Functions and Graphs – Seeing Maths in Pictures
Functions and graphs let us turn an equation into a picture – and a picture into understanding. Once a relationship between two quantities is written as a function, plotting it as a graph instantly reveals patterns that are almost invisible in a table of numbers: where a value is growing fastest, where it peaks, where it crosses zero, or where two competing quantities meet.
The bridge between algebra and pictures was built by the French mathematician and philosopher René Descartes, who in 1637 published his breakthrough insight that every equation corresponds to a shape on a grid, and every shape corresponds to an equation – the coordinate grid is still called the Cartesian plane in his honour. The function concept itself, and the f(x) notation, came a century later from Leonhard Euler, who treated a function as any well-defined rule connecting inputs to outputs. This section goes beyond the straight lines and parabolas covered in Algebra to explore exponential growth, logarithms, and the graph-reading skills used every day in science, finance, and the news.
What Is a Graph?
A graph is a visual picture of a function or a relationship: every point (x, y) on the graph represents a pair of values that make the equation true. Reading a graph left-to-right shows how the output changes as the input increases.
Families of Graphs
| Function Type | Shape | Real-World Example |
|---|---|---|
| Linear | Straight line | Cost that grows at a fixed rate per item |
| Quadratic | Parabola (U-shape) | The path of a thrown ball |
| Exponential | Rapidly curving growth or decay | Compound interest, viral spread, radioactive decay |
| Logarithmic | Fast rise that quickly levels off | The Richter earthquake scale, perceived loudness |
Why Graph Literacy Matters
- Science: Every experiment's results are checked against a predicted graph shape.
- Finance: Stock charts, interest growth, and inflation are all read as graphs.
- Health and news: Epidemic curves and climate charts are graphs the public is asked to interpret directly.
- Technology: GPS, computer graphics, and data visualisation all rely on the coordinate plane.
What You Will Learn
This section builds from exponential functions and logarithms – inverse operations of each other – through to plotting and identifying different graph families, transforming graphs by shifting and reflecting them, and finally reading the kind of real-life graphs you meet in the news, in science class, and in everyday decisions.
Key Takeaways
- A graph is a visual picture of every (x, y) pair that satisfies a function.
- Descartes's coordinate plane unifies algebra and geometry; Euler's f(x) notation describes the rule itself.
- Linear, quadratic, exponential, and logarithmic functions each have a distinctive graph shape.
- Reading graphs correctly is a real-world skill used in science, finance, and the news.