Understanding Mathematical Functions: How Equations, Tables and Graphs Work Together
When it comes to functions, many students quickly get confused. Between the equation, the graph, and the value table, it’s easy to lose track. This confusion often stays all the way until the final exams — but it doesn’t have to. Functions can actually be understood in a very visual and simple way.
What Is a Function?
Most students have an idea, but the definition often remains a bit blurry.
Put simply: a function describes how one value depends on another. There is always a cause and an effect.
We usually call the cause x, and the effect y or f(x).
A function can be represented in three ways:
Function equation
Value table
Graph
The Function Equation
The function equation is like the recipe of the function.
It could look like this, for example:
f(x) = 3 + 4·x
The variable x is the part that you can change — a kind of placeholder.
The letter f stands for the function itself, meaning the rule that assigns each number x to exactly one number y. This second number is what we call f(x).
Example:
If we substitute x = 2, we get:
f(2) = 3 + 4·2 = 11
That means for x = 2, the function value is y = 11.
A Simple Example: The Gumball Machine
Imagine a gumball machine.
You put in a coin (x = input) and one gumball comes out (f(x) = output).
Every coin you put in gives exactly one result — just like every x-value in a function gives exactly one y-value.
The Value Table
A value table is created by plugging several x-values into the function and writing down the corresponding y-values.
Usually, we choose consecutive numbers like 1, 2, 3, and so on.
For our example f(x) = 3 + 4·x, the table looks like this:
These values are calculated by using the function equation:
f(1) = 7, f(2) = 11, f(3) = 15, and so on.
The table shows the pattern that connects each x to its y.
It’s especially helpful when drawing the function’s graph.
The Graph
The graph is the visual representation of a function.
You plot the points from the value table into a coordinate system and connect them.
For f(x) = 3 + 4·x, the points from our table form a straight line that rises continuously upward.
Here, too, you can see the same cause-and-effect relationship: x is the input, f(x) is the output — just like with our gumball machine.
Anyone who understands the graph can immediately see how the function behaves, without calculating every single value.
Conclusion
A function is nothing more than a system that connects one value with another.
The function equation, the value table, and the graph are just three different ways of showing the same idea:
Equation = the recipe
Table = the individual values
Graph = the visual representation
Once you understand how these forms are connected, functions will no longer seem complicated — almost as simple as getting a gumball from a machine.