In the series: Note 3.

Subject: Polynomials & quadratic functions

Date : 16 Februari, 2016

Version: 0.5

By: Albert van der Sel

Doc. Number: Note 3.

For who: for beginners.

Remark: Please refresh the page to see any updates.

Status: Ready.

Maybe you need to pick up "some" basic "mathematics" rather

So really..., my emphasis is on "rather

So, I am really not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

This note: Note 3: Polynomials & quadratic functions.

Each note in this series, is build "on top" of the preceding ones. If you are not familiar with, for example,

"linear equations" in the form of "y=ax+b", then it's highly recommended to read Note 2 first.

1. Introduction quadratic equations y=ax

2. How "a", "b" and "c" determine the shape and location of y=ax

3. Quickly finding the "top" (or max/min) of the parabola.

4. Intersections of a parabola with the X-axis and Y-axis.

5. Intersection of a line with a parabola.

6. Linear and Quadratic Inequalities.

7. Quadratic functions compared to relations.

8. A few words on Higher order polynomials.

are very common to use to

As a simple example: If a car drives 50 mph, then in one hour, it has travelled 50 miles. In 2 hours

it has travelled 100 miles. In 3 hours time, it has travelled 150 miles etc..

This is a linear relation between "distance" and "time".

Usually, in physics, the distance covered in a certain time is abreviated by "S" (or "d"),

the time involved is usually abreviated by "t", and the (constant) speed is usually abreviated by "v".

It can be shown that "S = v x t", or "S = vt", since usually the multiplication operator (symbol "x") is left out.

You can plot this relation in a Coordinate system, like we have seen in Note 2, and in such Coordinate system,

if you would replace "y" with "S", and "x" with "t", you get a common line again.

As another simple example of a linear relation: if you earn 10 bucks per hour, and you work 5 hours, then you have

earned 50 bucks, if you have worked for 10 hours, you have 100 bucks etc.. etc..

The relation between "amount earned" and "hours worked" is linear too.

You can always represent them by "lines" in a (2 dimensional) Coordinate system.

higher order functions and relations as well.

So, what are we talking about? The general form of a quadratic equation is:

where a, b, and c, are some constant numbers. Each a, b, and c, defines another quadratic equation,

just like every

br> In a quadratic equation, the constants "b" and "c" can optionally be 0, since the "most" defining term

in the equation is the x

The associated curves in a (2 dimensional) Coordinate system, are

Let's take a=1, so we have y=x

As usual, we take a couple of "x-values", and see what the corresponding "y-values" are. See table 1.

Table 1: calculate "y" for a number of x values.

value of x: | value of y: |

x= -4, then | y= 16 (because -4 x -4 = 16) |

x= -3, then | y= 9 (because -3 x -3 = 9) |

x= -2, then | y= 4 (because -2 x -2 = 4) |

x= -1, then | y= 1 (because -1 x -1 = 1) |

x= 0, then | y= 0 (because 0 x 0 = 0) |

x= 1, then | y= 1 (because 1 x 1 = 1 |

x= 2, then | y= 4 (because 2 x 2 = 4 |

x= 3, then | y= 9 (because 3 x 3 = 9 |

x= 4, then | y= 16 (because 4 x 4 = 16) |

Let's plot the points now. Take a look at figure 1.

Figure 1. Some points of y=x

If you draw a smooth curve connecting the points, you get a parabola.

Note that such a curve (near x=0) has "a slow start", but then starts "climbing fast".

For example, if x=100, then y=10000. You can calculate that fact by yourself.

Note:

In physics (and other sciences), parabolic curves are very common. I will explain them later.

For example, a parabolic antenna (or reciever), which can focus radio- or light waves.

Or, if you fire a bullet with a certain angle, the trajectory will be a near perfect parabola

on the moon, or a litlle less perfect parabola on Earth (due to friction and other interactions).

With the latter example, that parabola will be "upside down" ofcourse, compared to figure 1.

Exercise 1: I would like you to Google on parabola images, for a couple of minutes or so.

Remember, "c" can be a positive number, or a negative number. So, we can have as example equations:

y=x

or, for example

y=x

Can you see what happens?

If we look at figure 1, and we now think of "y=x

is lifted upwards, by 3, over all x values.

You can also see that from table 1. Just add 3 to all "y-values" in that table.

If we now think of "y=x

over all x values. You can also see that from table 1. Just substract 3 from all "y-values" in that table.

Exercise 2: Can you analyze "y=x

With the example parabola "y=x

If we would have analyzed the function "y=

that curve, would be the mirror of "y=x

So, the "vertex" (the nose of the parabola) would be pointed

If you do not believe this, you only need to invert (make negative) all values of y in table 1.

So, what I want to achieve here, is that you immediately will reckognize that:

y=ax

y=ax

The coefficient "a" determines also the "steepness" of a parabola, which we will see a bit later on.

It's interesting to see the effect of "c", if it takes different values, while "a" and "b" stays the same.

So, suppose we have the parabola:

y= x

and

y= x

So, in the first equation, c=0, and in the second equation, c=8.

If you add a number "c" to y= a

with the value of "c".

If you substract a number "c" to y= a

with the value of "c".

Take a look at figure 2. The continuous curve is the parabola "y= x

The "dotted" curve is the parabola "y= x

You can check it by yourself, for the simple parabola "y= x

to all values of y in that table. You will see that the whole curve rises up by "3".

It's only logical, since you might have "y={some stuff}" and then "y={some stuff} + 3" for all x on the x-axis.

So, the constant "c" has the effect that a curve rises or drops by "c", depending of the fact you add "c",

or substract "c".

Figure 2. An example of the effect of adding c=8 to a parabola.

Figure 3. Some more examples.

Exercise 3: For which of them, is the "a" coefficient negative?

There is much more to say on how the constants a,b, and c determine the shape and location

of a parabola.

Very important points, are the intersections of the parabola with the x-axis and y-axis.

And, finding the "top" (or maximum or minimum) is important too.

You know that a parabola has a "maximum point", or a "minimum point", depending on the sign of "a".

Such a minimum or maximum is also often called the "top".

For example:

f(x) = -x

f(x) = x

In a later note, we will see how we must take "the derivative" of a function, which also

can help in finding the maxima or minima of a function.

For now, however, a simple formula will help us.

This maximum, or minimum point, is a unique point (x, y) which we will call (x

For that coordinate, it holds that:

x_{M} |
= | - |
b -- 2a |

Please note the "minus sign" in the formula x

And, once x

the equation of that particular parabola.

Suppose we have:

y=x

Then:

x

Then finding y

So the coordinate (x

particular parabola (this parabola has it's nose down).

Suppose we have:

y = -2x

Then:

x

Then finding y

So the coordinate (x

particular parabola (this parabola has it's nose up).

Our aim is to find out, where a parabola crosses the x- and y- axes.

Well, some do not cross the x-axis. Take a look at figure 3 again. The parabola in examples B and D,

do not cross the x-axis. Their vertex (the lowest point of the "nose") is above the x-axis, and

from that point, they only rise up more and more.

-A parabola in the form "y=ax

-A parabola in the form "y=ax

Let's try some calculations.

Key statement: At the intersection with the y-axis, then

Note the exclamation mark. Or take a look at figure 1. There you see a nice example of a Coordinate system.

Only at the point where the y-axis is located (vertically), there it holds that x=0.

So, for a general equation, then holds:

y=ax

So, where the parabola crosses the y-axis, is at the point (0,c).

Find the point where "y=0.3x

Solution: y= 0.3 x 0

So the point where that parabola intersects with the y-axis is (0,4).

Find the point where "y=3x

Solution: y= 3 x 0

So the point where that parabola intersects with the y-axis is (0,5).

Again, such a fact will help us enormously.

I like to split the theory in (1) a special case, and (2) a general solution.

Ofcourse, the general solution covers (1) too, otherwise it would not be a "general solution".

This is quite quickly to solve. First note that ax

This is based on the fact that "ab + ac" is the same as "a(b + c)".

Indeed, we know that "

It's the same with "ax

If y=0, then:

ax

Find the point where "y= x

Solution: x

For the last one, we may add "-3" to both sides of the "=" symbol, resulting in x=

So, we have found x=0, and x=

can only have 0, or 1, or 2, intersections with the x-axis (where y=0).

So, we are dealing with ax

Especially when a, b, and c are "whole" numbers (integers), you can often "expand" (or disolve)

the equation into factors.

Here is an example:

Suppose we have the equation "x

Then: x

It's really not always easy to see. But hopefully, you agree that (x-5)(x-1) = x

It's not much different from ordinary numbers only, for example: (2 + 3)x(4 + 2) = 2x4 + 2x2 + 3x4 +3x2.

Once you have found: x

(x-5) = 0 or (x-1)=0, thus:

x=5 or x=1

In this case, the points where that parabola intersects with the x-axis, are (5,0) and (1,0).

If you did not fully understood the reasoning above: not to worry at all!

It's still not a general solution that was presented here.

Next, we will present the true general solution of ax

It's this. If the parabola crosses the x-axis at two points, called x

x=_{1} |
-b + √(b^{2} -4ac)------------------- 2a |

x=_{2} |
-b - √(b^{2} -4ac)------------------- 2a |

We can "proof" those formula's, but that will "clutter" this note too much, so I will leave that out.

But for a less formal "proof", see the note below.

Note: Simple way on how to proof them:

But those formula's can be made "less hard to believe". Really. remember the simpler equation "ax

where "c" is nul?

I came up with:

"x = 0" as a solution, and

"x =

Well, in upper 2 formula's, you might like to substitute "c=0", then √(b

This means that for example x

This is exactly what we found before.

Suppose we have the equation

Then using the two formula's above:

x

x

If you take a look above, then indeed we found the same intersections (1,0) and (5,0) again.

Our two formula's for x

That cannot be correct. Please see figure 3 again.

Remember, this is always true:

-A parabola in the form "y=ax

-A parabola in the form "y=ax

Implicit, in using the 2 formula's for x

(b

It turns out that:

-If (b

-If (b

-If (b

Sometimes, the expression (b

how many intersections with x-axis exists.

at two points, or just one, or maybe not at all. It just depends on the slope of that line, for example.

How can we find the coordinates of the intersections?

Suppose the parabola is denoted by f(x).

Suppose the line is denoted by g(x).

Let's try an example:

Suppose we have:

f(x)=2x

g(x)=3x + 2

Find the point(s) where they intersect (if any, ofcourse).

In order to find the "y" values, we do this:

At the intersections, f(x) must be equal to g(x), so we must have:

2x

We are going to solve that equation. Bring "3x" to the left, and "2" to the left:

2x

x

x(x-3) = 0 =>

x=0 and x=3 (in math we actually should say x=0 OR x=3).

So, we have found the two x values of the intersections. We can next use one of both equations

to find the corresponding y values:

For x=0, then we have y=2.

For x=3, then we have y=11.

Thus the points where the line and the parabola intersect, are (0,2) and (3,11).

-2x+3 > 3x-4

-x

Let's take the second example above. Suppose f

If you see such an inequality as "-x

f

Figure 4. Inequalities: Where is the one graph, below or above, the other one?

If you look at the two pictures above:

- then where is the parabola below the line? Meaning:

If you simply look at the first figure, then you will see that for x in the interval [0,4],

the parabola is below the line. Usually, it is notated as: "0 < x < 4".

- then where is the red line above the blue one? Meaning: for which "x" is the the red line above the blue one?

If you simply look at the second figure, then you will see that for "x > 3", the red line, is clearly above the blue line.

What seems to work, is finding the intersecting point(s) of both graphs, and from then

it is rather easy to determine the "x" interval for which the inequality is true.

Simply use any "test value" for x on the left or right, for the "x" you have found

of the intersection.

Let's see a few examples.

Suppose we have:

f(x) = x

g(x) = -x + 3

See figure 5 below.

Figure 5. For what x, is f(x) < g(x)?

Now we need to solve: x

In other words: for which interval(s) of x, is the parabola below that line?

Let's find the intersection(s) of the line with quadratic function.

Where they intersect, we must have f(x)=g(x).

x

=>

x

=>

x(x-3)=0

Thus x=0 Or x=3.

These are the solutions for the "equation", which is f(x)=g(x).

Now, it's relatively easy "to see" from the graphs that for the interval

0 < x < 3, we indeed have that f(x) < g(x).

Suppose in Example 1, it was the other way around: That is: For what x, is f(x) > g(x)?

We already have found the two intersections. So we may conclude:

For x>3 and x<0, we have f(x) > g(x).

Again, let's stay with example 1.

We do not neccessarily

Let's stay with pure algebra this time.

f(x) = x

g(x) = -x + 3

For what x, is f(x) < g(x)?

x

=>

x

=>

x(x-3) < 0

We have two factors here. We only can get a negative result (< 0), if only one of the factors is negative.

For example, if both terms are negative, the product would be positive (> 0).

That means:

x>0 AND x-3<0 =>

Yes, but why not "x<0 AND x-3>0" ???

Good question. It's because "x-3"

This is why we only can have: "x>0 AND x<3" <=> which is more nicely notated as "0 < x < 3"

Only using algebra is fine, but then you need a tiny bit of additional reasoning....

You may see the left and right sides of the inequality symbol, as linear functions.

Suppose we have linear inequality:

2x - 6 > -3x + 4

Then you might say: where (for what x) is the line f(x)=2x - 6 above the line g(x)=-3x + 4?

-Ofcourse, you may first find the point of intersection of those lines. Suppose for

the intersection we have found "x=a". Next, we substitute some test values for x in the inequality,

for "x>a" or "xa" or "x We can also evaluate the graphs, to see if we need "x>a" or "x

So, let's find the intersection first:

2x - 6 = -3x + 4 => 5x = 10 => x=2.

If we fill in "x=2" in either of the equations, we find the corresponding "y".

Let's use the first one: "y=2x - 6". Then y=2*2-6=-2.

Thus, the lines intersect at the point (2,-2). However, we only need the "x" value,

we already have found before, namely "x=2". The question was: for which x values is

the line f(x)=2x-6, above the line g(x)=-3x + 4? So, finding the "y" value was not needed after all.

It's reasonable to use, say x=3

Now insert it in g(x)=-3x + 4. This gives -3*3 + 4 = -9 + 4= -7.

Thus for x=3, we have indeed that it is true that 2x - 6 > -3x + 4.

Thus the solution for "2x - 6 > -3x + 4" is "x>2".

So, the method above is not very complicated.

-But let's use algebra only:

For the example above, we have:

2x - 6 > -3x + 4

=>

5x > 10

=>

x > 2

You see that? Without any drawing of graphs, we immediately have found "x>2".

When we have some sort of inequality, and during solving, we need to multiply or divide

both sides of "<" or ">", by a negative number, then:

"<" becomes ">".

">" becomes "<".

Example:

Suppose we have:

7x + 5 < 9x + 4

add -5 to both sides (or some say: bring 5 to the other side, and flip sign).

7x < 9x – 1

add -9x to both sides (or some say: bring 9x to the other side, and flip sign).

Now we are left with:

–2x < –1 => -x < -½

As usual, we like to have it in the for "x=", instead of "-x=".

Just think about it: for what "-x", would it be smaller than -½?

Suppose "x=1" (the plus 1), then would it be true? Fill it in: -1 < -½? True.

-x < -½

multiply both side with "-1", and flip the "<" to ">".

x > ½

Figure 6. The costs that three companies charge you for phone services.

If you use a cell phone, the costs are dependent on how many hours you use your phone.

In the figure above, we see the graphs of "costs per hour" that 3 different providers

would charge you.

The y-axis represents the costs, and the x-axis represents the numbers of hours you

use you phone.

Question: which provider is the cheapest?

There is not "one" answer, because if you study the figure above, there are different x intervals

for which a certain provider would be the cheapest one.

Could you work this out, that is, finding the x intervals which identifies the cheapest provider?

The next two sections are optional reading. You may skip this, if you want to follow my series.

to exactly one "y". This is just an "agreement" among mathematicians, established a long time ago.

You can take a look at the parabola we have shown sofar, and it's true. For each x on the x-axis, you find only 1 "y".

By the way, if you have equations like "y=ax+b", or "y=ax

the "rule" that every "x" is mapped to exactly one "y" only.

You can see that very clearly with a line defined by "y=ax+b". There really is just one "y" for every specific "x".

In such cases, people also write y as y =

that we are dealing with functions.

But not every relation between "x" and "y" is a function. Take a look at figure 4:

Figure 7. An example of a "circle" and "ellips".

If you look at the circle, you will see that any "x", is mapped to 2 "y's".

For example, x=3 is mapped to y = 4 and y = -4.

For this reason, the circle is not described by a function, but by a "relation" between x en y.

For a circle centered around the "origin" (where x-axis and y-axis comes together), the relation is:

Where "r" is the radius of the circle. Since a circle is fully symmetrical, if I prove the equation for a few

x's en y's, then you probably "believe" the equation is right.

Well, for example, take a look at the circle in figure 4. For that circle, the radius "r" is 5. So r

For the point (5,0) holds: 5

For the point (-5,0) holds: -5

For the point (0,5) holds: 0

Alright, these were special points. But now, for example for the point (3,4). Does the relation then holds too?

If you look closely, you will see that (3,4) sits on the circle, so this is true. Now, let's calculate:

3

Right again.

For points on a circle (centered around the origin), "x

Maybe you see the "connection" with the famous Pythagorean theorem.

Let's focus on the point (3,4) again. It's on the circle. In figure 4, you can see that the length of y = 4,

and the lenght of x = 3. You notice the

For the point (3,4), the equation "x

Actually, if you think about it, you can always create (or imagine) such a right-angled triangle for any point on the circle.

y=ax+b

y=ax

But, ofcourse, higer powers of "x" are possible too. Let's see a simple function of "x" to the power of three.

We also often hear that the "degree is three".

y=ax

Or, in a general format:

y=ax

Sometimes, equations of a certain degree, are also called "polynomials" of that degree (like 3).

Usually, in basic math, these sorts of equations are "left alone", like in the sense of finding

intersections with the x- and y-axis.

So I won't tought them here either. But I like you to know about them.

By the way, the point where such higher-degree function crosses the y-axis, is very simple to calculate.

Because, at that point, neccessarily (as always) x just has to be 0.

So: for the intersection with the y-axis:

y=ax

So the point where the curve hits the y-axis is (0,d).

Since y varies mainly with ax

For example, if we look at the simplest one, namely "y=x

You might like to Google on the shape of polynomials of the 3rd degree. They often have one "top" (maxima),

and one "crest" (minimum), and tend to rise or sink steeply with larger (positive or negative) "x".

But depending on the values of the coefficients (a, b etc..) their shapes can vary enormously.

(Some polynomials of the 3rd degree, with specific coefficients, do not have a distinguished maxima or minima).

Figure 8. An example of a polynomial of the 3rd degree.

You don't need to memorize the following.

Indeed, the following might look pretty awkward at first sight. But really, it not so difficult at al.

We might thus also have "n" th degree polyniomials

For example, in a very general form:

y=a

So, we might have one of the fourth degree:

y=a

Note that the usual coefficients like a,b, and c, are replaced by numbers designated as a

The next note is a super quick intro in sinus and cosinus functions.