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Mathematical functions

15th January 2011 Paul Chris Jones

This will (hopefully) be the first of a series of posts on mathematical functions. The purpose is to help me learn the material to pass an A Level in Mathematics.

Contents
1. Introduction
2. Real-life examples of functions
3. Solving quadratic equations




Introduction

A function is: a variable (called 'y') whose value is completely determined by the value of a second variable (called 'x').

What does this mean?

If you change the value of x, you get a new value for y.
Let's look at an example to make it more clear.



Example: y=x


The simplest example of a function is y = x.
Here, the value of y is always the same as x.
E.g., say x is 3. y would equal 3 too, as y equals x.

Functions are drawn as graphs so that we get something to look at instead of a formula.
Below is the graph for the function y = x.
Notice that the values for x are on the horizontal axis (which is called the 'x-axis') and the values for y on the vertical axis (called the 'y-axis'). 

Where the x-axis and y-axis meet, x and y both equal 0.



 




Quadratic equations


When you introduce an to the function, it becomes a quadratic equation.


What does this mean?


Essentially, the graph becomes curved instead of being a straight line all the time.


Here's the graph for y = x²



Why is it curved?
Because each time you add 1 to x, y increases by a larger and larger amount.
Look at the following values for x and y to confirm this for youself:

x         function     y

1              1²        = 1

2              2²        = 4

3              3²        = 9

4              4²        = 16



Notice the area of the graph where there are negative x values but postive y values. Why is this?

Because when you square a negative number you get a positive number.

E.g.: -2² = -2 x -2 = 4

Again, here are the x and y values so you can see for youself:


x         function     y

-1            -1²        = 1

-2            -2²        = 4

-3            -3²        = 9

-4            -4²        = 16



Why are quadratic equations so important?
I don't really know yet. They feature heavily in A level Mathematics however!
They have x²'s, and the only purpose of a squaring a number is to find the area of a square. At least, that's all I can think of.
Pythagoras' theorem has squared numbers in it: hyp² = opp² + adj². But this is because it involves turning straight lines into squares.




Beyond the quadratic

To summarise so far:

in linear equations, the highest power x is raised to 1 (x¹)

in quadratics, the highest power x is raised to 2 (x²)


What happens if you introduce a ? The equation is no longer quadratic. It's called a cubic equation.

in cubic equations, the highest power x is raised to 3 (x³)

in quartic equations, the highest power x is raised to 4 (x?)


Going from quadratic to cubic equations introduces another curve in the graph. Essentially, the number of 'bends' in the graph is equal to the power that x is raised by, then subtracting 1.


E.g. a quadratic has (x)²?¹ = 1 curve.

A linear equation has (x)¹?? = 0 curves.



Let's look at a graph of the equation y = x³:




next: Real-life examples of functions

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Paul Chris Jones is a writer and dad living in Girona, Spain. You can follow Paul on Instagram, YouTube and Twitter.