And polynomial functions I would be looking at functions of X
that represent polynomials of varying degrees, including
cubics and quadratics and
Quartics. We will look at some of the properties of these
curves and then we will go on to look at how to deduce the
function of the curve if given the roots. So first of all, what
is a polynomial with a polynomial of degree add is a
function of the form F of X.
Equals a an X to the power N plus AN minus One X to the power
and minus one plus, and it keeps going all the way until we get
to. A2 X squared plus a 1X plus a
zero and here the AA represents real numbers and
often called the coefficients.
Now, this may seem complicated at first sight, but it's not and
hopefully a few examples should convince you of this. So, for
example, let's suppose we had F of X equals 4X cubed, minus
three X squared +2.
Now this is a function and a polynomial of degree three.
Since the highest power of X is 3 and is often called a cubic.
Let's suppose we hard F of X equals X to the Power 7 -
4 X to the power 5 +
1. This is a polynomial of degree Seven. Since the highest
power of X is A7, an important thing to notice here is that you
don't need every single power of X all the way up to 7. The
important thing is that the highest power of X is 7, and
that's why it's a polynomial
degree 7. And the final example F of X equals.
Four X squared minus two X minus 4.
So polynomial of degree 2 because the highest power of X
is a 2. Less often called a
quadratic. Now it's important when we're thinking about
polynomials that we only have positive powers of X, and we
don't have any other kind of functions. For example, square
roots or division by X.
So for example, if we had.
F of X equals
4X cubed. Plus the square root of X minus one.
This is not a polynomial because we have the square root of X
here and we need all the powers of X to be positive integers to
have function to be a polynomial. Second example.
If we had F of X equals 5X to the Power 4 - 2 X squared plus 3
divided by X. Once again this is not a polynomial and this is
because, as I said before, we need all the powers of X should
be positive integers and so this 3 divided by X does not fit in
with that. So this is not a
polynomial. OK, we've already met some basic polynomials, and
you'll recognize these. For example, F of X equals 2 is
as a constant function, and this is actually a type of
polynomial, and likewise F of X equals 2X plus one, which is a
linear function is also a type of polynomial, and we could
sketch those. So the F of X.
And X and we know that F of X equals 2 is a horizontal line
which passes through two on the F of X axis and F of X equals
2X plus one.
Is straight line with gradient two which passes through one.
On the F of X axis, stuff of X equals 2 and this is F of X.
Equals 2X plus
one. Important things remember is that all
constant functions are horizontal straight lines
and all linear functions are straight lines which
are not horizontal.
So let's have a look now at some
quadratic functions. If we had F of X equals X squared, we know
this is a quadratic function because it's a polynomial of
degree two and we can sketch. This is a very familiar curve.
If you have F of X on the
vertical axis. An X on our horizontal axis. We
know that the graph F of X equals X squared looks
something like this.
This is F of X equals
X squared. So what happens as we vary the coefficient of X
squared? That's the number that multiplies by X squared, so we
had, for example, F of X equals 2 X squared. How would that
affect our graph? What actually happens is you can see that all
of our X squared values are the F of X values have now been
multiplied by two and hence stretched in the F of X
direction by two.
So you can see we actually get.
Occurs that looks something like
this. And this is F of X equals 2 X squared.
And likewise, we could do that for any coefficients. So if we
had F of X equals 5 X squared.
It would be stretched five times from the value it was
when it was X squared, and it would look something like
this.
So this would be F of X equals 5 X squared.
So this is all for positive coefficients of X squared. But
what would happen if the coefficients were negative?
Well, let's have a look.
It starts off with F of X equals minus X squared. Now,
compared with F of X equals X squared would taken all of
our positive values and we've multiplied them all by
negative one. So this actually results in a
reflection in the X axis because every single value is
now, which was positive has become negative. So F of F of
X equals minus X squared will look something like this.
So this is F of X equals minus X squared.
Likewise, F of X equals minus two X squared will be a
reflection in the X axis of F of X equals 2 X squared, and
so we get something which looks like this.
F of X equals 2 X squared 3 - 2 X squared and Leslie are going
to look at F of X equals minus five X squared and you may well
have guessed by now. That's a reflection of F of X equals 5 X
squared in the X axis, which gives us a graph which would
look something like.
This.
We F of X equals minus five X squared.
And in fact, it's true to say that for any polynomial, if you
multiply the function by minus one, you will always get the
reflection in the X axis.
I will have a look at some other quadratics and see what happens
if we vary the coefficients of X as opposed to the coefficient of
X squared. And to do this I'm going to construct the table.
So we'll have RX value at the
top. And the three functions I'm going to look at our X squared
plus X. X squared
plus 4X. And X squared plus 6X.
And for my X file use, I'm going to choose values of minus 5 - 4
- 3 - 2 - 1 zero. I'm going to go all the way up to
a value of two.
Now for each of these functions, I'm not necessarily going to
workout the F of X value for every single value of X. I'm
just going to focus around the turning point of the quadratic,
which is where the quadratic
dips. So just draw some lines in on my table.
On some horizontal lines as well.
And now we can fill this
in. So for RF of X equals X squared plus X, I'm just going
to focus on minus 2 + 2. So when I put X equals minus two in here
we get minus 2 squared, which is 4 takeaway two which gives Me 2.
When we put X is minus one in we get minus one squared, which is
one. Take away one which gives
us 0. Report this X is zero in here we just got 0 + 0 which is
0. I put X is one in here. We've got one plus one which
gives Me 2.
And finally, when I pay taxes 2 in here, the two squared, which
is 4. +2, which gives me 6 and in fact just for symmetry we
could put X is minus three in here as well, which gives us
minus 3 squared 9.
Plus minus three. Sorry, which gives us 6.
OK, for my next function, X squared plus 4X.
I'm going to look at values from minus four up to 0.
So if I put minus four in here, we get 16 takeaway 16
which gives me 0.
Put minus three in here.
We get 9 takeaway 12 which gives us minus three.
For put minus two in here, we get four takeaway 8 which is
minus 4. If I put minus one in here, we get one takeaway 4
which is minus three, and if I put zero in here 0 + 0 just
gives me 0.
And Lastly, I'm going to look at F of X equals X squared plus 6X.
And I'm going to take this from minus five all the way up to
minus one. So we put minus five in first of all minus 5 squared
is 25. Take away 30 gives us
minus 5. Minus four in here 16 takeaway 24 which
gives us minus 8.
For X is minus three in. Here we get 9 takeaway 18, which is
minus 9. For X is minus two in.
We get minus two square root of 4 takeaway 12, which is minus 8.
And we put minus one in. We get one takeaway 6 which is minus 5.
And you can actually see the symmetry in each row here, to
show that we've actually focused around that turning point we
talked about. So let's draw the graph of these functions.
So in a vertical scale will
need. F of X and that's going to take us from all those values
minus nine up 2 - 6.
So. If we do minus
9. Minus 8 minus Seven 6 - 5.
For most 3 - 2 -
1 zero. 12345 and we can just about squeeze 6 in
at the top.
On a longer horizontal axis.
We've gone from minus five to
two. So we just use one
2. Minus 1 - 2 - 3 -
4. And minus 5.
So first of all, let's look at our function F of X equals X
squared plus X. We've got minus three and six.
Which is appear.
We've got minus two and two.
Which is here.
Minus one and 0.
We've got zero and zero.
One and two.
And two 16.
And you can see clearly that this is a parabola
which, as we expected, we can draw a smooth curve.
Through those points.
So this is F of X equals X
squared. Plus X.
Next, we're going to look at the function F of X equals X squared
plus 4X. And the points we had with minus four and zero.
Minus three and minus
three. Minus two and minus 4.
Minus one and minus three.
And zero and zero which is already drawn. And once
again you can see this is a smooth curve is a parabola
as we expected.
And this is F of X equals X squared plus
4X. And the final function we're going to look at F of X equals X
squared plus 6X. And so we've got minus 5 - 5.
So over here minus 4 - 8.
Which is down here.
Minus three and minus 9.
Minus 2 - 8.
And minus one and minus five and once again you can see smooth
curve in the parabola shape.
And this is F of X equals X squared plus 6X.
So we can see that as the coefficient of X increases from
here to one to four to six, the curve the parabola is actually
moving down and to the left.
But what would happen if the coefficients of X was negative?
Well, let's have a look and will do this in the same way as we've
just on the previous examples. And we look at a table, and this
time we'll look at how negative values of our coefficients of X
affect the graph. So we look at X squared minus XX squared minus
4X. And the graph of F of X equals X squared minus 6X.
Now as before, will.
Put a number of values in for X, but we won't use. All of them,
will only use the ones which are near to the turning point of the
quadratic, but the values will put in the range will be from
minus two all the way up to
five. Four and five.
I just put in the lines for
our table. OK, so for the first one, F of X equals X squared
minus X. We look at what happens when we substitute in X is minus
2. So minus 2 squared is 4 takeaway, minus two is the same
as a plus two, which gives us six when X is minus 1 - 1
squared, is one takeaway minus one is the same as a plus one
which just gives us 2.
Zero is just zero takeaway
00. When we put one in, we've got 1. Take away, one which is
0. And two gives us 2 squared four takeaway two which gives us
2. And again, for symmetry will
do access 3. So 3 squared is 9 takeaway, three is 6.
So our second function F of X equals X squared minus 4X. We're
going to look at going from zero up to five.
And we put zero and we just get zero takeaway 0 which is 0.
When we put X is one.
We get one takeaway 4 which is
minus 3. When we put X is 2, two squared is 4 takeaway
eight, which gives us minus 4.
When X is 3, that gives us 3 squared, which is 9 takeaway 12
which gives us minus three and you can see the symmetry
starting to form here. Now a Nexus 4 gives us 4 squared 16
takeaway 16 which as we expect it is a 0.
And for our final function, FX equals X squared minus six X,
we're going to look at coming from X is one all the way up to
X equals 5, X equals 1, gives US1 takeaway six, which gives us
minus 5. X equals 2 gives us 2 squared. Is 4 takeaway 12
which gives us minus 8.
I mean for taxes, three in three squared is 9 takeaway 18, which
gives us minus 9.
X is 4 + 4 squared.
Take away 24. Which 16 takeaway 24 which is minus
8. I'm waiting for X is 5 in we get 25 takeaway 30, which
is minus five and once again symmetry as expected. So as
with our previous examples we want to draw this graphs
of these functions so we can see what's going on as our
value of the coefficients of X or Y is changing.
So if we got F of X on our vertical axis, this time we're
going from Arlo's values, minus nine. Our highest value is 6.
So we're going from minus 9 - 8
- 7. 6 - 5 - 4 - 3 - 2
- 1 zero.
12345 I will just squeeze in sex and are horizontal
axis. We've got one.
2. 3. Four and five letter VRX axis, and we
go breakdown some minus 2 - 1 - 2. So first of all, let's look
at the graph F of X equals X squared minus X. So minus two
and six was our first point, so minus two and six, which is
here. And we've got minus one and two which is here.
We've got the origin 00.
And we've got 10.
22
I'm 36 And as we expected, you can see we
can join the points of here. Let's make smooth curve which
will be a parabola.
So this is F of X equals X squared minus X.
Our second function was F of X equals X squared minus 4X. So
first point was 00, which we've
got. One and negative 3.
Say it. We've got two and minus 4.
Just hang up three and negative
3. Which is here.
And four and zero, which is here and straight away. We can see a
smooth curve which is a
parabola. Coming through those points.
And we can label up this is F of
X. Equals X squared minus 4X.
Lost function to look at is the function F of X equals X squared
minus 6X. So our first point was one and minus five, which is
here. We are two and negative 8.
Yep. Three and minus 9.
4 - 8.
And five and minus five, which is hit, and once again we can
just draw a smooth curve through
these points. To give us the parabola we wanted, this is F
of X equals X squared minus
6X. So what's happening as the coefficients of X is getting
bigger in absolute terms. So for instance, we can from minus one
to minus four to minus six, and we can see straight away that
the actual graph this parabola is moving down and to the right
as the coefficients of X gets bigger in absolute terms.
And that's where the coefficients of X is negative.
OK, so we know what happens when we varied coefficients of X
squared and we know what happens when we vary the coefficients of
X and that's both of them. For quadratics, what happens when we
vary the constants at the end of a quadratic well? Likewise table
of values is a good way to see what's going on.
So this time I'm going to use X again. I'm going to go from
minus two all the way up to +2, so minus 2 - 1 zero one and two,
and for my functions I'm going to use X squared plus X.
X squared plus X
plus one. X squared plus X
+5. And X squared plus X minus 4.
Now this table is
particularly. Easy for us to workout compared with the other
ones because we have a slight advantage in that we have a Head
Start because we already know the values for X squared plus X
that go here and we can just take them directly from Aurora,
the table and you'll remember that they actually gave us the
values 200, two and six.
And in fact, we can use this line in our table to help us
with all of the other lines. The second line here, which is X
squared plus X plus one.
His only one bigger than every value in the table before. So
all we need to do.
Is just add 1 answer every value from the line before, so this
will be a 311, three and Seven.
Likewise, for this line, when we've got X squared plus X +5.
All this line is just five bigger than our first line, so
we can just 7.
557 and
11. And likewise, our last line is minus four in the
end with exactly the same thing before hand, so it's just take
away for from every value from our first line, which gives us
minus 2. Minus 4 - 4 - 2 and
2. So let's put this information on a graph now so we can
actually see what's happening as we vary the constants at the end
of this quadratic. So you put F of X and are vertical scale so
that we need to go from minus four all over to 11 so it starts
at minus 4 - 3.
Minus 2 - 1.
0123456789, ten and we can just
about squeeze already
left it. I no longer horizontal axis. We're
going from minus 2 + 2 - 1 - 2 plus one
and +2. So first graph is the graph of the function F of X
equals X squared plus X, so it's minus two and two.
Which is a point here, minus one and 0.
So first graph is the graph of the function F of X equals X
squared plus X, so it's minus two and two.
Which is a .8 - 1
and 0. Because the origin next 00 at the .1 two
and we got the .26.
For our next graph, actually sorry, now we should.
Draw a smooth curve for this one
first. And Labor Lux F of X equals X squared plus X.
For an X graph which is F of X equals X squared plus X plus
one, we need to put the following points minus two and
three. So minus one and one you can see straight away
that all of these points are just want above what we
had before zero and one.
One and three.
Two and Seven.
So you can imagine our next parabola that we draw
through these points.
Is exactly the same but one above previous, that's F of X
equals. X squared plus X plus
one. So what about F of X equals X squared plus X +5?
Well, let's have a look minus two and Seven.
Gives us a point up here.
Minus one and five.
Zero and five.
One and Seven.
And two and 11.
So we draw a smooth curve through these points.
See, that gives us a parabola and this is F of X equals X
squared plus X +5.
So finally we look at the function F of X equals X squared
plus X minus four. So we've got minus two and minus 2.
Minus 1 - 4.
0 - 4.
One 1 - 2.
Two and two.
I once again we can draw a smooth curve.
Through these points and this is F of X equals X squared plus
X minus 4.
So we can see that what's actually happening here is from
our original graph of F of X equals X squared plus X, and if
you like, you could have put a plus zero there. And when we
added one, it's moved one up when we added five, it's moved 5
or. And when we took away four, it moved 4 down. So it's quite
clear to see the effect that the constant has on our parabola.
When looking at the graph of a function, a turning point is the
point on the curve where the gradient changes from negative
to positive or from positive to
negative. And we're thinking about polynomials. Are
polynomial of degree an has at most an minus one turning
points. So for example, a quadratic of degree 2.
Can only have one turning
points. And if we draw just a sketch of a quadratic, you can
see this point here would be at one turning point.
We think about cubics, obviously cubic as a polynomial of degree
3, so that can have at most two turning points, which is why the
general shape of a cubic looks something like this.
We see the two turning points are here and here.
But as I said, it can have at most 2. There's no reason it has
to have two, and a good example is this of this is if you look
at F of X equals X cubed.
And in fact, that would look if I do a sketch over here.
F of X&X that would come from the bottom left.
Through zero and come up here
like this. That's F of X equals X cubed, and so we can see that
the function F of X equals X cubed does not have a turning
point. Another example, if we were looking at a quartic
curve I a polynomial of degree four, we know can
have up to at most three turning points, which is why
the general shape of a quartic tends to be
something like this.
With the three, turning points are here, here and here.
Now let's suppose I had the function F of X equals.
X minus a multiplied by X minus B.
Now to find the roots of this function, I want to know the
value of X when F of X equals 0.
So when F of X equals 0, we have 0 equals X minus a
multiplied by X minus B.
So either X minus a equals 0.
Or X minus B equals 0, so the roots must be X
equals A. Or X equals
B. Now we can use the converse of this and say that if
we know the roots are A&B, then the function must be F of X
equals X minus a times by X minus B or a multiple of that,
and that multiple could be a constant. Or it could be in fact
any polynomial we choose. So for example, if we knew that the
roots were three and negative.
2. We would say that F of X would be X minus 3
multiplied by X +2 or multiple so that multiple
could be three could be 5, or it could be any polynomial.
OK, another example. If my roots were one 2, three
and four. Then my function would have to be F of X equals X
minus One X minus 2.
X minus three.
At times by X minus four, or it could be a multiple of that, and
as I keep saying that multiple could be any polynomial.
Right so Lastly, I'd like to think about the function F of X.
Equals X minus two all
squared. Now if we try to find the roots of this function I
when F of X equals 0, look what happens. We had zero equals and
I'll just rewrite this side as X minus two times by X minus 2.
Which means either X minus two must be 0.
Or X minus 2 equals 0 the same thing. So X equals
2. Or X equals 2. So there are two solutions and two
routes at the value X equals 2 as what we call a repeated root.
OK, another example. Let's suppose we have F of
X equals X minus 2
cubed. Multiply by X +4 to the power 4.
Now, as before, we want to find out what the value of X has to
be to make F of X equal to 0.
And we can see that if we put X equals 2 in here, we will
actually get zero in this bracket. So X equals 2 is one
route and actually there are three of those because it's
cubed. So this has a repeated route, three of them, so three
repeated roots. So three roots for X equals 2.
But also we can see that if X is minus four, then this bracket
will be equal to 0, so X equals minus four is a second route and
there are four of them there. So there are four repeated roots
there. Now what we say is that.
If a route.
Has an odd number of repeated roots. For instance, this
one's got 3 routes, then it has an odd multiplicity.
If a root, for instance X equals minus four, has an
even number of roots, then it hasn't even multiplicity.
Why are we interested in multiplicity at all? Well, for
this reason. If the multiplicities odd, then that
means the graph actually crosses the X axis at the roots.
If the multiplicity is even, then it means that the graph
just touches the X axis and this is very useful tool when
sketching functions. For example, if we had.
F of X equals.
X minus 3 squared multiplied
by. X plus one to the power 5.
By X minus 2 cubed times by X +2 to the power 4.
Now, first sight, this might look very complicated, but in
fact we can identify the four roots straight away. The first
One X equals 3 will make this brackets equal to 0, so that's
our first roots. X equals 3 and we can see straight away there
are. Two repeated roots there, which means that this has an
even multiplicity. Some
even multiplicity. Our second route.
He is going to be X equals minus one because that will make this
bracket equal to 0. So X equals
minus one. I'm not actually five of those repeated root cause.
The power of the bracket is 5, which means that this is an odd
multiplicity. We look at the next One X is 2 will give us a
zero in this bracket. So X equals 2 is a root there and
there are three of them, which means is an odd multiplicity.
And finally, our last bracket X +2 here. If we put X
equals minus two in there.
That will give us zero in this bracket and there are four of
those repeated roots, which means it isn't even.
Multiplicity. So as I said before, we can use this to help
us plot a graph or helper sketch the graph. So for instance, and
even multiplicity here would mean that at the roots X equals
3. This curve would just touch the access at the roots, X
equals minus one. It would cross the Axis and at the River X
equals 2 is an odd multiplicity, so it will cross the axis.
And at the roots for X equals minus two is an even more
simplicity, so it would just touch the axis. So let's have a
look at a function where we can actually sketch this now.
So if we look at the function F
of X. Equals X minus two all squared multiplied by X plus
one. We can identify the two
routes immediately. X equals 2 will make this bracket zero and
we can see that this hasn't even
multiplicity. So even multiplicity.
And X equals negative one would make this make this bracket 0.
So this will be an
odd multiplicity. Because the power of this bracket
is actually one.
Now we talked before about the general shape of a cubic, which
we knew would look something
like this. So how can we combine this information and this
information to help us sketch the graph of this function?
Well, let's first of all draw
axes. Graph of X here.
Convert sleep X going horizontally. Now we know the
two routes ones X equals 2 to the X equals negative one.
Now about the multiplicity. This tells us that the even
multiplicity is that X equals 2, so that means it just touches
the curve X equals 2.
But it crosses the curve X equals minus one.
I should add here that because we've got a positive value for
our coefficients of X cubed, if we multiply this out, we're
definitely going to get curve of
this shape. So we can see it's going to cross through
minus one.
It's going to come down.
And it's just going to touch the access at minus 2.
So this is a sketch of the curve F of X equals X minus
two all squared multiplied by X plus one.
We do not know precisely where this point is, but
we do know that it lies somewhere in this region.