Sinusoidal Functions
Introduced| English | Meaning |
|---|---|
| sinusoidal function/ˌsaɪnəˈsɔɪdl ˈfʌŋkʃn/ | A function with the shape of a sine or cosine wave, possibly shifted or scaled. |
| amplitude/ˈæmplɪtjuːd/ | The maximum displacement from an equilibrium position. |
| midline/ˈmɪdlaɪn/ | A reference line halfway between the maximum and minimum values of an oscillating graph. |
| period/ˈpɪərɪəd/ | The time taken for one complete oscillation or cycle. |
| phase shift/feɪz ʃɪft/ | A horizontal displacement of a periodic function's graph. |
One wave, four dials
- The parent sine wave is fixed, but real waves come in every size and speed.
- Four numbers let you reshape it: taller or shorter, faster or slower, shifted up or sideways.
- Master those four and you can match any smooth periodic curve.
- This flexible family is called the sinusoidal functions.
The general form
- A sinusoidal function 正弦型函数 is $y = a\sin(b(x - c)) + d$.
- The amplitude 振幅 is $|a|$ — how far the wave rises above and below its centre.
- The midline 中线 is $y = d$ — the horizontal line the wave oscillates around.
- Cosine fits the same template; the two are interchangeable with a shift.
In $y = a\sin(b(x - c)) + d$, the amplitude is…
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The coefficient $a$ stretches the wave vertically, so the amplitude is $|a|$.
In $y = a\sin(b(x - c)) + d$, the midline is the line $y =$ ____.
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Adding $d$ shifts the whole wave up, so the midline sits at $y = d$.
Amplitude and midline
- Change $a$ to stretch the wave taller or squash it flatter (or flip it if $a < 0$).
- Change $d$ to slide the whole wave up or down to a new midline.
- The maximum is $d + |a|$ and the minimum is $d - |a|$.
- So $a$ and $d$ together fix the top and bottom of the wave.
Reshape a sinusoid with four controls
y = a·sin(b(x − c)) + d
a sets the amplitude, b sets the period, c shifts it sideways, and d sets the midline. Change each and watch.
The period of $y = a\sin(b x) + d$ is…
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The factor $b$ compresses the wave horizontally, so the period shrinks to $\dfrac{2\pi}{b}$.
What is the period of $y = \sin(2x)$? (Use $\pi \approx 3.14$.)
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Period $= \dfrac{2\pi}{2} = \pi \approx 3.14$ — doubling $b$ halves the period.
Period from b
- The number $b$ controls how quickly the wave cycles.
- The period 周期 is $\dfrac{2\pi}{b}$ — a bigger $b$ packs cycles closer together.
- So $y = \sin(2x)$ repeats twice as fast, with period $\pi$.
- Frequency (cycles per unit) is the reciprocal of the period.
In $y = a\sin(b(x - c)) + d$, match each role. Select all correct.
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$b$ controls the period, not the amplitude. A phase shift of $c$ slides the wave sideways. The other three are correct.
Phase shift from c
- The value $c$ slides the wave left or right — a phase shift 相位平移.
- Inside the function, $(x - c)$ moves the graph right by $c$ (the usual backwards rule).
- A phase shift lets a sine model line up with data that peaks at a chosen time.
- Together $a, b, c, d$ pin down any sinusoid exactly.
Remember $b$ sets the period, not the amplitude, and it works inversely: a larger $b$ gives a shorter period. And the shift $(x - c)$ moves the graph right, not left — the inside-the-function reversal trips many students.
Describe $y = 3\sin(2x) + 5$.
- Amplitude $|a| = 3$; midline $y = 5$; so it swings between $2$ and $8$.
- Period $= \dfrac{2\pi}{2} = \pi$ — one cycle every $\pi$ units.
- No phase shift ($c = 0$), so it starts a cycle at $x = 0$ on the midline, rising.
A sinusoidal function $y = a\sin(b(x-c)) + d$ has amplitude $|a|$, midline $y = d$, period $\dfrac{2\pi}{b}$, and a phase shift of $c$. These four numbers reshape the parent wave to fit any smooth periodic pattern.
Write from a period of pi over 2
- The function is y equals sine of 4x.
- The period is 2 pi divided by 4.
What is the period of y equals sine of 4x?
2 pi over 4 is pi over 2.
What is the period?
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The inside coefficient divides the parent period.