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# Square-root functions & their graphs

Sal is given a drawing with four graphs and four formulas of square-root functions. He uses transformations to match each formula with its appropriate graph. Created by Sal Khan.

Video transcript

Match each function
with its graph. And we have graph
D, A, B, and C. And let's just start
with the graph of B because, actually, this
one looks the closest to the square root of x, which
would look something like that. But it's clearly shifted. And it's flipped over
the horizontal axis. The fact that it's flipped over
the square horizontal axis, that means that we're taking
the negative square root. So it's going to be one of
these two cases right over here. We're shifting it down by 2. So we should have a negative
2, and both of these cases have a negative 2 here. And we're shifting it relative
to the square root of x. We're shifting it
to the right by 1. So if we're shifting
to the right by 1, what we see under the
radical should be x minus 1. And both of these have
an x minus 1 on it. So which of these is B? And which of these is C? And I encourage you
to think about that. Pause the video if you'd like. Well, the difference between
these two is this one has a scaling factor of 2 here
while this one doesn't. And of course, this
is going to turn into more and more
negative values. This is going to be
getting negative faster. And you see that graph C
here gets negative faster. And you could even
try some points. In either case, when
x is equal to 1, in either case, what we have
under the radical becomes 0. x minus 1 is 0. x minus 1
is 0 when x is equal to 1. And in either case, our
y value's negative 2. But then, you see,
as x increases, this one right over here
gets negative twice as fast. You see that right over here. This one right over here,
y has gone from negative 2 to negative 4 here. For graph C, y has gone from
negative 2 to negative 6. So it's gone down by 4. This has only gone down by 2. So it's pretty clear
that graph B corresponds to the one that doesn't
have the 2 out front. So this is graph
B right over here. And graph C
corresponds to the one that has a negative 2 out front. So let me try that. The negative 2 out front. So now we just have to think
about these two equations and match them to
these two graphs. And so both of
these, they haven't been flipped around
the horizontal axis. They've been flipped
around the vertical axis. And that's why whatever
we have under the radical we've essentially taken
the negative of it. And actually, we
could figure out which ones these are just by
looking at how much they've been shifted in the y direction
relative to the square root of x-- which would look
something like that. I know I can't draw on
this one right over here. Well, this one has
been shifted up by 2. D has been shifted up by 2. This one over here, g of x
has been shifted up by 2. And you could also see that it's
been shifted to the left by 1. If you're shifting
to the left by 1, normally, under the radical,
you would have x plus 1. But then, we flipped it over
around the vertical axis. And so that takes
the negative of that. That's why it's
negative x minus 1. You could view this as
the negative of x plus 1. So either way, that
is D. Throw that under the D. And just
deductive reasoning tells us A. This is
A, and it makes sense. We see it has shifted up by 1. So it's plus 1. And we could view this as
the negative of x plus 4. So it's been shifted
4 to the left. And we see that is
definitely the case relative to the
square root of x. We got it right.