Periodic Functions and Repeating Patterns
- Many real situations repeat in a regular way, for example day and night cycles, tides, seasons, sound waves, and rotating machinery.
- A function that repeats the same values over equal intervals is called periodic, and the most important periodic models in mathematics are based on sine and cosine.
Periodic function
A function whose values repeat after a fixed input interval $T$, so $f(x+T) = f(x)$ for all $x$ in its domain. The smallest positive such $T$ is the period.
In this topic you build the sine and cosine graphs, describe their key features (amplitude, period, midline), and learn how the equation changes when you translate, reflect, or dilate (stretch) the curve.
Sine and Cosine Curves
The graphs of $y = \sin x$ and $y = \cos x$ (with $x$ measured in degrees) have the same smooth wave shape.
Key Points over One Cycle (Degrees)
For $0^\circ \le x \le 360^\circ$:
| $x$ | $0^\circ$ | $90^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|
| $\sin x$ | 0 | 1 | 0 | -1 | 0 |
| $\cos x$ | 1 | 0 | -1 | 0 | 1 |
- When sketching, plot these key points first, then draw a smooth curve through them.
- The curve is not made of straight line segments.
Range, Midline, and Symmetry
- Both basic functions satisfy:
- Range: $-1 \le \sin x \le 1$ and $-1 \le \cos x \le 1$
- Midline (mean value): the horizontal line $y = 0$
- They also have useful symmetry:
- $\sin x$ is an odd function: $\sin(-x) = -\sin x$ (symmetry about the origin)
- $\cos x$ is an even function: $\cos(-x) = \cos x$ (symmetry about the $y$-axis)
Period and Frequency
The sine and cosine curves repeat every $360^\circ$.
Period
The horizontal length of one complete cycle of a periodic graph.
So for $y = \sin x$ and $y = \cos x$ (degrees), the period is $360^\circ$.
Frequency
The number of complete cycles per unit interval. In this course, the coefficient $b$ is often referred to as the frequency, representing the number of cycles in $360^\circ$ (distinct from the physics definition of cycles per unit time).
For the basic curves, the frequency is 1 cycle per $360^\circ$.
- To predict the graph beyond $360^\circ$, copy the shape and repeat it.
- To extend to negative angles, use the symmetry properties (odd/even) to reflect appropriately.
Amplitude and Vertical Transformations
- Real data rarely oscillates exactly between $-1$ and $1$.
- Multiplying or shifting the function changes the vertical features.
Amplitude
The maximum distance from the midline (mean value) to a peak or a trough.
- For $y = a \sin x$ or $y = a \cos x$:
- Amplitude $= |a|$
- Range becomes $[-|a|, |a|]$
- If $a < 0$, the graph is also reflected in the $x$-axis.
- If you add a constant $k$:
- $y = \sin x + k$ shifts the whole curve up by $k$.
- The midline becomes $y = k$.
- The range becomes $[k-1, k+1]$ (or $[k-|a|, k+|a|]$ if there is also a factor $a$).
- Students often confuse amplitude with maximum value.
- If the midline is not $y = 0$, the maximum value is $k + |a|$, but the amplitude is still $|a|$.
Horizontal Dilations and Phase Shifts
In $y = \sin(bx)$ or $y = \cos(bx)$, the factor $b$ changes how quickly the graph cycles.
Period for $y = \sin(bx)$ and $y = \cos(bx)$ (Degrees)
- Because the basic period is $360^\circ$, we need $bx$ to increase by $360^\circ$ to complete one cycle: $$b(x+T) - bx = 360^\circ$$
- So: $$T = \frac{360^\circ}{|b|}$$
- If $|b| > 1$, the graph is horizontally compressed (shorter period).
- If $0 < |b| < 1$, the graph is horizontally stretched (longer period).
- If $b < 0$, there is also a reflection in the $y$-axis (since replacing $x$ by $-x$ flips left-right).
For $y = 0.4 \sin(2x)$:
- amplitude $= 0.4$
- period $= \frac{360^\circ}{2} = 180^\circ$
So you should see 2 full cycles between $0^\circ$ and $360^\circ$.
Phase Shift (Horizontal Translation)
- A horizontal translation is written inside the function:
- $y = \sin(x-h)$ shifts the graph right by $h$.
- $y = \sin(x+h)$ shifts the graph left by $h$.
- Sine and cosine are shifts of each other:
- Translating the sine graph $90^\circ$ left produces the cosine graph.
- Translating the cosine graph $90^\circ$ right produces the sine graph.
- Equivalently, $\cos x = \sin(x+90^\circ)$ and $\sin x = \cos(x-90^\circ)$.
When sketching $y = a \sin(bx) + k$ or $y = a \cos(bx) + k$, find the midline $y = k$ first, then mark peaks and troughs at $k \pm |a|$, and finally space them using the period $\tfrac{360^\circ}{|b|}$.
The General Sinusoidal Model
A common way to write a transformed sine or cosine graph is: $$y = a \sin(b(x-h)) + k \quad \text{or}\quad y = a \cos(b(x-h)) + k$$ where:
- $|a|$ is the amplitude
- $k$ is the vertical shift (midline $y = k$)
- $\frac{360^\circ}{|b|}$ is the period
- $h$ is the phase shift
The most reliable approach is to read the equation for features (amplitude, period, midline, phase shift) rather than trying to apply transformations one-by-one.
Order of Transformations
- Yes, if you literally perform transformations in sequence, the order can change intermediate steps.
- However, for sinusoidal functions written in the standard form above, you can extract the main features directly and sketch reliably.
- A common mistake is to treat $\sin(b(x-h))$ as "shift right $h$ then compress by $b$" without adjustment.
- If the function is written as $\sin(bx-c)$, rewrite as $\sin\bigl(b(x-\tfrac{c}{b})\bigr)$ so the phase shift is $\tfrac{c}{b}$.
Recognizing a Sinusoidal Equation from a Graph
When you are given a graph and asked to find its equation, use a consistent checklist.
Step-by-Step Method
- Identify the midline (average of max and min): $$k = \frac{y_{\max} + y_{\min}}{2}$$
- Find the amplitude: $$|a| = \frac{y_{\max} - y_{\min}}{2}$$
- Find the period $T$ by measuring the horizontal length of one cycle.
- Compute $b$ using $T = \frac{360^\circ}{|b|}$, so $|b| = \frac{360^\circ}{T}$.
- Choose sine or cosine based on a convenient starting point:
- If the graph has a maximum or minimum at some $x = h$, cosine often fits naturally.
- If the graph crosses the midline going upward at some $x = h$, sine often fits naturally.
- Determine the phase shift $h$ using a key point (max, min, or midline crossing).
There can be many correct equations for the same graph because sine and cosine are shifts of each other, and adding a whole number of periods to the phase shift does not change the function.
A graph oscillates between 2 and 8, and one full cycle takes $120^\circ$.
- Midline: $k = \tfrac{8+2}{2} = 5$
- Amplitude: $|a| = \tfrac{8-2}{2} = 3$
- Period $T = 120^\circ$ gives $|b| = \frac{360^\circ}{120^\circ} = 3$
A possible model is $y = 3 \sin(3x) + 5$ (the phase shift depends on where the graph starts).
- State the amplitude and period of $y = -2 \sin(4x)$.
- Write the range of $y = 0.5 \cos x - 1$.
- A sinusoidal graph has midline $y = 3$ and amplitude 5. What are its maximum and minimum values?
- Rewrite $\sin(2x-60^\circ)$ in the form $\sin(2(x-h))$ and state the phase shift.
