​Functions and graphs

​Functions and graphs

Functions is a branch of mathematics that includes an input, a function, and an output. It works like a printer. The blank piece of paper goes into the machine and will come out a different piece of paper. The numbers are the same.

​Functions are very useful

​Functions can be beneficial for practising algebraic skills and using graphs. There will be more things it will improve but we will just focus on evaluating functions and plotting functions.

​Function notation

All functions have one thing in common. Their structure is the same.

​Evaluating Functions

​Example 1: Evaluate x(n)=3n+1 when n=2. You should always substitute first. So substituting will turn the function into x(n)=3(2)+1=6+1=7.

​Example 2: You can also set the input value as an algebraic expression. Here's an example:

​Evaluate t(a)=3x+1 when a=2x+b. Substituting will turn your function into f(x)=3(2x+b)+1=6x+3b+1

​Example 3: You can also set the input as another function. Example:

​Evaluate f[n(a)]=3[n(a)]+1 when n(a)=3a+1 if a=4

​Plotting functions

​When plotting functions, our x-axis is our domain or input. Y-axis is our range or output. Our input and output will have to be numerals because they will form a coordinate. For example:

​Plot f(x)=3x+1 when x=4

f(x)=13

Therefore, the coordinates are (4,13)

​(An estimate of where the coordinates are)

​Making parabolas

​Firstly, what's a parabola? A parabola is a U-shaped plane curve where any point is at an equal distance from a fixed point (known as the focus) and from a fixed straight line which is known as the directrix.

​So now you know what a parabola is, you need to know how to make them using functions. To make a parabola, you need a table to get organised.

​Parabola example (do it yourself)

​Input (domain)	​Function:f(x)=3x+1	​Output (range)	​Co-ordinate
​2	​f(x)=3x+1 =3(2)+1 =7	​7	​(2,7)
​1	​​f(x)=3x+1 =3(1)+1 =4	​4	​(1,4)
​0	​f(x)=3x+1 =3(0)+1 =1	​1	​(0,1)
​-1	​​f(x)=3x+1 =3(-1)+1 =-2	​-2	​(-1,-2)
​-2	​f(x)=3x+1 =3(-2)+1 =-7	​-7	​(-2,-7)