Algebra to Geometry: Functions, Cartesian Coordinates, and Graphs

Algebra to Geometry: Functions, Cartesian Coordinates, and Graphs


In mathematics, a function represents a causal relationship between “input” and “output.” For example, the area \(A\) of a square of side length \(s\) is given by the formula \(A = s^{2}\), read “\(A\) equals \(s\) squared.” If \(s\) is known, the area is determined. The name of the algebraic operation, squaring, matches its geometric meaning.

We may view the area of a square as “a function of” the side length \(s\). We use a letter such as \(f\) to denote the relationship by which the side length determines the area, and write \(A = f(s)\). Here, concretely, \(f\) stands for the operation of multiplying a real number by itself. We may signify this with the formula \(f(s) = s^{2}\), read “\(f\) of \(s\) equals \(s\) squared.”

To separate notation from a particular application, we generally denote the input of a function by \(x\) and the corresponding output by \(y\). The formula \(y = f(x)\), read “\(y\) equals \(f\) of \(x\),” says that if \(x\) is an input of \(f\), then \(y\) is the corresponding output.

Cartesian Coordinates

A function can be represented geometrically by a picture known as its “graph.” To describe the graph, we'll first develop a way to translate between algebra and geometry by representing pairs of real numbers as points in a plane.

We start by drawing a horizontal number line and a vertical number line, meeting at their common \(0\) points, below left. These number lines define Cartesian coordinates whose origin is the crossing point.

Cartesian coordinates in the plane

A pair of real numbers \((x, y)\) is the “Cartesian address” of the point located at position \(x\) on the horizontal axis and at position \(y\) on the vertical axis, right.

The pairs \((x, y)\) and \((y, x)\) are different unless \(x = y\). For this reason, mathematicians usually speak of ordered pairs.

Cartesian coordinates are a “dictionary” between pairs of real numbers (algebra) and points in the plane (geometry). Every pair \((x, y)\) specifies a unique plane location. Every point in the plane has a unique Cartesian address.

We now have the means to convert a function (loosely, an algebraic rule) into geometry (its graph).


The graph of a function \(f\) is the set of all pairs \((x, y)\) with \(x\) an input of \(f\) and \(y = f(x)\) the corresponding output. The graph of a function encodes, in picture form, information about all possible inputs and their corresponding outputs.

For the area function, the inputs and outputs are positive real numbers. For illustration, let's restrict our attention to squares whose side is between \(0\) and \(1\) unit. Practically, we select a possible input \(x\), calculate the corresponding output \(y = f(x)\), here \(y = x^{2}\), and plot the resulting point \((x, y)\). Once we see the general shape of the graph we sketch in a curve. For example, we might create a “table of values”: \[ \begin{array}{l|*{7}{l}} x & 0 & 1 & 0.5 & 0.25 & 0.75 & 0.125 & \cdots \\ \hline x^{2} & 0 & 1 & 0.25 & 0.0625 & 0.5625 & 0.015625 & \cdots \\ \end{array} \] Successively plotting gives pictures something like:

Graphing a function by plotting points

The final curve, mathematically comprisingly infinitely many pairs, one for each of infinitely many possible inputs, is the graph. The horizontal axis represents inputs and the vertical axis represents outputs.

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