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- 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.