Transformations of Sinusoidal Functions
Introduced| English |
|---|
| amplitude/ˈæmplɪtjuːd/ |
| midline/ˈmɪdlaɪn/ |
| period/ˈpɪərɪəd/ |
| phase shift/feɪz ʃɪft/ |
| sinusoidal function/ˌsaɪnəˈsɔɪdl ˈfʌŋkʃn/ |
Bending the parent wave
- The parent $\sin x$ is fixed, but real waves are taller, faster, and shifted.
- Four transformations reshape it into any sinusoid you need.
- Two act vertically (on the output), two act horizontally (on the input).
- Learn what each dial does, and you can graph or write any wave.
Vertical changes
- Multiplying by $a$ stretches the wave vertically: the amplitude 振幅 becomes $|a|$.
- A negative $a$ also flips the wave upside down.
- Adding $d$ raises or lowers the whole wave, setting the midline 中线 to $y = d$.
- Together these fix the peak ($d + |a|$) and trough ($d - |a|$).
Increasing $|a|$ in $y = a\sin(bx) + d$ makes the wave…
$|a|$ is the amplitude, so a larger $|a|$ stretches the wave taller.
Increasing $d$ in $y = a\sin(bx) + d$ …
$d$ is the midline level, so adding to it lifts the whole wave up.
Horizontal changes
- The factor $b$ squeezes the wave horizontally: the period 周期 becomes $\tfrac{2\pi}{b}$.
- A bigger $b$ means faster cycling — more waves in the same space.
- The constant $c$ produces a phase shift 相位平移, sliding the wave left or right.
- As always, $(x - c)$ moves the graph right by $c$.
Stretch, shift, and lift a sine wave
y = a·sin(b(x − c)) + d
Grow a to raise the amplitude, grow b to shorten the period, change c to slide sideways, and change d to raise the midline.
For $y = \sin(3x)$, what is the period? (Use $\pi \approx 3.14$; give the value of $2\pi/3$.)
Period $= \dfrac{2\pi}{3} \approx 2.09$ — tripling $b$ shrinks the period to a third.
The constant $c$ inside $\sin(b(x - c))$ produces a horizontal ____ shift.
A phase shift slides the wave left or right without changing its shape.
Combining all four
- A full sinusoidal function 正弦型函数 uses all four at once: $y = a\sin(b(x-c)) + d$.
- Apply them in a sensible order: stretch, then shift.
- Each parameter acts independently, so you can read them off one at a time.
- The same four transform a cosine identically.
Match each parameter of $y = a\sin(b(x-c)) + d$. Select all correct.
$c$ controls the phase shift, not amplitude. The other three are correct.
Reading a transform from a graph
- Measure the midline: average the highest and lowest points → that is $d$.
- Measure the amplitude: half the distance from peak to trough → that is $|a|$.
- Measure the period: the horizontal length of one cycle → gives $b = \tfrac{2\pi}{\text{period}}$.
- Find where a cycle "starts" to read the phase shift $c$.
Amplitude and midline come from outside the sine (they scale and lift the output); period and phase shift come from inside (they act on the input). Mixing up inside and outside — especially the backwards horizontal shift — is the classic sinusoid error.
Describe the transformations in $y = 4\sin\!\big(2(x - 1)\big) + 3$.
- Amplitude $|a| = 4$; midline $y = 3$; so it swings between $-1$ and $7$.
- Period $= \tfrac{2\pi}{2} = \pi$.
- Phase shift: right by $1$ (from the $x - 1$ inside).
Transform the parent sine with four dials: $a$ sets the amplitude, $d$ the midline, $b$ the period ($\tfrac{2\pi}{b}$), and $c$ the phase shift. Outside-the-sine values act vertically; inside values act horizontally and run backwards.