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Welcome to Crystal Clear Mathematics where it is easier than you think.
I'm your host, Graeme Henderson.
This particular video is a rather special and unique one.
In the last few videos I've been showing you how to draw the graphs of polynomial equations
(when they're factorised) by studying their roots.
And I've also shown you how to construct polynomial equations given their graph ...
by studying their roots and their y-intercept.
All of this skill has been based on our understanding of zeros.
That is, that when we have a product of a number of factors making zero,
then each of those factors may be zero.
We're going to use that information,
plus an understanding of another simple curve, and another principle (a third principle),
to construct what I hope will be a rather unique thing for you.
And that is, I'm going to attempt in this one video
and perhaps even without editing (as an uninterrupted video)
to construct the words "I LOVE YOU" on graph paper with just one equation.
So, follow me carefully if you will.
The first principle is this.
If I wish to construct an ellipse (a long, thin one),
that goes through a and -a, and b and -b, it has something like the equation for a circle,
x-squared plus y-squared equals radius-squared (I'm going to use 1).
And, the way we get x to go from a to -a, is to do this.
You can see that, when y is worth zero, when x is worth a we get
a-squared over a-squared which is one.
And, when x is -a, we get -a all squared which is plus a-squared over a-squared,
which is one also.
This is a rather clever way of distorting the x-axis.
And similarly, to distort the y-axis, we do this.
So, I think you can see that, when y is equal to b
(and x is zero because we're on the y-axis),
when y = b, we get b-squared over b-squared is one ... and the same occurs at -b.
So, the question is,
"How can I construct a line that appears to be a single, vertical straight line?"
Well, obviously, I need to bring these in, and I need a to be a very, very tiny number.
So, if a is a very tiny number, like one thousandth for example,
then you can see that, when I'm dividing by a fraction,
this denominator appears up the top so I would have something like
1000-squared, x-squared, plus y-squared over ...
(I haven't decided what I want to do with b yet ... let's make it go from 2 to -2) ...
then this would be 2-squared equals 1.
I can probably write this a little more compactly
by changing a thousand squared, which is a million, into 10-to-the-power-6,
because I'm going to need every bit of room I can get.
And, I think you'll agree that, if I bring the 1 over to this side,
that that also is a true form for the equation of this particular ellipse.
So, this ellipse now goes from -2 to +2
and from plus-one-thousandth to minus-one-thousandth,
so it's basically a line interval ... at least, to the naked eye ...
and that's what I'm trying to do [achieve].
So, that's step number one ... to create a line interval,
like the letter I or the descender on the letter L.
That's what we need.
The second thing is,
"How do we move graphs around the coordinate plane (the Cartesian plane)?"
The fact is that we have to distort the x-axis again,
simply by adding and subtracting values.
To move it down to here so the line would appear down here (for example) through -5,
our adjustment would be, not to do anything on the y-axis,
but simply to do this ... and this particular ellipse has now moved here.
So, we now have a stratagem, or a strategy, whereby we can create what appears to be a line interval
and we can move it. So far, so good!
Well, I did have to stop and clean the board, but let's continue.
I'm going to plan out, on a very long x-axis, where my letters will go.
I'm going to need quite a few gaps
... this is unplanned unrehearsed, and I've never done it before ...
it's just an idea I had a few days ago to share with you ... but let's try it.
Minus 2,3,4,5,6,7,8,9,10,11,12,13,14 ... we'll do that ... -14,-12,-10,-8,-6,-4,-2 ...
just marking every second one.
1,2,3,4,5,6,7,8,9,10,11,12,13 ...
Ok ...
Let's try [to] put ... we want "I LOVE YOU" ... we can probably get "LOVE" here ...
let's try to put the "E" there, for example.
Now I told you the "U" at this stage will have ... the "V" will have to look a bit like a "U."
There's our "O." There's our "L." Leave a gap of perhaps three ... "I LOVE."
Leave another gap of three between the words and we have Y ... O ... (it'll go up to 14) ... U.
And let's see if we can generate one equation that produces all of this material.
Now you can see that I'm going to have to write fairly small.
This vertical line ... y = (you're going to enjoy this) ...
this vertical line is that same ellipse that we were taking about before.
So, it's going to be 10^6 ...
and to move the x-value down to -11, it'll be (x+11)^2 ...
and it's going to be y^2 on 2^2 minus 1.
If this expression is zero
... sorry, I don't need that (you can see this is very unrehearsed) ...
but if this expression is zero, then it will construct that graph.
The next one, I want this line, and it's going to look the same ... 10^6
... going through -8, so it's (x+8)^2 plus y^2 on 2^2 minus 1.
If that expression there ... if that factor is equal to zero, it will create that line.
Let's do the next vertical line. This one goes through -5.
You can see this is going to get a little bit boring,
so I might actually speed this up on the camera [I chose not to].
The next vertical line is through -3 ... (x+3)^2 plus y^2 on 2^2 minus 1.
Where are we ... we've done that one, we've got the one through -2 ...
the one through zero (that's just going to be our ordinary vertical ellipse) ...
one through 1 ... I'll just do the long ones first ...
so, one through 8 ... if I can squeeze it in here we'll get the one through 9 ...
one through 11, 12 and 14.
That's got all those vertical ones done. Now, this one's a bit more tricky.
We want it to start down at the origin going from
... instead of +2 to -2 we just want a +1 to -1 ...
and we're going to shift it so its centre moves from the origin to there.
And it's going to look like ... same width (so the x-value [coefficient] won't change)
... it's going to move to x=6 ... the y-value is going to move up to y=1 ...
and (I'll write it in but it's going to go
from plus to minus one, so that's a bit redundant).
That has now got that descender there so, we've got all of our descenders in place.
Let's now deal with the horizontals.
If we want a horizontal, then we want the ellipse to have a very small vertical dimension
and to be quite long horizontally.
So, we're going to swap these two coefficients around
and I want it only to be two units long so x^2 will be over 1^1 (I'll write it in).
And this time we're going to have 10^6 y^2 minus one ...
but, I want it moved to here, for example, where the centre is x=-7 and y=-2.
So, I want (x+7)^2 and (y+2)^2. I hope that made sense to you.
I haven't really taught you how to do this yet.
I'll be creating some videos describing it.
And now we want to create these other horizontal lines.
So, we've got one centred at (-4, +2), so it's going to be (x+4)^2
... I won't write the 1^2 ... plus 10^6 (y-2)^2 -1.
I should be using parentheses and brackets, but I won't start now.
We'll just embed parentheses.
This one here (let's speed up) is going to be (x+4)^2 plus 10^6 (y+2)^2 -1,
which makes it go through (-4, -2). There it is.
This one here is through (-1, -2) so we're going to have (x+1)^2 + 10^6 (y+2)^2 -1.
We've done that one. We've still got a few to go.
This one here's at (2, 2) ... (x-2)^2 and 10^6 (y-2)^2 -1.
This one here at (2, -2) ... (x-2)^2 + 10^6 (y+2)^2 -1.
I'll come back to this one.
This one here has only moved up to x=7, y=0, so it'll be (x-7)^2 + 10^6 y^2 -1.
This one here at (7, -2) ... (x-7)^2 + 10^6 (y+2)^2 -1.
Three and a half to go!
This one here's at (10, 2) ... so, we've got ...
(x-10)^2 + 10^6 (y-2)^2 -1.
(10, -2) ... y ... x ... oh, I'm obviously getting tired
(this is a long equation),
ah ... (x-10)^2 ... that's correct ... plus 10^6 (y+2)^2 -1.
This one here is at (13, -2) so it'll be ... (x-13)^2 + 10^6 (y+2)^2 -1.
And, we've just got this one to do.
Now this one, the centre is going to be at (1.5, 0)
and it's going to be extended horizontally half a unit in each direction.
So (I think this'll just show on the video), we want ... (x-1.5)^2 ...
that will locate it correctly.
To get the right dimensions we want to divide by half-squared,
so I'm going to put a 4 out the front ... + 10^6 y^2 -1.
And I'll set it all equal to zero. That is one equation!
Now, why does it work?
Well, the principle is this: if I choose any particular x or y value,
every single one of these terms, every single one of these factors
has a value!
That is, the entire expression is fully defined for every x and y value.
That's important.
But, if you take any particular factor ...
because here we have a multitude of factors multiplying to make zero
... so, any one of them could be zero ... if you made that one zero,
then it creates (or it defines) a graph where x is 12 and y is zero
... 12 and zero ... and it creates that vertical line.
If we chose this one here, it's when x is -4 and y is -2
... -4, -2 ... and it creates this horizontal line.
So, every part of this equation has a job to do, and creates a separate little part of the graph.
And I hope you found that an interesting exercise.
I don't recommend it ... well, you can do what you like with it!
But it does show you the principle that, when you have an equation fully factorised,
each factor when it equals zero can create an interesting part of the curve for you.
Now, I know that's been a bit of a fumble.
I decided to do it so you could see me 'ad lib.'
I hope you've enjoyed the experience and learned from it.
There'll be an interesting worksheet created which will encourage you to draw graphs
like circles with crosses through [them] and all that sort of business.
If you're interested in doing that and interested in learning the skills,
then please look at the description below the video and download the worksheet.
It's a PDF file ... there's no charge.
Just have fun and enjoy your mathematics.
Thank you for watching.