Look for the smallest horizontal distance over which the graph exactly repeats. Practically: - Pick an easy feature that repeats each cycle (peak to peak, trough to trough, or two identical midline crossings with the same direction). - Measure the x-distance between two consecutive occurrences of that feature. That distance is the period k (CED 3.1.B.1–3.1.B.2: the smallest positive k with f(x+k)=f(x)). - If the graph shows one full cycle drawn, you can also measure from the leftmost start of that cycle to its rightmost end—any interval of width k describes the whole function (CED 3.1.B.1–3.1.B.2). - For sin/cos models, once you have period k, the horizontal dilation factor b = 2π/k (so standard form is A sin(b(x−c))+d or A cos(b(x−c))+d). On the AP exam you’ll often have to read period from graphs to construct sinusoidal models (FRQ 3) or answer behavior questions—so be precise about which repeating point you pick. For more examples and practice, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and the AP Precalculus practice bank (https://library.fiveable.me/practice/ap-pre-calculus).

Period and frequency describe the same repeating behavior but from two different angles: - Period (k in the CED): the smallest positive input change that repeats the output. Formally f(x + k) = f(x) for all x (CED 3.1.B.1). Example: y = sin(bx) has period = 2π/b. In the ride example on the AP sample, if the next peak is 8 seconds later, the period is 8 s. - Frequency: how many cycles happen per unit input = 1/period. Using the same examples: y = sin(bx) → period = 2π/b, frequency = b/(2π). For the ride with period 8 s, frequency = 1/8 cycles per second. Related terms: angular frequency (often called ω) equals b in y = sin(bx) and measures radians per unit input (CED keywords: frequency, angular frequency, period). On the AP exam, remember angle units—many items use radians (calculator in radian mode for Part A). For more practice and the Topic 3.1 study guide, see (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh). For unit review and lots of problems, visit (https://library.fiveable.me/ap-pre-calculus/unit-3) and (https://library.fiveable.me/practice/ap-pre-calculus).

A function is periodic when its outputs repeat in a regular way as the input increases—there’s a smallest positive length k (the period) so that f(x + k) = f(x) for every x in the domain. That means one full cycle (width k) completely determines the whole graph: copy that cycle left and right to build the function. For sinusoids, period, amplitude, midline (vertical shift), and phase shift control the shape; amplitude is half the peak-to-trough distance and the midline is the vertical center. To find a period from words or data, look for successive equal-length input intervals where the outputs repeat (CED 3.1.A.1–3.1.B.2). On the AP exam, expect to construct sinusoidal models from contexts and use the smallest k so f(x+k)=f(x) (FRQ 3 models this; see Topic 3.1). For a quick review, check the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and practice many problems (https://library.fiveable.me/practice/ap-pre-calculus).

Start with that one cycle and copy it left and right by the period. Steps that match the CED language: 1. Find amplitude and midline from the cycle: amplitude = (max − min)/2, midline = (max + min)/2. 2. Measure the period (the length of that one cycle). The period k is the smallest positive horizontal distance so f(x+k)=f(x). 3. Choose a base trig form (sine or cosine) and note any phase shift so the single-cycle graph aligns with the usual sine/cosine shape (or just treat the drawn cycle as the template). 4. Plot the single cycle precisely (key points: start, quarter, half, 3-quarter, end) with correct y-values and x-positions. 5. Repeat (translate) that cycle horizontally by adding or subtracting multiples of the period k to cover the domain you want. Make sure endpoints match so the graph is continuous. On the AP, remember EK 3.1.A.2 and 3.1.B.1: the whole function is determined by any interval of width k, and FRQ 3 requires you to construct two cycles and report parameters (no calculator). For more examples and practice, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

f(x + k) = f(x) means: whenever you shift the input x forward by k, the output is exactly the same. That k is the period—the smallest positive number for which the equality holds for every x in the domain (CED 3.1.B.1). Practically, it says the function repeats its behavior every k units, so knowing one cycle (an interval of width k) tells you the whole graph (CED 3.1.A.2). Why it’s important: periodic models (like sinusoids) let you predict repeating real-world behavior—days of daylight, tides, or motion. On the AP exam you’ll use this idea to build graphs from one cycle and to find period/phase shifts when writing sinusoidal models (see FRQ 3 in the CED). If you want a quick topic review, check the Periodic Phenomena study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh). For more practice, use the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Use periodic functions whenever the situation repeats over equal intervals—when one quantity cycles back and forth (CED 3.1.A). Look for a clear cycle you can describe with period, amplitude, and midline (CED 3.1.B keywords). Common real-life word-problem contexts: daylight minutes and seasons, ocean tides, Ferris wheels or elevator bounces, rotating fan blades or wheels, sound waves and EM waves, circadian rhythms, and anything with harmonic motion (springs, pendulums). How you model it: identify one cycle from the verbal description (vertical shift = midline, half the peak-to-trough = amplitude, time between repeats = period, and any horizontal shift = phase). Then write a sinusoid (a·sin(b(x+c))+d or a·cos(b(x+c))+d) and justify parameter choices—exactly what AP FRQ 3 asks you to do (model a periodic context; no calculator). For step-by-step help and examples, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

The period k is, by definition, the smallest positive number with f(x + k) = f(x) for all x. How you find it depends on the form of f: - For a basic sinusoid y = A sin(Bx + C) or A cos(Bx + C): period = 2π/|B|. So the smallest k is 2π/|B|. - For a sum of sinusoids (e.g., sin(m x) + cos(n x)): find each component’s period T1 = 2π/|m|, T2 = 2π/|n|, etc., then find the least common multiple (LCM) of those periods. Practically, solve for k so k/T1 and k/T2 are both integers. Example: sin(2x) has T1 = π, cos(3x) has T2 = 2π/3; k must satisfy k = a·π = b·(2π/3) for integers a,b → smallest k = 2π (here). - For periodic data or word problems: locate the shortest repeating interval in outputs (use successive equal-length intervals as in the CED 3.1.B.2). Tip for AP: work in radian mode on the exam when you use a calculator (CED guidance). If you want step-by-step examples and practice, check the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and try problems from the unit page (https://library.fiveable.me/ap-pre-calculus/unit-3) or the practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Look for the same features repeating over equal-length input intervals. Concretely: - Identify one cycle: pick a clear feature (consecutive peaks, consecutive troughs, or identical crossings through the midline with the same direction). The distance between two identical features = period k (smallest positive k with f(x+k)=f(x)). - Check multiple cycles: measure that distance for several successive cycles and average to reduce noise. AP CED calls this estimating period by “successive equal-length output values” (3.1.B.2). - Note amplitude and midline: amplitude = (max − min)/2, midline = (max + min)/2. Those plus period let you build one-cycle sinusoid and then repeat it to get the whole graph (3.1.A.2, 3.1.B.1). - Also watch phase shift (horizontal offset) and vertical shift; signs (phase inversion) change whether you use sine or cosine. On the AP exam you’ll often need to construct a model from one cycle (FRQ 3 is a common task—no calculator), so practice extracting period, amplitude, midline, and phase from data or a verbal description. For a quick review, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and more unit resources/practice problems at (https://library.fiveable.me/ap-pre-calculus/unit-3) and (https://library.fiveable.me/practice/ap-pre-calculus).

1) Read the context and identify what repeats—name the input (time, angle) and the repeating output pattern. Confirm it’s periodic (EK 3.1.A.1). 2) Find one full cycle: determine the period k (smallest positive repeat). Use units from the context (e.g., “every 8 seconds” → period = 8). (CED 3.1.B.1–2) 3) Determine midline (vertical shift) and amplitude: midline = average of max and min; amplitude = (max − min)/2. (use “highest/lowest” values in the description) 4) Pick a basic sinusoid (sine or cosine) whose shape matches where the cycle starts: cosine if it starts at a max/min, sine if it starts at midline going up/down. 5) Compute horizontal scale and phase: b = 2π/period; phase shift c found by matching a key point (solve b(x + c) so the basic sine/cosine hits that point). 6) Write model: f(x) = A·sin(b(x + c)) + d or A·cos(b(x + c)) + d, where A = amplitude, d = midline. 7) Sketch one cycle using five key points (midline crossings, max, min), then repeat every period across domain (CED 3.1.A.2). On the AP FRQ, show these steps clearly and label units; use the context to justify choices. For a topic refresher and examples see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh). For extra practice, try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

Look for the repeating-piece inside the formula and use the definition: a function is periodic if there’s a smallest positive k with f(x + k) = f(x) for all x in the domain (that k is the period). For equations this usually means: - Trig functions: sine and cosine are periodic. f(x)=sin(bx+c) or cos(bx+c) has period 2π/|b|; tan(bx+c) has period π/|b|. Amplitude = |a| if f(x)=a·sin(...)+d, midline = d, phase shift from c. (AP exam expects angle work in radians unless told otherwise.) - Reciprocal trig (sec, csc) inherit trig periods but watch domain holes (vertical asymptotes). - Nontrig basics: polynomials, exponentials, logs are not periodic (no repeating pattern). Rational functions aren’t periodic in general unless they include a trig factor. - Composites: if x only appears inside a periodic subfunction (like sin or cos), the whole function is periodic with that subfunction’s period (adjusted by horizontal stretch). Quick check: test f(x + k) = f(x) symbolically for candidate k (common choices 2π/|b| or π/|b|). For more examples and AP-aligned practice, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and unit resources (https://library.fiveable.me/ap-pre-calculus/unit-3). For extra practice, try problems at (https://library.fiveable.me/practice/ap-pre-calculus).

Because a periodic function repeats its outputs after a fixed horizontal shift, every feature you find in one cycle must appear in every other cycle. Formally, the period k is the smallest positive number with f(x + k) = f(x) for all x (CED 3.1.B.1). That equation means the entire rule for outputs—maxima/minima, zeros, intervals of increase or decrease, concavity, and instantaneous rates of change—is identical when you shift x by k. So if on [a, a+k] the function rises, then falls, and is concave up on some subinterval, those exact behaviors occur on [a+k, a+2k], [a+2k, a+3k], etc. Think of y = sin x (period 2π): one hump from 0 to π with specific slopes and concavity will reappear on [2π, 3π], [4π, 5π], etc., because sin(x+2π)=sin x. This is why you only need one full cycle to construct the whole graph (CED 3.1.A.2). For AP-style modeling and FRQ 3 (sinusoidal context), use one cycle to get amplitude, midline, period, and phase shift, then tile it across the domain. For more practice, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and AP Precalculus practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

For trig sinusoids: if you have y = A·sin(B(x − C)) + D (or cosine), the period T = 2π / |B|. So y = sin(x) has T = 2π, y = sin(3x) has T = 2π/3. Frequency = 1/T = |B|/(2π); engineers call B the angular frequency (ω). For other periodic functions (any type): the period is the smallest positive k such that f(x + k) = f(x) for every x in the domain (CED 3.1.B.1). There’s no universal algebraic formula like 2π/|B| unless you can express the function as a transformed sinusoid or know its generating rule. Practically you estimate T by finding successive equal-length intervals where the output pattern repeats (CED 3.1.B.2). On the AP exam you’ll often model real-world cycles with sinusoids (FRQ 3), so learn to read period from data/graphs and use T = 2π/|B| when you’ve written a sinusoidal formula. For a quick Topic 3.1 review see the study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Look for a repeating feature in the outputs (peak, trough, midline crossing in the same direction). Measure the x-distance between successive occurrences of that feature—that distance is one period. Example: peaks at x = 2 and x = 8 → period ≈ 6. To be more accurate, measure over several cycles and average: if peaks occur at x = 2, 8, 14, average period = (8−2 + 14−8)/2 = 6. If data are noisy, use zero crossings in the same direction (they’re less sensitive to amplitude noise) or average peak-to-peak and trough-to-trough distances. On the AP, justify your choice of feature and show the equal-length interval idea from the CED (3.1.A.1 and 3.1.B.2). If allowed, a graphing calculator or regression can fit a sinusoid to refine the period estimate; note FRQ 3 (modeling a periodic context) gives points for two full cycles without a calculator. For a quick review, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Think of amplitude as “how tall” the waves are and period as “how wide” each cycle is. - From a formula y = A sin(B(x − C)) + D: amplitude = |A| (vertical stretch), period = 2π/|B| (horizontal length of one cycle). Frequency = 1/period; angular frequency = B. - From a graph or context: amplitude = (max − min)/2 and the midline (vertical shift) = (max + min)/2. Period = the horizontal distance between repeating features (peak→next peak, trough→next trough, or a full cycle of the pattern). That matches the CED definition: the smallest positive k with f(x + k) = f(x). - Quick checks: if your sinusoid is y = 5 cos(3x) + 2 → amplitude = 5, period = 2π/3. If a real-world description gives “time between repeats = 8 sec,” period = 8 and B = 2π/8 = π/4. This exact stuff shows up on AP items (FRQ 3 models periodic contexts). For extra review and practice, see the Topic 3.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) and the unit/practice pages (https://library.fiveable.me/ap-pre-calculus/unit-3, https://library.fiveable.me/practice/ap-pre-calculus).