Algebra & Functions
Variables, equations, and functions — the language used to describe every algorithm.
Algebra is using symbols to stand for numbers so we can describe relationships
that hold for any value. A function f(x) is a rule that turns an input x
into an output — exactly how we describe what a piece of code computes and how its
cost grows.
Pick a function and drag the coefficients. Watch how the shape of the graph encodes the formula.
Variables and equations
A variable is a placeholder. An equation says two expressions are equal,
e.g. 2x + 3 = 11; solving it means finding the x that makes it true (x = 4).
The moves are simple: do the same thing to both sides until x is alone.
Functions and their graphs
f(x) = a·x + b is a line: a is the slope (steepness), b is where it
crosses the y-axis. f(x) = a·x² + b·x + c is a parabola — the curve behind
projectile motion and the O(n²) cost of simple sorts. The graph is just every
(x, f(x)) pair plotted.
Why it matters for CS
When we say an algorithm is O(n²), we’re describing a function of the input
size n. Reading graphs and manipulating expressions is how you compare growth
rates, solve for break-even points, and reason about performance.
Going deeper
For a full ground-up course, Khan Academy: Algebra is free and excellent.
Takeaways
- Variables let one statement describe infinitely many cases.
- A function maps inputs to outputs; its graph is the picture of that rule.
- Big-O notation is algebra — functions of the input size
n.