Foundations — Transformations of trig graphs — A·sin(Bx + C) + D (amplitude, period, phase, vertical shift)

This page assumes you know nothing. We build every symbol the parent note uses, one at a time, each resting on the one before it. If a piece already makes sense to you, skim it — but do not skip, because the last pieces lean on the first.

0. What is a "function" and why the letters $x$ and $y$?

The picture: a horizontal $x$-axis (inputs) and a vertical $y$-axis (outputs). Each input $x$ has a dot floating at height $y$ above it. Join the dots → a graph.

Why the topic needs this: the whole note is about one function, $y=A\sin (Bx+C)+D$, and how moving its dots up/down/sideways reshapes the curve.

1. Angles, and why we measure them in $\pi$

The picture: stand at the centre of a circle of radius $1$ and walk once around the edge. The distance you walk is the radian measure of the turn. Since the edge of a radius-$1$ circle has length $2\pi$, one full turn $=2\pi$ radians.

• $\pi \approx 3.14159\dots$ = half a turn.
• $\frac{\pi }{2}$ = a quarter turn. $\frac{\pi }{4}$ = an eighth turn.

Whenever you see $\pi$ in the parent note, silently translate it to "a half-turn's worth".

2. The unit circle and the birth of $\sin$

This is the single most important picture on the page. Everything else is decoration on top of it.

The picture: the dot starts at the far right (height $0$). As it climbs to the top, its height rises to $1$. Coming down the left side its height falls back through $0$ to $-1$ at the bottom, then back to $0$. Unroll that height against the angle and you get the familiar wave.

See Unit circle and sine definition for the full construction.

3. Reading the basic wave $y=\sin x$

The picture: the plain $\sin x$ wave crosses the origin going upward, peaks at height $+1$, dips to $-1$, and returns — all within a horizontal distance of $2\pi$. Its midline is the flat $x$-axis.

Why the topic needs this: these four features (range height, cycle length, wobble-centre, starting spot) are exactly the four things $A,B,C,D$ will change. Know the "before" picture and every transformation is just "how did the after differ?"

4. Multiplying and adding — the four moves

Now the four letters. Each is a piece of ordinary school arithmetic you already own; the only new idea is what it does to the picture.

Picture: grab the top and bottom of the wave and pull them apart (if $A>1$) or squash them together (if $0<A<1$). If $A$ is negative, you also flip the whole thing upside-down.

Picture: the wave gets squished together side-to-side (if $B>1$) so more cycles fit in the same width.

Picture: the entire wave slides sideways. (Which way is the famous trap — see §5.)

Picture: the whole wave floats up (if $D>0$) or sinks (if $D<0$); its midline moves from $y=0$ to $y=D$.

For the general rules behind "multiply/add inside vs outside" applied to any function, see Function transformations.

5. Inside vs outside — why sliding feels backwards

This is the concept that trips everyone, so it gets its own picture and its own name now, before the parent note assumes it.

Why: the graph of $\sin (x+C)$ reaches its peak when the inside equals a quarter-turn, i.e. when $x+C=\frac{\pi }{2}$, i.e. when $x=\frac{\pi }{2}-C$. A bigger $C$ makes the peak happen at a smaller $x$ — earlier, to the left. So "$+C$ inside" slides the graph left.

6. Absolute value — the $||$ bars

Picture: distance along a number line from $0$ — always positive (or zero), because a distance can't be negative.

Why the topic needs it: amplitude is how tall the swing is — a height, a distance. A negative $A$ flips the wave but doesn't make it "$-3$ tall". So amplitude is written $|A|$, always non-negative.

7. Notation you'll meet: $\Delta$, $T$, factoring

Why: the parent note derives the period from "$B\Delta x=2\pi$" — read as "$B$ times the change in $x$ equals one full turn." Solving gives $T=\Delta x=\frac{2\pi }{|B|}$.

Why the topic needs it: only after factoring can you see the real horizontal slide $\frac{C}{B}$, measured in $x$, instead of the fake number $C$ measured in $Bx$.

8. Cosine — the same wave, shifted

Picture: at $x=0$ the dot sits at the far right — full horizontal distance $1$ — so $\cos$ starts at its maximum, whereas $\sin$ starts at $0$. In fact $\cos x=\sin \left(x+\frac{\pi }{2}\right)$: cosine is sine slid a quarter-turn left.

See cos as shifted sin. This is why Example 2 of the parent (a $\cos$) fits the same $A,B,C,D$ machinery.

Prerequisite map

Equipment checklist

Test yourself — each line hides its answer.

Connections

• Parent topic — where these symbols get put to work.
• Unit circle and sine definition — the source of $\sin$, its range and period.
• Function transformations — the general inside/outside rules for any function.
• cos as shifted sin — why cosine slots into the same formula.
• Simple harmonic motion — where these waves describe real oscillations.
• Solving trig equations — using these graphs to find where $y$ hits a value.