Foundations — Graphs of functions — plotting, reading key features

This page assumes you have seen nothing. Every squiggle, letter, and arrow the parent topic uses is built here, one at a time, each earning the right to appear before the next.

1. The number line — where everything lives

Before points, before graphs, there is a single straight road of numbers.

Figure 1 — A horizontal line marked at each whole number from $-5$ to $5$. Zero (amber dot) sits in the middle; a cyan arrow to the right is labelled "bigger", a cyan arrow to the left "smaller". This picture defines what "positive" and "negative" look like as positions.

Why the topic needs it. The horizontal axis and the vertical axis of every graph are just two number lines. If you can't picture "3 is to the right of 1" you can't picture any curve.

2. The symbol $x$ and the symbol $y$ — names for unknown numbers

3. The Cartesian plane — two number lines meeting

Now cross two number lines at right angles. Stand the second number line upright: on it, positive numbers go up and negative numbers go down (the vertical mirror of "right is bigger"). Fix this direction firmly — it is the convention every quadrant sign below depends on.

Figure 2 — Two crossed number lines. Right = positive $x$, up = positive $y$. The four regions are labelled I, II, III, IV with their sign pairs. An amber path shows how the point $(3,2)$ is reached: first "right 3" along the horizontal, then "up 2" along the vertical, landing the dot in the top-right region (Quadrant I).

The plane splits into four quadrants. Because right is positive-$x$ and up is positive-$y$, the signs follow automatically:

Why the topic needs it. A graph is a collection of coordinate pairs. Every "point $(0,-3)$" in the parent note is an instruction: stand on the origin, move right $0$, move down $3$.

4. The symbol $f(x)$ — the function machine

Figure 3 — A cyan box labelled $f$. A white arrow feeds "input $x=4$" into the left side; a cyan arrow carries "output $f(4)=5$" out of the right side. Amber text underneath reminds us: exactly ONE output per input. This picture is the meaning of the notation $f(x)$.

Why "exactly one" matters. This is why the Vertical line test works: because each input gives one output, a vertical line (which fixes $x$) can only ever touch the curve once. See more in Types of functions.

The link between machine and picture (here the long arrow $⟶$ just means "then leads to", a flow direction — not "approaches", and the double arrow $⟹$ means "and therefore you do this next"): $\text{input }x⟶\boxed{f}⟶\text{output }f(x)⟹\text{plot the dot }(x,f(x))$

5. Sets and braces — collections of numbers

Before we can talk about which inputs are allowed, we need a way to write "a collection of numbers".

Domain and range — the allowed inputs and reachable outputs

Rules that create forbidden inputs. The parent note only meets the first two, but for completeness here is the full family of "banned input" rules you will keep meeting:

• No dividing by zero. For $\frac{1}{x-2}$, the input $x=2$ is banned because $x-2$ would be $0$. This ban is why a vertical asymptote appears there.
• No even root of a negative (in real numbers). Square roots, fourth roots, etc.: for $\sqrt{x}$, inputs below $0$ are banned. (Odd roots like cube root are allowed on negatives.)
• No fractional exponent of a negative (same reason as even roots — $x^{1/2}$ is a square root).
• No logarithm of zero or a negative. For $\log x$ or $\ln x$, only inputs $x>0$ are allowed.

Full detail lives in Domain and range.

6. The symbols of sets: $\in$, $\mathbb{R}$, $\setminus$, $\to$, $\infty$

The parent note quietly uses these. Each is just shorthand for a plain English phrase.

7. Interval brackets: $(a,b)$ vs $[a,b]$

8. Symbols that describe the curve's behaviour

These are the words the parent note uses in its "key features" list — now grounded. One new symbol appears here, the double arrow $\Rightarrow$: it means "which forces / therefore" — read "$P\Rightarrow Q$" as "if $P$ is true, then $Q$ must be true too." (This is the same "therefore" arrow met in §4, not "approaches".)

Why gather these? Together they let you narrate a function's whole life story — "climbs, peaks at $(2,-1)$, dives, crosses zero at $3$" — which is exactly what Solving equations graphically and later Calculus: derivatives and graphs build on. Mirror-image behaviour (a curve identical on both sides) is the subject of Symmetry (even and odd functions), and shifting/stretching a known curve is Transformations of functions.

Prerequisite map

Read the map below as a set of "you need this before that" arrows. Here each arrow means simply "feeds into". In plain English: the number line is the root — it feeds both the Cartesian plane and the idea of variables $x$ and $y$. Variables let us build the function $f(x)$. The plane plus the function together let us plot a point, and the function plus set/interval shorthand together give us domain and range. Plotting points and domain/range then combine into the full graph of a function, from which we finally read key features and apply the vertical line test.

Equipment checklist

Test yourself — answer before revealing.