Graphs of sine and cosine (amplitude, period, phase shift)

1. Anatomy of the basic graphs: domain, range, midline, and key points (zeros, max, min)

Anatomy of the Basic Graphs

The functions $y = \sin(x)$ and $y = \cos(x)$ are periodic waves with domain $(-\infty, \infty)$ and range $[-1, 1]$. The midline is the horizontal line $y = 0$ that bisects the graph, representing the average value of the function.

Both functions oscillate symmetrically about their midline, creating a predictable wave pattern.

Core structural features:

• Period: Both complete one full cycle over an interval of length $2\pi$
• Zeros of sine: $x = n\pi$ for any integer $n$; zeros of cosine: $x = \frac{\pi}{2} + n\pi$
• Maximum value: $1$ occurs at $x = \frac{\pi}{2} + 2n\pi$ for sine, $x = 2n\pi$ for cosine
• Minimum value: $-1$ occurs at $x = \frac{3\pi}{2} + 2n\pi$ for sine, $x = \pi + 2n\pi$ for cosine

These key points divide each period into four equal quarters of length $\frac{\pi}{2}$, creating the characteristic wave shape.

Example: For $y = \sin(x)$ on $[0, 2\pi]$, zeros occur at $0, \pi, 2\pi$; maximum at $\frac{\pi}{2}$; minimum at $\frac{3\pi}{2}$.

2. Calculating and graphing amplitude changes ($|a|$) and reflections (handling negative amplitude)

Amplitude Changes and Reflections

For $y = a\sin(x)$ or $y = a\cos(x)$, the amplitude is $|a|$, representing the maximum distance from the midline to a peak or trough. This stretches or compresses the graph vertically without affecting the period.

Amplitude determines the range: $[-|a|, |a|]$ when the midline is $y = 0$.

Core rules for coefficient $a$:

• If $|a| > 1$: vertical stretch (taller waves)
• If $0 < |a| < 1$: vertical compression (flatter waves)
• If $a < 0$: reflection across the $x$-axis occurs in addition to the amplitude change
• The midline and period remain unchanged by amplitude transformations

When $a$ is negative, every output value is negated, flipping peaks to troughs and vice versa.

Example: For $y = -3\sin(x)$, amplitude is $|{-3}| = 3$, range is $[-3, 3]$, and the graph is reflected so the first quarter-period descends from $0$ to $-3$ instead of ascending to $3$.

3. Calculating and graphing period changes ($2\pi/b$) and handling negative inside coefficients

Period Changes and Inside Coefficients

For $y = \sin(bx)$ or $y = \cos(bx)$, the period is $\frac{2\pi}{|b|}$, representing the horizontal length of one complete cycle. The coefficient $b$ compresses or stretches the graph horizontally.

The period formula uses $|b|$ because negative values of $b$ affect symmetry, not period length.

Core rules for coefficient $b$:

• If $|b| > 1$: horizontal compression (more cycles in $2\pi$)
• If $0 < |b| < 1$: horizontal stretch (fewer cycles in $2\pi$)
• If $b < 0$: for sine, this reflects the graph across the $y$-axis; for cosine, no visible change occurs due to even symmetry
• Key points are spaced at intervals of $\frac{\text{period}}{4} = \frac{\pi}{2|b|}$

Always compute period using the absolute value to avoid negative lengths.

Example: For $y = \cos(3x)$, period is $\frac{2\pi}{3}$, so three complete cycles fit in $[0, 2\pi]$, with key points at $0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \ldots$

4. Combining transformations including phase shifts ($-c/b$) and vertical translations ($d$)

Combining Multiple Transformations

The general form $y = a\sin(b(x - c)) + d$ or $y = a\cos(b(x - c)) + d$ combines four transformations: amplitude $|a|$, period $\frac{2\pi}{|b|}$, phase shift $c$ (horizontal translation), and vertical shift $d$ (midline translation). When written as $y = a\sin(bx - c) + d$, the phase shift is $\frac{c}{b}$.

The midline becomes $y = d$, and the range shifts to $[d - |a|, d + |a|]$.

Transformation order (applied to parent function):

• Horizontal: Apply phase shift $c$ units right (if $c > 0$) or left (if $c < 0$)
• Horizontal: Apply period change via factor $b$
• Vertical: Apply amplitude scaling $|a|$ and reflection if $a < 0$
• Vertical: Apply vertical shift $d$ to move the midline

Always extract $b$ as a factor before identifying the phase shift to avoid errors.

Example: For $y = 2\sin(3x - \pi) + 1$, rewrite as $y = 2\sin(3(x - \frac{\pi}{3})) + 1$: amplitude $2$, period $\frac{2\pi}{3}$, phase shift $\frac{\pi}{3}$ right, midline $y = 1$, range $[-1, 3]$.

5. Applications: Modeling sound waves in acoustics, spring-mass systems, and pendulum oscillations

Applications in Physical Systems

Sine and cosine functions model periodic phenomena where quantities oscillate regularly over time. In acoustics, sound waves are modeled as $y = a\sin(2\pi ft)$ where $a$ is amplitude (loudness), $f$ is frequency in hertz, and period $\frac{1}{f}$ represents the time for one cycle.

These models capture the repetitive nature of oscillatory motion in physical systems.

Key application domains:

• Sound waves: Amplitude corresponds to volume; frequency to pitch; a 440 Hz tone (musical A) has period $\frac{1}{440}$ seconds
• Spring-mass systems: Displacement $x(t) = a\cos(\omega t + \phi)$ where $\omega = \sqrt{k/m}$ depends on spring constant $k$ and mass $m$
• Pendulum motion: Angular displacement $\theta(t) = \theta_0\cos(\sqrt{g/L} \cdot t)$ for small angles, where $g$ is gravity and $L$ is length
• Phase shift $\phi$ represents initial conditions (starting position)

Example: A spring with mass 2 kg and constant 50 N/m oscillates as $x(t) = 0.1\cos(5t)$ meters, with amplitude 0.1 m and period $\frac{2\pi}{5}$ seconds.