Sinusoidal transformations

In $f(x) = a\sin(b(x-h)) + k$ each parameter owns exactly one feature:

1. a: vertical stretch. Amplitude = |a|. A negative a reflects the wave over its midline; the amplitude stays positive.
2. b: horizontal compression. Period = 2π/|b|. The b counts how many basic cycles are squeezed into the old 2π window; it is not the period itself.
3. h: phase (horizontal) shift, right when the argument reads (x − h), left for (x + h). Readable only when b has been factored out.
4. k: vertical shift. The midline is y = k, and the range is k ± |a|.

Read them in the order a, k, b, h: height and center first, then timing, then position. Mixing the roles, quoting b as the period, or k as the amplitude, is the fastest way to lose a feature question.

Multiplying the input by b makes the graph cycle b times faster, so the period shrinks by that factor: period = 2π/|b|. For $y = \sin(3x)$, the period is $2\pi/3$; three full waves now live where one used to. For $y = \sin(x/2)$, b = 1/2, so the period stretches to $4\pi$.

Run the law both directions. Want a period of π/2? Solve $2\pi/|b| = \pi/2$ to get b = 4. And keep amplitude away from the calculation entirely: $4\sin(x)$ is tall, not fast; its period is still $2\pi$.

The phase shift can only be read from the form $b(x - h)$. An expression like $\sin(2x + \pi)$ hides its shift until you factor: $\sin(2x + \pi) = \sin(2(x + \tfrac{\pi}{2}))$, a shift LEFT by $\pi/2$, not by $\pi$. Reading the unfactored constant is the single most common transformation error in this unit.

Direction is the second half of the discipline: $(x - h)$ moves the graph right h, and $(x + c)$ moves it left c, the reverse of what the sign suggests. Combine the two habits into one procedure: factor out b, then read the sign inside the parentheses backward.

A negative a flips the wave over its midline: $y = -\cos(x)$ starts at its minimum where cosine starts at its maximum. The amplitude is still |a|; "amplitude −2" is not a thing, and a reflected wave still swings equally far above and below its midline.

Because sine and cosine are the same wave a quarter cycle apart, every sinusoid has both a sine form and a cosine form: $\cos(x) = \sin(x + \tfrac{\pi}{2})$, and in general a cosine form is just a sine form with the shift adjusted by a quarter period. Choosing the form that matches a graph's starting position (crest for cosine, rising midline for sine) usually saves a shift entirely.