IB Maths AA HLTopic 2 β FunctionsPaper 1 & 2~6 min read
Periodic Functions
A function is periodic if its graph keeps repeating the same pattern. The width of one full pattern is called the period. The classic examples are sin, cos, and tan β but lots of real-world things behave periodically too: tides, sound waves, daylight hours through the year. The key skills here are short: spot a period from a graph, compute the period of f(bx) with the divide-by-b rule, and use periodicity to find function values that aren’t directly given.
π What you need to know
Definition: f is periodic with period k if f(x + k) = f(x) for all x. The period is the smallest positive such k.
Standard periods: sin x and cos x have period 2Ο (or 360Β°). tan x has period Ο (or 180Β°).
Scaling rule: sin(bx) and cos(bx) have period 2Ο/|b|. tan(bx) has period Ο/|b|.
Linear combinations: if two functions share the same period k, their sum or difference is also periodic with period k (or a divisor of it).
Translational symmetry: the graph is unchanged when shifted horizontally by k (or any integer multiple of k).
Using periodicity: f(x) = f(x + nk) for any integer n. Add or subtract whole periods to land in a known interval.
Periodic graphs cross any horizontal line at infinitely many points (or none) β never finitely many.
The defining property
Periodic with period kf(x + k) = f(x) for all x
The period is the smallest positive k for which this works. Bigger multiples (2k, 3k, β¦) also satisfy the equation, but those aren’t the period.
A periodic graph β the period is the width of one cycle
Periods of the standard trig functions
sin x
period 2Ο
or 360Β°
cos x
period 2Ο
or 360Β°
tan x
period Ο
or 180Β° β half of sin/cos
The scaling rule for f(bx)
Multiplying the input by b compresses the graph horizontally β making cycles shorter:
Scaling rule
sin(bx), cos(bx) βΉ period = 2Ο|b|
tan(bx) βΉ period = Ο|b|
π€ Why divide by b?
If sin(x) takes 2Ο to complete one cycle, then sin(bx) does it whenever bx covers 2Ο β i.e. when x covers 2Ο/b. Doubling b halves the period; that’s why the cycles get squeezed.
Linear combinations with the same period
If two functions share a common period, any linear combination has that same period (or sometimes a smaller one):
Example: 4 sin(3x) and β7 cos(3x) both have period 2Ο/3, so their sum 4 sin(3x) β 7 cos(3x) is also periodic with period 2Ο/3.
If the periods don’t match, the combination is usually not periodic at all (or has a much larger period). Stick to same-period combinations unless the question gives you something specific.
Using periodicity to find function values
The big practical use: if you know f on one cycle, you know it everywhere. To find f at a point outside that cycle, just add or subtract whole periods until you land back inside.
π§ Recipe β finding f(a) using periodicity
Identify the period k from the graph or formula.
Add or subtract k repeatedly until you land in a region where the value is known.
Read off the value β it’s the same as the original input’s value.
Worked examples
WE 1
State the period of standard trig functions
State the period of each function:
(a) y = sin(2x) (b) y = cos(x/3) (c) y = tan(Οx) (d) y = 4 sin(x) + 3
(a) sin(bx): period = 2Ο/|b|, b = 2(a) period = Ο(b) cos(bx): period = 2Ο/|b|, b = 1/3period = 2Ο / (1/3) = 6Ο(b) period = 6Ο(c) tan(bx): period = Ο/|b|, b = Οperiod = Ο/Ο = 1(c) period = 1(d) 4 sin x + 3 β vertical shift & stretch don’t change period(d) period = 2Οvertical transformations (stretching, translating up/down) NEVER change the period β only horizontal scaling does
WE 2
Find the period from a graph description
A periodic function f has a graph that repeats the same pattern between consecutive peaks. Two adjacent peaks occur at x = 4 and x = 11. State the period of f.
Period = horizontal distance between matching points on consecutive cyclesperiod = 11 β 4 = 7period = 7use ANY pair of matching features β peak to peak, trough to trough, or zero-crossing to next zero-crossing of the same type
WE 3
Verify a period using the definition
Verify, using the definition f(x + k) = f(x), that f(x) = sin(4x) is periodic with period Ο/2.
Compute f(x + Ο/2)f(x + Ο/2) = sin(4(x + Ο/2))= sin(4x + 2Ο)Use sin(ΞΈ + 2Ο) = sin ΞΈ= sin(4x) = f(x) βf(x) = sin(4x) is periodic with period Ο/2multiplying x by 4 and adding Ο/2 inside gives 4Β·Ο/2 = 2Ο β a full revolution for the inside argument
WE 4
Period of a linear combination
Find the period of g(x) = 3 sin(2x) β 5 cos(2x).
Identify period of each piece3 sin(2x): period = 2Ο/2 = Ο5 cos(2x): period = 2Ο/2 = ΟSame period β combination has same periodperiod of g = Οif the two periods had been different, the combination would generally not be periodic
WE 5
Use periodicity to find function values
The function f is periodic with period 6 and f(2) = 9. Find:
(a) f(20) (b) f(β4) (c) f(38)
Strategy: subtract or add multiples of 6 to land at x = 2(a) f(20)20 β 6 β 6 β 6 = 2 β f(20) = f(2) = 9(a) f(20) = 9(b) f(β4)β4 + 6 = 2 β f(β4) = f(2) = 9(b) f(β4) = 9(c) f(38)38 = 2 + 6(6) β subtract 36 β f(38) = f(2) = 9(c) f(38) = 9shortcut: divide (input β 2) by 6 β if it’s a whole number, the value matches f(2)
WE 6
Find all solutions of a periodic equation
Find all real solutions of cos(x) = 12, expressing answers in degrees.
Step 1: Find solutions in one cycle (0Β° β€ x < 360Β°)cos x = 1/2 β x = 60Β° or x = 300Β°Step 2: Add multiples of 360Β° (the period of cos)x = 60Β° + 360kΒ° or x = 300Β° + 360kΒ° for k β β€x = 60Β° + 360kΒ° or x = 300Β° + 360kΒ°, k β β€because cos is periodic, there are infinitely many solutions β always express the full set using a parameter like k β β€
π‘ Top tips
Memorise the three standard periods: sin and cos give 2Ο, tan gives Ο. Everything else is built from these.
Divide by b for sin(bx), cos(bx), tan(bx). Big b β short period; small b β long period.
Vertical changes don’t affect period. 5 cos(x) β 3 still has period 2Ο. Period only depends on horizontal scaling.
To read a period off a graph, find any two adjacent matching features (two peaks, two troughs, two zero-crossings going up). Distance between them = period.
For function values, add or subtract whole periods to land on something you know.
For periodic equations, find solutions in one cycle, then tag on “+ nk, n β β€” for the full set.
Use the GDC graphically to verify a period β zoom out and look for the repeating pattern.
β Common mistakes
Confusing the period of tan with sin/cos. Tan repeats every Ο, not 2Ο.
Multiplying by b instead of dividing. sin(3x) has period 2Ο/3, not 6Ο.
Letting vertical shifts confuse you. 7 sin(x) + 4 still has period 2Ο β the +4 just moves the graph up.
Reporting only the principal solution to a periodic equation. The full answer is an infinite family β say so explicitly.
Mixing up degrees and radians in the period. 2Ο in radians = 360Β° in degrees. Match the units the question is using.
Reading the period as the amplitude from a graph. Period = horizontal repeat distance; amplitude = vertical max minus mid-line.
Using a non-minimum value for the period. The period is the smallest positive k β 4Ο is also a “period” of sin x, but the answer is 2Ο.
Periodic functions form the entire backbone of trig in IB β and the rules here apply directly when you start sketching a sin(bx + c) + d in the trig topic. The next note covers self-inverse functions: a special class where the function is its own inverse β applying it twice gives back the original input.
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