Amplitude, period and phase in models

1. Read vertical and phase parameters separately.

   $$|A|=3,\quad D=1$$

2. Divide the phase shift by frequency and compute a full cycle.

   $$h=\pi/4,\quad T=\pi$$

Amplitude 3; midline 1; period π; right shift π/4.

Checks and common pitfalls: A phase of −π/2 inside the sine is not the horizontal displacement.

1. Determine vertical parameters.

   $$D=2,\quad A=3$$

2. Determine frequency but retain free phase.

   $$\omega=2,\quad\phi\text{ undetermined}$$

D=2, A=3, ω=2; phase is not fixed.

Checks and common pitfalls: Matching extreme heights and cycle length is insufficient to locate a cycle horizontally.

1. Recover scale and frequency.

   $$D=2,\quad A=4,\quad\omega=3$$

2. Place an upward crossing at the given input.

   $$y=2+4\sin(3(x-\pi/6))$$

y=2+4sin(3(x−π/6)).

Checks and common pitfalls: The direction of crossing removes the half-cycle ambiguity.

1. The range is symmetric about zero.

   $$-2\le y\le2$$

2. Half the range width gives amplitude.

   $$(2-(-2))/2=2$$

2.

Checks and common pitfalls: The minus sign reflects the curve vertically.

1. Set a full phase increase.

   $$4T=2\pi$$

2. The amplitude does not change the cycle length.

   $$T=\pi/2$$

π/2.

Checks and common pitfalls: Frequency scales the horizontal input, not the vertical amplitude.

1. Expose the displaced input.

   $$2x+\pi/3=2(x+\pi/6)$$

2. Read h with its sign.

   $$h=-\pi/6$$

Left by π/6.

Checks and common pitfalls: The phase must be divided by 2 before reading horizontal distance.

1. Multiply the bound by 2.

   $$-2\le2\sin x\le2$$

2. Add 3; both extreme sine values occur.

   $$1\le y\le5$$

Midline 3; range [1,5].

Checks and common pitfalls: The vertical shift changes the midpoint, not the amplitude.

1. Evaluate the same input in both formulas.

   $$f(\pi/2)=0,\quad g(\pi/2)=2$$

2. Identify horizontal versus vertical scaling.

   $$T_f=\pi,\quad T_g=2\pi$$

No; one changes frequency, the other amplitude.

Checks and common pitfalls: A counterexample and parameter explanation serve different purposes and both are useful.

1. Identify the constant function.

   $$y=4$$

2. Every positive shift preserves it, so no smallest positive shift exists.

   $$T>0\Rightarrow f(x+T)=f(x)$$

No least positive period.

Checks and common pitfalls: The nonconstant-sinusoid period formula requires nonzero amplitude and frequency.

1. Measure the cycle.

   $$T=5\pi/8-\pi/8=\pi/2$$

2. Invert the period-frequency relation.

   $$\omega=2\pi/T=4$$

4.

Checks and common pitfalls: Nonadjacent maxima would leave a whole-number cycle ambiguity.

1. Locate upward rather than all zero crossings.

   $$3x-\pi=2k\pi$$

2. Choose the smallest nonnegative representative.

   $$x=\pi/3$$

π/3.

Checks and common pitfalls: x=0 is a zero but crosses downward, so direction is essential.

1. Read the period in hours.

   $$T=2\pi/(\pi/6)=12$$

2. Convert the phase interval to time and repeat once.

   $$t\in[4,8]\cup[16,20]$$

Period 12 h; height≤3 m on [4,8] and [16,20] h.

Checks and common pitfalls: The horizontal variable is time; the phase is the dimensionless angle πt/6.

1. Construct two compatible models.

   $$f(t)=2,\quad g(t)=2+\sin(2\pi t)$$

2. Verify every integer sample.

   $$g(n)=2+\sin(2\pi n)=2$$

3. Choose an unsampled quarter-cycle time.

   $$f(1/4)=2,\quad g(1/4)=3$$

Models f=2 and g=2+sin(2πt); measure at t=1/4.

Checks and common pitfalls: Equal samples do not prove a constant continuous signal; the new measurement is a discriminating test.