Transforming Trigonometric Graphs
The same outside/inside toolkit that stretches and shifts any curve applies directly to sine, cosine and tangent — giving each transformation a trig-specific name: amplitude, period and phase shift.
The toolkit, applied to trig
Trig graphs transform exactly like any other function: outside changes touch $y$ and behave intuitively; inside changes touch $x$ and behave in reverse. What's new here is that each transformation has its own trig vocabulary — amplitude, period, and phase shift — because the wave shape gives them a physical meaning a generic curve doesn't have.
$y = a\sin\big(b(\theta - c)\big) + d$ packs all four transformations of $y=\sin\theta$ into one equation: $a$ is an outside stretch (amplitude), $d$ is an outside shift (vertical shift), $b$ is an inside stretch (period), and $c$ is an inside shift (phase shift) — exactly the same four slots as any transformed function, just with trig-specific names.
Vertical transformations — amplitude and vertical shift
These are the outside changes, so they behave exactly as they look:
$y = a\sin\theta$: an outside stretch by factor $a$. The wave still oscillates about $0$, but now reaches up to $a$ and down to $-a$. This maximum reach is called the amplitude — for $y=\sin\theta$ and $y=\cos\theta$ the amplitude is normally $1$, so $a$ scales it directly.
$y = \sin\theta + d$: an outside shift up by $d$. The wave still has amplitude $1$, but now oscillates about the line $y=d$ instead of $y=0$, so its range becomes $[d-1,\ d+1]$.
Combined, $y = a\sin\theta + d$: amplitude $a$, oscillating about $y=d$, with range $[d-a,\ d+a]$.
Both $a$ and $d$ are outside changes — they only ever touch $y$-values. The $x$-values where the wave completes a cycle are entirely unaffected, so the period stays exactly $360°$ (or $180°$ for tan) no matter what $a$ and $d$ are. Only an inside change can alter the period.
Horizontal transformations — period and phase shift
These are the inside changes, so — as always — they behave in reverse of how they look:
$y = \sin(b\theta)$: an inside stretch by factor $\tfrac1b$. A larger $b$ squashes the wave horizontally, fitting more cycles into $360°$; the new period becomes $\dfrac{360°}{b}$. (For tan, the base period $180°$ becomes $\dfrac{180°}{b}$.)
$y = \sin(\theta - c)$: an inside shift right by $c$ (inside changes reverse the intuitive direction, so $-c$ inside the bracket moves the graph right). This horizontal slide is called a phase shift.
The reflection rules carry over unchanged too: $-\sin\theta$ (outside) flips the wave in the $\theta$-axis, turning peaks into troughs; $\sin(-\theta)$ (inside) flips it in the $y$-axis. Since sine has rotational symmetry through the origin, $\sin(-\theta) = -\sin\theta$ — for sine specifically, both reflections give the same graph. Cosine is different: it's symmetric about the $y$-axis, so $\cos(-\theta) = \cos\theta$, meaning an inside reflection does nothing to a cosine graph at all.
Worked example 1
The period only ever depends on $b$ — never on $a$ or $d$. It's a common slip to think a "bigger" equation somehow stretches the period too; keep the four slots ($a$, $b$, $c$, $d$) mentally separate and each one only affects what it's supposed to.
Worked example 2
Just as with any transformed function, track the key point through each transformation one at a time — inside changes move $\theta$, outside changes move $y$.
It's tempting to shift the maximum left by $30°$ because of the minus sign — but inside changes always reverse, just as $f(x-a)$ shifts right. Sketch $y=\cos\theta$ mentally: subtracting from $\theta$ before evaluating means you need a larger $\theta$ to reach the same output, which is a shift right.
Worked example 3
Order matters here only in how you describe it, not in the final equation — a stretch then a shift, applied to the whole graph in that order, always lands on $a\sin\theta + d$ with the stretch factor multiplying before the shift is added, since that's exactly how outside operations compose.
Worked example 4
Reflecting in the $\theta$-axis swaps every maximum with a minimum, but doesn't change how far either reaches from the centre line — so the amplitude, and therefore the total range width, is identical whether $a$ is $+2$ or $-2$. Only the sign of $a$ decides whether the graph starts by rising or falling from its centre line.
Reading $b$ backwards is the single most common error here: $y=\sin(3\theta)$ has period $360°/3=120°$ — a smaller period, i.e. a more compressed graph — even though $3$ is a "bigger" number. Inside multipliers greater than $1$ always compress; between $0$ and $1$, they stretch.
Check yourself
State the amplitude, period, and range of $y = 2\cos\left(\dfrac{\theta}{2}\right) + 3$, and find the coordinates of the point that $(0°, 1)$ — the starting point of $y=\cos\theta$ — maps to on this new graph.
(Hint: b = ½ here, so the period formula gives 360° ÷ ½.)