lot faster. It's scaled up. Then we could shift that one
up a little bit by 4. And then we graph it
and there you go. And notice when we graph these,
it's not a sideways parabola because we're
talking about the principal square root. And if you did the plus or minus
square root, it actually wouldn't even be a valid
function because you would have two y values for
every x value. So that's why we have to just
use the principal square root. Anyway, hopefully you found
this little talk, I guess, about the relationships with
parabolas, and/or with the x squared's and the principal
square roots, useful. And how to shift them. And that will actually be really
useful in the future when we talk about inverses
and shifting functions.