More LaTeX Experiments: Transformation of Graphs

If you take a graph and transform it in various ways, here is what you get:

A reflection in the y-axis turns a graph of y=f(x) into the graph of y= f(-x).

A reflection in the x-axis turns a graph of y=f(x) into the graph of y= -f(x)

- note the difference -

A stretch of factor s in the y- direction turns the graph of y= f(x) into the graph of

y = s f(x).

A stretch of factor m in the x-direction turns the graph of y = f(x) into the graph of \[y = f(\frac{x}{m})\]

A translation by \[ \left( \frac{k}{l} \right) \] turns the graph of y = f(x) into the graph of y = f(x-k) + l

A Recommendation

Normally I would not advise students of maths under degree level to go anywhere near Wikipedia. It's not the reliability issue as such, it's that it gets very complicated very quickly.

An exception is the page on Trig. It gives some useful basics and some really handy animations which illustrate the key trigonmetric functions.

Geometric Series Part II

Because timesing is a funny operation, a scaling up or down operation - repeated addition or however you like to look at it, it does strange things to series.

A common ratio of 2, like in the previous post, gives you a series which just keeps getting more and more massive, into infinity. It therefore has no sum - because it just keeps going on. You can find the sum of the first n terms of course.

But what if you had a common ratio of 1? Well then your series stays exactly the same

2,2,2,2,2,2,2,2,2,.....into infinity. It's the same sort of thing as above. Finding a complete sum of the series isnot going to be possible, because there will simply be an infinite load of 2s.

But if your common ratio is a fraction:

1/2, 1/4, 1/8, 1/16

Here the common ratio is 1/2. The terms of this sequence get smaller all the time, but will never reach 0.

When we think of the sequence properly as a series:

1/2 + 1/4 + 1/16 + 1/32 + 1/64

Then we can see that it is an infinite series, and will go on halving ad infinitum.
If we add the first few terms, we get 1/2, 3/4, 13/16, 27/32....

The sum is getting larger each time but by progressively smaller amounts.

The sum is getting closer and closer to 1, but without ever quite reaching it. If it had an infinite number of terms, then its sum would indeed reach 1. Its sum to infinity is 1.

It converges on 1, and is therefore called a convergent series.

This happens with certain types of common ratio, which we will come back to.

Geometric Series

These are a bit like arithmetic series but instead of there being a common difference between terms there is a common power-type difference.

Consider 2,4,8,16,32,64...

This series appears to double each time, which means that there is no common number which gives you the nth term when added to the n-1th term.

But the common difference is "doubling" - or rather, powers of 2.

The series goes 2¹, 2², 2³, 2⁴.....

With geometric series, therefore, we don't talk about a common difference, but a common ratio - the thing you times each term by to get the next one. So this series has common ratio 2.

Sum of an Arithmetic Series

This was supposedly demonstrated by Gauss, aged eight or something, when he was set a problem by a teacher desperate to get on with something more interesting to add together all the numbers from 1-100. He's supposed to have realised that if you paired up the numbers, they all had the same sum ie 101. It was then a matter of spotting the number of pairs and multiplying the two (5050).

The sum of an arithmetic series is done in a similar way. Think of a series as being:

a₁ + (a₁ + d) + (a₁ + 2d) +......

since d is a constant...

You want to find the sum of the first n terms.

S_n = a₁ + (a₁ + d) + (a₁ + 2d) +....

....(a₁ + (n-1)d)

This is probably quite a few terms (otherwise you wouldn't bother trying to find the sum, would you?).

So the wisest option would be to pair up the terms, like Gauss did, and find the sum of each pair.

The formula is derived from writing the above series out from beginning to n, and the opposite way, and adding the terms.

However, just as Gauss's addition always gave him 101, in our case we will always get 2a + (n-1)d

Since we want the sum to n of the series, there will be n of these pairs. So the sum is n(2a + (n-1)d)

BUT -

We have just added TWO series! Because we wrote it out twice to make adding the terms easier.

Therefore, the final formula will be: n(2a + (n-1)d)/2.


Quite fiddly to prove, but easy enough to use.

Series and Sequences 1 - nth terms

There are a few of these.

Let's just start with a definition. A sequence is a load of numbers in a list, with there being a common difference between the numbers.

A series is the same thing, but added together.

Here is an arithmetic sequence: 0,2,4,6,8,10....

Here is an arithmetic series: 0+2+4+6+8+10.....

In an arithmetic series or sequence, there is always a common difference between the numbers. In the ones above, the common difference is 2. It is always a constant (and not anything weird like n² - that's more like geometric series).

So finding the nth term of an arithmetic series or sequence is easy enough.

You need the first term a₁ and the common difference d.

a_n = a₁ + (n-1)d

So say I wanted to find the 42nd term of the above sequence.

a₄₂ = 0 + 41d
= 0 + 41x2
= 82

The 42nd term of the sequence 0,2,4,6,8...is 82


Because the difference is a constant with arithmetic sequences, this formula is simple enough to grasp. To find the nth term you need the first term, and then the number of terms before the nth one timesed by the common difference because there are that many lots of the common difference.

Long Time, No Blog

It's been a tricky old year so far, like a particularly knotty equation, so I haven't found the time for maths blogging, or even for maths.

So it was with some trepidation that I looked at my calendar about a month ago and saw the date for the AQA C2 exam. At that point I knew nothing of radians, logarithms I couldn't even spell, and geometric series I thought were just lots of pretty pictures (lots of geometry type stuff). So I spent a month fairly hectically doing the last four chapters of C2, missing out a few questions on the way and also, disappointingly, not really bothering to look at the proofs for the various formulae but just learning them.

C2 was yesterday. It was ok, though exams are always hard to assess until you get the marks. I answered all qs, but I think the differentiation with fractional indices one might have cost me a few %.

As before, the exam was thrilling. No I really mean this. I had bags of adrenalin, lots of excitement, masses of determination to show what I could do. I am like this with things that don't matter. Only important things have me quivering in the corner like a wobbly jelly who's just been made professor of wobbling at Oxford (ok ok that's a Blackadder joke sort of). Lining up beforehand with fifty sixth formers all pointing at me and whispering was a bit strange but it just reminded me of how utterly uninterested in it I was at their age. You do it for reasons, to get to uni, or because you happen to be good it, but you rarely do it just because you love it - you learn that later. Sometimes.

More Basic Trig

Every angle has a sin or cosine or tan (well, not quite every angle, in the case of tan). You can use this value to find the angle, doing it easily on a calculator.

You just press sin^-1 .456 or whatever it is, and that gives you the angle (if you have the calc. set to degrees that is).

A Proper Go At Explaining Trig

Alright, well, clearly it is the study of angles and triangles and by use of the unit circle, of circles too.

We talk about sin, cosine, and tan a lot in the basic stuff.

Although we learned SohCahToa at school, it really only applies to right angled triangles. But by using the unit circle (radius 1, centre at the origin) and drawing a right angled triangle inside it, we can calculate basic values of sine, cosine and tan, which hold for any angle and can be used in calculations involving any triangle.

The sine of an angle is a measurement of its distance from the horizontal axis, which can be negative or positive and which can be between 1 and -1 inclusive. If you think of an angle sweeping up around the circle, it reaches its furthest point from the axis (ie sin of 1) at 90 degrees.

The cosine of an angle is a meaurement of the distance from the vertical y axis of a point on the end of the angle; again, between 1 and -1.

The tan of an angle is a measurement of whether a line of angle x would meet a tangent to the circle drawn at right angles to the radius (the x axis) near, far, or not at all. The tan can have any value. Calculators really don't like being asked tan 90 or tan 270 because the lines of these angles will never meet a tangent.

There are some cool diagrams on the wikipedia page that show these things in action.

General AS Advice

Know your volumes and areas from GCSE.

I don't.

The Really Basic Basics of Trig

Trig is basically to do with triangles and circles. It works from the properties of angles relative to the sides of triangles and how these can be seen in circles (unit circles - with a radius of 1). It's used to tell you the size of angles and sides using other information you already know about the triangle.

The trig identities, I reckon, are probably best defined as ratios.

All fractions are ratios, as well as being processes.

For a right angled triangle (where we tend to begin with these)

SohCahToa

Sin =opposite/hypotenuse

Cos = adjacent/hypotenuse

Tan = opposite/adjacent


For each angle, there will be a different value for each one of these. So sin 42 will be the same whatever triangle you have. It regulates what the sizes of the opposite and hypotenuse will be only in terms of the ratio between them.

Sin and Cos (not sure about Tan) will always give a value between 1 and -1 inclusive.

Indices And Differentiation

Yesterday, courtesy of my beautiful and awesomely intelligent maths tutor, I suddenly worked out a way of making differentiation with negative indices slightly easier.

When inputting the x value to find the gradient at that point, to make the final stage of adding it all up much easier, turn your x^-2 into 1/x². It makes calculations a lot, lot easier.

Laws of Indices

Are pretty simple.

a^m x aⁿ = a^m+n

a^m / aⁿ = a ^m-n

(a^m)ⁿ = a ^mxn


But a^m + aⁿ DOES NOT equal anything other than what it says.
There isn't much you can do to collect the terms if the indices are different, or if the constant is different (ie x^m + y^m)

a^m + a^m = 2(a^m), of course.

Fractional Indices (2)

Well yes it gets a bit harder.

You see...a ^n/m


means ^m√aⁿ


...but I will try to explain this later.

Fractional Indices (1)

Think about it.


Think about it.


x is not just x. x is x¹. x is x raised to the power of one: x multiplied by itself no times at all; x just being x. So then x ¹ is x on its own. x is, therefore, x.

So then, x^1/2 is going to be a number that, multiplied by itself, makes x.

In other words, the square root of x.


√x = x ^1/2


It does get a bit more complicated than this....

Negative Indices

Negative Indices

Now these seem simple enough: you raise a number to a power and it usually means multiplying a number by itself n times.

a³ = a x a x a

All numbers to the power of 1 are themselves, and all numbers to the power 0 are one.

a⁰ = 1



Easy enough.


But then you can also have negative powers.

a^-3

What? You can’t multiply a number by itself a negative number of times!

Well, no, clearly. But you can see how negative powers come about.


It’s to do with the powers rule.

aⁿ x a^m = a^n+m

You can see this if you write out aⁿ and a^m in full.

aⁿ x a^-n = a^{n+ -n}

= a^n-n
=a¹
=a

Therefore, the negative powers are used to denote reciprocals (the number you multiply n by to get 1 – so 1/6 is the reciprocal of 6 – and it is always 1/n.

So.


3³ =27

3^-3 = 1/27 (or 1/33 – the reciprocal).


Negative powers are easy to manipulate, they just seem a bit weird until you think that positive powers are going up by multiplying the number by itself:

2⁴ is 2 x 2 x 2 x 2

But if you go down, towards 0, the same process is division by two.

64...32...16..8...4..2..

This continues as you go down below zero:

......2, 1, ½, ¼, 1/8, 1/16......

(2¹, 2⁰, 2^-1, 2^-2, 2^-3, 2^-4)

And you can see that these are reciprocals of the powers of 2.

The same applies for the powers of each number.