3.7 Circular functions and their graphsIB Maths: Analysis and Approaches HL: Revision notes

Think of each graph in words, since you will not be asked to draw it:

• $y = \sin x$: starts at $(0, 0)$, maximum 1 at $x = \frac{\pi}{2}$, back to 0 at $\pi$, minimum $-1$ at $\frac{3\pi}{2}$, 0 at $2\pi$. Period $2\pi$, amplitude 1.
• $y = \cos x$: the same wave shifted left by $\frac{\pi}{2}$; starts at its maximum $(0, 1)$. Period $2\pi$, amplitude 1.
• $y = \tan x$: period $\pi$, zeros at $0, \pi, 2\pi, \ldots$, vertical asymptotes at $x = \frac{\pi}{2} + k\pi$. It has no amplitude because it is unbounded.

These are periodic functions: $\sin(x + 2\pi) = \sin x$ for all $x$.

Each constant controls one feature:

• $|a|$ is the amplitude; a negative $a$ reflects the graph in the principal axis.
• Period $= \frac{2\pi}{b}$ (or $\frac{360^\circ}{b}$ in degrees).
• $c$ is a horizontal translation of $-c$: $x + c$ moves the graph left by $c$.
• $d$ is the principal axis $y = d$ (the vertical translation).

So the maximum value is $d + |a|$, the minimum is $d - |a|$, and the range is $d - |a| \le f(x) \le d + |a|$. The same reading works for $a\cos(b(x + c)) + d$.

Example: $f(x) = 3\sin\left(2\left(x - \frac{\pi}{6}\right)\right) + 1$ has amplitude 3, period $\pi$, principal axis $y = 1$ and range $[-2, 4]$.

Starting from $y = \sin x$:

• $y = a\sin x$: vertical stretch, scale factor $a$.
• $y = \sin bx$: horizontal stretch, scale factor $\frac{1}{b}$.
• $y = \sin(x + c)$: translation by $-c$ in the $x$-direction.
• $y = \sin x + d$: translation by $d$ in the $y$-direction.

For example, $y = \sin x \to y = 3\sin 2x$ is a vertical stretch with scale factor 3 and a horizontal stretch with scale factor $\frac{1}{2}$.

Order matters for horizontal changes: stretching by $\frac{1}{2}$ and then translating right $\frac{\pi}{3}$ gives $\sin\left(2\left(x - \frac{\pi}{3}\right)\right)$, but translating first and then stretching gives $\sin\left(2x - \frac{\pi}{3}\right)$.

Repeating real-life quantities (tides, Ferris wheels, daylight hours, temperature over a year) can be modelled by $h(t) = a\cos(bt) + d$ or $a\sin(b(t + c)) + d$:

1. $d$ = the middle value = $\frac{\text{max} + \text{min}}{2}$.
2. $|a|$ = $\frac{\text{max} - \text{min}}{2}$ (for a wheel, the radius).
3. $b = \frac{2\pi}{\text{period}}$.
4. Choose sin or cos and the sign of $a$ from the starting value: $\cos$ starts at a maximum, $-\cos$ at a minimum, $\sin$ on the principal axis going up.

Example: a wheel of radius 20 m, centre 22 m high, one turn every 10 minutes, starting at the bottom: $h(t) = -20\cos\left(\frac{\pi t}{5}\right) + 22$.