General sine function

What is it about?

The general sine function

f(x) = a \cdot \sin\big(b\,(x - c)\big) + d

describes periodic processes such as oscillations, tides or the length of the day over a year. Each parameter has a clear meaning:

Parameter	Meaning	Effect on the graph
$a$	amplitude $\lvert a \rvert$	vertical stretch; $a < 0$ reflects in the midline
$b$	period $p = \frac{2\pi}{b}$	horizontal compression ( $b > 1$ ) or stretch ( $b < 1$ )
$c$	horizontal shift	by $c$ to the right ( $c > 0$ ) or left ( $c < 0$ )
$d$	vertical shift	midline $y = d$

The range is $W = [\,d - |a|,\; d + |a|\,]$ .

The x-values are in radians: $\pi$ corresponds to $180^\circ$ , and a full period of the sine, $2\pi$ , corresponds to $360^\circ$ .

Notation: some books write $f(x) = a \cdot \sin(bx + c) + d$ . There the horizontal shift is $-\frac{c}{b}$ , not $c$ .

The cosine is a shifted sine: $\cos(x) = \sin\!\left(x + \frac{\pi}{2}\right)$ . That is why the simulation can treat both base functions the same way.

Explore

Change $a$ and $d$ . Both act only vertically. Amplitude 3 Midline y = 1
Change $b$ . Does the period get longer or shorter for larger $b$ ? b = 2 b = 0.5
Shift the graph with $c$ or drag the point $(c, d)$ on the midline where the period starts. c = π/4
Drag the maximum (amplitude) and the hollow point at the end of the period directly in the coordinate system.
Show that the cosine is a shifted sine. Sine shifted π/2 to the left – compare with the cosine.

Exercises

Exercise 1: Read off the parameters

Load exercise Determine the amplitude, period, shifts, range and the equation. The dashed curve is $y = \sin x$ .

Show solution

Midline $y = 1$ , so $d = 1$ . Amplitude 2, so $a = 2$ . A full period is $\pi$ long, so $b = \frac{2\pi}{\pi} = 2$ . The graph starts shifted $\frac{\pi}{4}$ to the right:

$f(x) = 2 \sin\!\left(2\left(x - \tfrac{\pi}{4}\right)\right) + 1, \qquad W = [-1,\, 3]$

Any shift that differs by whole periods also works, e.g. $c = \frac{5\pi}{4}$ . In the notation $a \cdot \sin(b(x + c)) + d$ , $c = -\frac{\pi}{4}$ . Check your result under Display → Show values and guides.

Exercise 2: Period

What is the period of $f(x) = \sin(3x)$ ? How many complete oscillations fit into $[0, 2\pi]$ ?

Show solution

$p = \frac{2\pi}{3}$ . The interval $[0, 2\pi]$ contains exactly 3 oscillations. Show in simulation

Exercise 3: Length of the day (modelling)

In Munich the longest day (around 21 June) is about 16 hours long, the shortest (around 21 December) about 8 hours 20 minutes. Describe the difference between the length of the day and 12 hours with a sine function ( $t$ : time in months from 1 January).

Show solution

Mean: $\frac{16 + 8.33}{2} \approx 12.2$ , so midline $d \approx 0.2$ (difference from 12 h).
Amplitude: $a \approx \frac{16 - 8.33}{2} \approx 3.8$ .
Period 12 months: $b = \frac{2\pi}{12} \approx 0.52$ .
Maximum at the end of June ( $t \approx 5.7$ ). For the sine, the maximum lies a quarter period (3 months) after $c$ , so $c \approx 2.7$ .

$f(t) \approx 3.8 \cdot \sin\big(0.52\,(t - 2.7)\big) + 0.2$

Show the model in the simulation

Ideas for the classroom

Order: first the “vertical” parameters $a$ and $d$ , then $c$ , and $b$ last – many students find its effect the least intuitive.
Common stumbling blocks:
- “Larger $b$ means a longer period”: in fact the period gets shorter.
- The sign of $c$ is misread, just as with parabolas.
- In $\sin(bx + c)$ , $c$ is read directly as the shift.
π axis: by default the x-axis is labelled in multiples of $\pi$ and $c$ snaps to steps of $\frac{\pi}{12}$ . Turn this off under Display for modelling tasks; $c$ can then be set as a decimal.
Exercise mode: switch off Show values and guides. Sliders, guides and results are then hidden, and the class reads the parameters off the graph. You can pass such a state on as a link via Share.
Introduction: the simulation “Unit circle” shows how the sine curve arises from motion on a circle.
Notation: some curricula and textbooks write $a \cdot \sin(b \cdot (x + c)) + d$ . You can switch the simulation to this form under Display → Notation; a positive $c$ then shifts to the left.