Foundations — Transformations — vertical - horizontal shifts, reflections, stretches - compressions

This page assumes nothing. We build every symbol the parent note uses one at a time, each from the picture it stands for. Read top to bottom — each idea leans on the one above it.

1. The number line and a point's address

Before functions, before graphs, there is the number line: a straight road with $0$ in the middle, positive numbers to the right, negative to the left.

To describe a dot on a flat page we need two number lines crossing at right angles.

Why the topic needs this: every rule in the parent note is phrased as "the point $(x,y)$ becomes $(x,y+k)$" or "$(x+h,y)$". If you cannot read $(x,y)$ as an address, none of those rules mean anything.

2. What a function actually is

The notation $f(x)$ is read "$f$ of $x$" — it means "the output the machine $f$ produces when you feed it the input $x$." The $x$ is just a placeholder for "whatever you put in."

Why the topic needs this: transformations are edits to this machine. $f(x)+k$ edits the output (add $k$ after the machine runs). $f(x-h)$ edits the input (change the number before it enters). You cannot tell inside from outside without knowing where the machine's parentheses are.

3. From machine to graph

If you feed the machine every possible input and plot each dot $(x,f(x))$, the dots join into a curve — that curve is the graph.

Why the topic needs this: the parent note says output edits ($+k$, $\times a$) move things vertically and input edits ($-h$, $\times b$) move things horizontally. That is only "obvious" once you see that height = output and sideways position = input.

4. The parent functions — the shapes we transform

A parent function is the simplest, un-transformed member of a family. The parent note transforms these four again and again, so meet them now.

See Parent Functions for the full family.

Why the topic needs this: you can only transform a shape you already recognise. Every example ("shift the parabola up 3", "compress cosine") starts from one of these.

5. Constants, symbols, and the $||$ bars

The parent note uses four letters for the four transformation types. Meet them before they appear:

Why the topic needs this: the rules say "$|a|>1$ stretches, $0<|a|<1$ compresses." The bars are essential because a negative $a$ both flips and scales — its sign controls the flip and its size $|a|$ controls the scaling.

6. Substitution: the trick behind "inside-out"

The parent note's hardest idea — that $f(x-h)$ shifts right — relies on substitution: temporarily renaming a chunk of an expression with a new letter to see it clearly.

This is exactly composition — feeding one expression into another machine.

7. Domain, range, and the arrow $\to$

Why the topic needs this: stretching $\sin x$ by $3$ changes its range from $[-1,1]$ to $[-3,3]$; shifting $\sqrt{x}$ left changes its domain. You track transformations partly by watching domain and range move.

8. How it all feeds the topic

Equipment checklist

Test yourself — cover the right side and answer before revealing.