Notes on Period of Modified Sine Equations

Understanding the Period of Sine Functions

The period of a function is the horizontal length over which the function completes one full cycle.
For the basic sine function, the period is the length needed for the input to increase by a full $2\pi$ in radians.

Consider the general sine form with horizontal scaling and vertical shift:
- y = A \sin(Bx) + D
Period formula: T = \frac{2\pi}{|B|}
Effects:
- Amplitude $A$ changes the height of the wave but not its period.
- Vertical shift $D$ moves the wave up or down but does not affect the period.
If the inner argument includes a phase shift, the period remains determined by $B$.

Phase shift is introduced via the term inside the sine's argument.
- For example: y = A \sin(Bx - C) + D = A \sin\big(B\big(x - \frac{C}{B}\big)\big) + D
The horizontal shift (phase shift) is \Delta x = \frac{C}{B} to the right if $-C/B$ is positive, but this shift does not change the period.
Therefore, even with phase shifts, the period remains:
- T = \frac{2\pi}{|B|}

For the most common sinusoidal form with both amplitude and vertical shift:
- y = A \sin(Bx + C) + D
- Period: T = \frac{2\pi}{|B|}
If $B$ is negative, the period is still the same because of the absolute value:
- T = \frac{2\pi}{|B|}

If the sine is composed with a nonlinear inner function, e.g., y = \sin(g(x)), the period depends on $g\,$:
- If $g(x)$ is linear, $g(x) = Bx + C$, then the period is still T = \frac{2\pi}{|B|}.
- If $g(x)$ is nonlinear (e.g., $g(x) = x^2$) or not strictly linear, the resulting function may not be periodic at all.
Key takeaway: The period is tied to how the inner argument grows with $x$; linear inner arguments yield a clear, constant period.

Frequency (cycles per unit of x) is the reciprocal of the period:
- f = \frac{1}{T} = \frac{|B|}{2\pi}
Angular frequency (radians per unit of x) is simply the absolute value of $B$:
- \omega = |B|

Example 1: y = \sin(3x)
- $B = 3$; Period T = \frac{2\pi}{3}; Frequency f = \frac{3}{2\pi}
Example 2: y = \sin\left(\frac{1}{2}x\right)
- $B = \tfrac{1}{2}$; Period T = \frac{2\pi}{1/2} = 4\pi
Example 3: y = \sin(2x - \tfrac{\pi}{6}) + 4
- $B = 2$; Period T = \frac{2\pi}{2} = \pi
- Phase shift: inner is $2x - \tfrac{\pi}{6} = 2\left(x - \tfrac{\pi}{12}\right)$, so shift right by\Delta x = \frac{\tfrac{\pi}{6}}{2} = \tfrac{\pi}{12}, but period remains T = \pi.
Example 4: y = -2\sin(-4x + \pi) - 3
- $B = -4$; Period T = \frac{2\pi}{|-4|} = \frac{\pi}{2}
- Phase shift: inner $-4x + \pi = -4\left(x - \tfrac{\pi}{4}\right)$, a rightward shift of \Delta x = \tfrac{\pi}{4}; period unaffected.

Step 1: Identify the inner argument of the sine function.
Step 2: If the inner argument is linear, i.e., of the form Bx + C(or equivalently B(x - \phi)), compute the period as T = \frac{2\pi}{|B|}.
Step 3: If the inner argument is not linear, assess periodicity by checking if there exists a T such that for all x, g(x+T) - g(x) is a multiple of 2\pi; if not, the function may be non-periodic.
Step 4: Note that amplitude and vertical shift do not affect the period.
Step 5: Distinguish phase shift from a change in period; phase shift affects horizontal position, not the length of one cycle.

Do not confuse the phase shift with the period; phase shift changes the starting point, not the cycle length.
Always use absolute value in the denominator: T = \frac{2\pi}{|B|}, since negative $B$ reverses the direction but not the length of a period.
When reading an equation like y = A \sin(Bx - C) + D, remember the equivalent form y = A \sin\big(B\big(x - \frac{C}{B}\big)\big) + D to read off the phase shift.
If the inner function is something like g(x) = kx^2 or g(x) = \sin(x), carefully check periodicity; not all $g$ yield a periodic $\sin(g(x))$.

The period is a fundamental aspect of sinusoidal models used in physics, engineering, and signal processing (e.g., wave motion, AC signals).
Amplitude, vertical shift, and phase shift are orthogonal to period in the sense that they modify height, baseline, and starting point, but not cycle length.
In graphs, horizontal compression (larger $|B|$) shortens the period; horizontal expansion (smaller $|B|$) lengthens the period.
The same period rule applies to cosine: y = A \cos(Bx + C) + D has period T = \frac{2\pi}{|B|}, mirroring sine.

Problem 1: Find the period of y = 5\sin(7x) - 2.
- Answer: T = \frac{2\pi}{7}
Problem 2: Find the period and phase shift of y = \sin\left(\tfrac{1}{3}x - \tfrac{\pi}{4}\right).
- Period: T = \frac{2\pi}{|1/3|} = 6\pi; Phase shift: \Delta x = \frac{C}{B} = \frac{\tfrac{\pi}{4}}{1/3} = \tfrac{3\pi}{4} to the right.
Problem 3: Is y = \sin(x^2) periodic? Explain.
- Answer: No, it is not periodic because the inner argument grows nonlinearly and does not repeat with a fixed interval.

The period of a sine function with horizontal scaling and phase shift is determined entirely by the coefficient of x inside the sine, specifically T = \frac{2\pi}{|B|}, regardless of amplitude or vertical shift.
Phase shifts affect where a cycle starts but not how long a cycle lasts. - Nonlinear inner functions require extra checks to determine periodicity.