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In my last video, I demonstrated to you
how you use your knowledge of the roots of a fully factorised polynomial function
to graph that function.
I mentioned that there were three variations on the theme
and I was going to deal with them all in one video but I realise that's going to take too long —
so this is the first of three videos dealing with those three variations.
And the theme for today's video is "what happens if one of those factors
(or more than one of those factors in the polynomial)
appears to be reversed?" —
particularly if there's a minus sign in front of the x-value.
And to do so I want to first of all discuss a simple quadratic equation.
That is, y = (x+2)(4-x)
We can see that the first term is exactly as we like it, with the x-value first,
but the second term is not.
It has the x-value last and it has a minus sign in front of it.
Now, if you're used to finding roots, this is perfectly acceptable,
but I want to show a particular principle to you and for that we need to tidy up this factor.
We leave the first factor untouched, but the second factor we have to factorise further —
and we notice that there's a -1 coefficient in front of the "x" and we remove that.
Now, normally, we don't write the "1" but we remove that from this term.
-1 times x gives the -x and -1 times the -4 gives us the +4.
Now, you can see that removing the negative one (or the minus one)
has had the effect of reversing the factor.
Now, this term is multiplied by the -1 which is multiplied by this term.
So, we can, in fact, remove the -1 and place it out the front.
Now, of course, we don't write the "1" — but we do need the minus sign.
And what we've successfully done in a couple of lines is —
we've rearranged the equation so that the term that appears to be reversed
is now in the form we like, but a minus sign has appeared at the front.
It's this minus sign that I want to talk about in terms of graphing.
But, first of all, let's find the roots of this equation.
We set "y" equal to zero to find the roots.
And the roots must be when this term (or this factor) is zero, at x = -2,
and when this factor is zero when x = 4.
You could have seen that back here and even here, when 4 minus 4 makes zero.
We now graph the quadratic equation.
A root at negative two; a root at plus four; but what does this minus sign do?
Well, when we expand this quadratic equation the largest term is going to be the x squared term,
and it's going to have a minus sign in front of it.
There will be other terms but, as "x" gets very, very large,
this term will dominate.
And you can see that, as "x" gets very, very large,
x squared gets huge and the minus [sign] in front makes it a huge negative number.
In other words, the further we go towards large positive values of "x",
the larger a negative value y takes on,
and, therefore, the graph goes downwards at this end.
As "x" increases, y decreases.
Now, the rest of the graphing takes place as normal;
that is, the graph has to come back through this root
and you can see that we have an upside down parabola
because that minus sign has had the effect of making all the positive values negative
and all the negative values positive for "y."
So, what was in the negative part of the y-axis has now come up to the top,
and what was in the positive part of the y-axis has now moved to the bottom.
Let's have a look at a cubic equation and see how that same principle applies.
So, let's try a cubic equation.
These two factors appear exactly as we would like them.
This one appears in reverse. So, we can take the minus sign out.
I'm going to put it straight at the front — leave that one untouched —
this is going to become x-2. You can just check that.
Minus "x" gives us the -x, and minus -2 gives us the +2.
To find the roots, we set y equal to zero,
and our roots will be at four, two and negative three.
When we graph the cubic equation, the roots are going to be
the four, two and negative three.
But remember this minus sign means
that the graph is going downwards when we go to the right (instead of upwards).
And that means the rest of the graph must go like this.
Be aware with polynomials, too, that
the closer together the roots the less distance the hump will rise,
the further apart the roots, the greater the loop will be.
Now, in the last video, we did graph a cubic equation
and you saw that its basic shape was that — and this, in fact,
is a complete vertical reversal of that graph.
Now, if you were very fast,
you would have noticed that this would also apply to straight lines —
that is, polynomials where there is only one term.
We could, of course, write this as -x+4 but we could also take out a negative sign
so we have -x and minus -4 is +4.
To find its root we set y equal to zero. So, the root is at four.
And, when we come to graph the line, there's the root at four.
If you understand about finding y-intercepts (when x is zero),
you will understand [that] the y-intercept is +4 but, even if you didn't know that,
you would know that the graph will head downwards to the right
(because of this minus sign).
And, in fact, that's exactly what it does. We call these negative gradients.
And there's one other thing you may have noticed. And that is ... that
you don't have to rearrange those factors anyway if you don't want to.
The fact that I rearranged them was to draw your attention to that minus sign.
So, let's draw a very complicated polynomial (2-x)(x+4)(x-3)(7-x)(-2-x).
This has five factors, so it's going to be an x to the power five polynomial.
But we could simply find the roots immediately.
I'm not going to be able to line my "=" up so I'll write the roots here.
There's going to be a root when x is two (because two minus two makes zero),
at minus four, at three, at seven,
and here x would have to be -2 (so -2 minus -2 would make zero).
And if you're happy reading the roots from factors like these
then there's no need to tidy it up too much.
And, if we wish to graph this polynomial, there's a root at two, minus four,
at three, at seven (I didn't leave room), and at minus two.
Now, how do I know whether the graph goes up or down at the end
(because I haven't bothered rearranging it)?
All I do is I look at the x terms as I multiply along —
and I've got quite a few minus "x"s.
I have a (-) ? (+) ? (+) ? (-) ? (-). I have three "-x"s and, when you multiply
three negative numbers together you get a negative.
So if we rearranged all of this
we would have had a minus sign at the front and know that the graph would go downwards
and we've managed to do that without the rearranging.
And, of course, the rest of the graph would follow like so.
So, this is a very powerful graphing tool — and THAT is variation number one!
Watch out for your minus signs!