Graphical Representation of Functions
Imagine you have a machine. You drop a number into it, it does something, and a new number comes out. That's a function: a rule that takes an input and gives you exactly one output. The graph is just a picture of that machine's behaviour — a way to see, at a glance, what output every possible input produces.
The Intuition: A Picture of the Rule
Suppose the rule is "double the input and add 1". If you put in 0, you get 1. Put in 1, you get 3. Put in 2, you get 5. You could write these as little pairs: (0,1), (1,3), (2,5). Now take a piece of paper. Draw a horizontal line (call it the x-axis) for the input, and a vertical line (the y-axis) for the output. For each pair, put a dot where the input value meets the output value. Connect those dots, and you have a graph.
That's all a graph is: the set of all points (x, y) such that y equals whatever the function does to x. It turns an abstract rule into something you can see.
The Precise Statement
A function f from a set X (the domain) to a set Y (the codomain) assigns to each x∈X exactly one y∈Y, written y=f(x). The graph of f is the set of all ordered pairs:
{(x,f(x))∣x∈X}
When X and Y are real numbers (which they usually are in school), this set of pairs becomes a curve or a collection of points on the Cartesian plane. The horizontal axis (x-axis) represents the input; the vertical axis (y-axis) represents the output.
The single most important thing: a vertical line can never cross the graph more than once. If it does, you don't have a function — because one input would be giving two different outputs, which breaks the definition.
What the Graph Tells You
Once you have the picture, you can read off properties instantly:
- Where is the function zero? Look where the graph touches the x-axis (where y=0). Those are the roots.
- Is it increasing or decreasing? If the graph climbs as you move right, it's increasing. If it falls, it's decreasing.
- What's the maximum? The highest point on the graph.
- What happens far to the left or right? That's the end behaviour — does the graph shoot up, drop down, or level off?
A Simple Example …