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

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Step 0 — The only thing we assume: a point on a circle

Before any letters, we need the raw sine wave itself. Everything grows from here.

WHAT. Picture a dot walking anticlockwise around a circle of radius $1$ (a "unit circle"; see Unit circle and sine definition). Its height above the centre is what we call $\sin$.

WHY a circle? Because going round and round is the simplest repeating motion there is — and a wave is just repetition drawn out sideways. The circle is the machine; the wave is its shadow.

PICTURE. On the left, the dot at angle $x$. Its vertical height is the red segment. On the right, we plot that height against the angle $x$ — and a wave falls out.

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Step 1 — Add $A$: stretch the height

WHAT. Multiply the whole output by a number $A$: $y=A\sin x$.

WHY. The raw wave only reaches height $1$. Real waves come in all sizes. Multiplying every height by $A$ scales the picture vertically without touching how fast it wiggles — exactly the "how tall?" freedom.

PICTURE. Every red height in Step 0 gets stretched by the factor $A$. The dashed grey wave is the original; the orange wave is $A=2$ times taller.

$y=\underset{\text{stretch factor}}{\underbrace{A}}\cdot \underset{\text{height in }[-1,1]}{\underbrace{\sin x}}$

• $A$ multiplies each height, so the new range is $[-A,A]$.

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Step 2 — Add $B$: squish the time

WHAT. Replace $x$ by $Bx$ inside the sine: $y=A\sin (Bx)$.

WHY. We want to control how fast the wave repeats. The sine finishes one lap when its input has increased by $2\pi$. If the input is $Bx$ instead of $x$, then $x$ only has to move a fraction of the way to push $Bx$ through a full $2\pi$.

PICTURE. Watch the input speedometer: with $B=2$, by the time $x$ reaches $\pi$, the input $Bx$ has already hit $2\pi$ — a whole cycle done in half the space.

Solve for the $x$-length of one cycle. One cycle means the input rises by $2\pi$:

$\underset{\text{input speed}}{\underbrace{B}}\cdot \underset{\text{x-length of one cycle}}{\underbrace{\Delta x}}=\underset{\text{one full lap}}{\underbrace{2\pi }}\Rightarrow \Delta x=\frac{2\pi }{B}$

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Step 3 — Add $C$: slide sideways (and the sign trap)

WHAT. Add a constant $C$ inside: $y=A\sin (Bx+C)$.

WHY. So far every wave starts its rise at $x=0$. We want to choose where the cycle begins. Adding $C$ pre-loads the input: the sine already thinks $x$ is a bit further along.

The key move is to factor $B$ out so the shift is measured in real $x$-units:

$Bx+C=B\left(x+\frac{C}{B}\right)$

• $B$ ::: pulled outside, unchanged — it still controls speed.
• $\frac{C}{B}$ ::: the actual horizontal distance the wave slides. This is the whole point of factoring — $C$ alone is added to $Bx$, not to $x$, so $C$ by itself is not the shift.

WHY does $+\frac{C}{B}$ slide the graph left? Because a $+$ inside a function reaches the target height earlier. If the wave already peaks when $x+\frac{C}{B}$ hits a certain value, then $x$ itself hits that peak sooner — the whole picture moves toward smaller $x$, i.e. left.

PICTURE. The teal wave is the shifted one. The plum arrow shows the slide: a positive inside offset drags the pattern to the left by $\frac{C}{B}$.

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Step 4 — Add $D$: lift the whole thing

WHAT. Add $D$ outside everything: $y=A\sin (Bx+C)+D$.

WHY. The wave still wobbles around the level $y=0$. Adding $D$ to every output raises the whole picture, so it now wobbles around a new horizontal line — the midline $y=D$.

PICTURE. The grey wave sits on $y=0$; the orange wave is the same shape lifted onto midline $y=D$. Peaks and troughs ride $|A|$ above and below it.

$y=A\sin (Bx+C)+\underset{\text{midline height}}{\underbrace{D}}$

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Step 5 — The degenerate & edge cases (never skip these)

A picture is only trustworthy if it survives the extreme inputs. Here is every corner.

Case $A<0$ (flip). The stretch factor is negative, so heights that were positive become negative. The wave turns upside-down — it now falls first instead of rising. Its height $|A|$ is unchanged; only the direction flips (this is exactly cos as shifted sin territory: a flip is a half-period slide in disguise).

Case $B<0$. $\sin$ has the property $\sin (-u)=-\sin u$, so a negative $B$ is a flip plus the same period $2\pi /|B|$. We use $|B|$ in the period so it stays a positive length.

Case $A=0$ (dead wave). Every height is $0\cdot \sin (\dots )=0$, so $y=D$: a flat horizontal line. No wobble at all — amplitude zero. The formula still works; it just draws the midline itself.

Case $B=0$ (frozen input). The input never moves: $\sin (0\cdot x+C)=\sin C$ is a constant, so $y=A\sin C+D$, again a flat line. The period $2\pi /|B|$ blows up to infinity — one "cycle" would take forever, which is a flat line. Consistent.

PICTURE. All four oddballs side by side, so you have literally seen each one.

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The one-picture summary

Here is the whole journey in a single frame: raw circle-shadow → stretch by $A$ → squish by $B$ → slide by $-C/B$ → lift by $D$. Follow the coloured labels top to bottom and you have re-derived $y=A\sin (Bx+C)+D$ from a dot on a circle.

Recall Feynman retelling — say it like a story

We start with a dot going round a circle; its height traced against the angle gives the basic wave. Then I turn four knobs. Knob $A$ stretches the wave taller — but "tall" is a distance, so its size is $|A|$, and a negative $A$ just flips it over. Knob $B$ speeds up the dot; the faster it laps the circle, the tighter the waves bunch up, so the period is $2\pi$ divided by $B$ — big $B$, brief wave. Knob $C$ pre-loads the angle so the pattern slides sideways; but because $C$ is bolted onto $Bx$ and not $x$, I must first factor out $B$ to see the real slide $C/B$, and because "inside acts backwards" it slides left for a plus. Knob $D$ just lifts the whole rope onto a shelf at height $D$. Turn all four and any sine wave in the universe appears.

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Connections

• Unit circle and sine definition — the circle-and-height picture Step 0 is built on.
• Function transformations — the general "inside vs outside", "acts backwards" rules used in Steps 2–4.
• cos as shifted sin — why a flip / slide can turn one into the other (Step 5).
• Simple harmonic motion — where these waves come from in physics.
• Solving trig equations — using the finished form to find $x$ for a given $y$.