Finding Amplitude from an Equation

Find the amplitude of $y = 3\sin(x)$

Identify the coefficient A: Comparing $y = 3\sin(x)$ with $y = A\sin(Bx)$ = $A = 3$

Calculate amplitude: Amplitude $= |A| = |3|$ = $\text{Amplitude} = 3$

Interpret the result: The wave reaches 3 units above and below the midline = Maximum: 3, Minimum: $-3$

Answer: The amplitude is 3. The wave oscillates between $y = 3$ and $y = -3$.

Finding Period from an Equation

Find the period of $y = \sin(2x)$

Identify the coefficient B: Comparing $y = \sin(2x)$ with $y = A\sin(Bx)$ = $B = 2$

Apply the period formula: Period $= \frac{2\pi}{|B|} = \frac{2\pi}{|2|}$ = $\text{Period} = \frac{2\pi}{2}$

Simplify: $\frac{2\pi}{2} = \pi$ = $\text{Period} = \pi$

Interpret the result: Instead of one cycle every $2\pi$, there are two cycles = The wave completes twice as fast

Answer: The period is $\pi$. The wave completes one full cycle every $\pi$ units.

Finding Both Amplitude and Period

For $y = 4\cos(3x)$, find the amplitude and period.

Identify coefficients: Comparing with $y = A\cos(Bx)$: $A = 4$, $B = 3$ = $A = 4$, $B = 3$

Calculate amplitude: Amplitude $= |A| = |4|$ = $\text{Amplitude} = 4$

Calculate period: Period $= \frac{2\pi}{|B|} = \frac{2\pi}{3}$ = $\text{Period} = \frac{2\pi}{3}$

Describe the transformation: Compared to $y = \cos(x)$: stretched vertically by 4, compressed horizontally by 3 = Taller wave with more cycles

Answer: Amplitude = 4, Period = $\frac{2\pi}{3}$

Writing an Equation from a Graph

A sine wave has maximum value 5, minimum value $-5$, and completes one cycle from $x = 0$ to $x = 4\pi$. Write the equation.

Find amplitude from max/min: Amplitude $= \frac{\text{max} - \text{min}}{2} = \frac{5 - (-5)}{2} = \frac{10}{2}$ = $A = 5$

Identify the period: One cycle from $x = 0$ to $x = 4\pi$ = $\text{Period} = 4\pi$

Find B from the period: $\frac{2\pi}{|B|} = 4\pi$, so $|B| = \frac{2\pi}{4\pi} = \frac{1}{2}$ = $B = \frac{1}{2}$

Write the equation: Substitute $A = 5$ and $B = \frac{1}{2}$ = $y = 5\sin\left(\frac{x}{2}\right)$

Answer: $y = 5\sin\left(\frac{x}{2}\right)$ or equivalently $y = 5\sin\left(\frac{1}{2}x\right)$

Mistake: Thinking amplitude can be negative

Why: Amplitude is always positive because it represents distance. When you see $y = -3\sin(x)$, the amplitude is still 3.

Correct: Amplitude $= |A|$. A negative $A$ causes reflection, not negative amplitude.

Mistake: Confusing $B$ with the period

Why: Students often think $B$ is the period. Actually, $B$ affects how compressed the wave is.

Correct: Period $= \frac{2\pi}{|B|}$. As $B$ increases, period decreases (inverse relationship).

Mistake: Forgetting the $2\pi$ in the period formula

Why: The standard sine/cosine wave has period $2\pi$, not 1. All period calculations are based on this.

Correct: Period $= \frac{2\pi}{|B|}$, not $\frac{1}{B}$. The $2\pi$ comes from the natural period of sine and cosine.

Mistake: Calculating period of $\sin(x/2)$ as $\frac{2\pi}{1/2} = \pi$

Why: Division by a fraction requires flipping and multiplying.

Correct: $\frac{2\pi}{1/2} = 2\pi \times 2 = 4\pi$. When $B < 1$, the period is LONGER than $2\pi$.

Sound Engineering

Audio engineers adjust amplitude (volume) and period (pitch) to mix music and design speakers.

A bass drum produces waves with long period (low frequency ~60 Hz), while a cymbal has short period (high frequency ~10,000 Hz).

Electricity and Power

AC (alternating current) electricity follows a sine wave pattern. In the US, the standard is 120V at 60 Hz.

The voltage can be modeled as $V(t) = 170\sin(120\pi t)$. The amplitude is 170V (peak voltage), and the period is $\frac{1}{60}$ second.

Q: What happens when amplitude is 0?

A: If $A = 0$, the function becomes $y = 0$ (a flat horizontal line). There is no wave at all.

Q: Can the period be negative?

A: No. Period is a distance (length of one cycle), so it is always positive. That's why we use $|B|$ in the formula.

Q: What is the period of $y = \sin(x)$?

A: Since $B = 1$, the period is $\frac{2\pi}{1} = 2\pi$. This is the 'standard' period for sine and cosine.

Q: How do I find amplitude from a graph?

A: Find the maximum value and minimum value, then: Amplitude $= \frac{\text{max} - \text{min}}{2}$