Mathematics: Analysis & Approaches HL · Unit 2: Functions · 45 min read · Updated 2026-05-11
Function properties: parity and periodicity — IB Mathematics Analysis and Approaches HL
IB Mathematics Analysis and Approaches HL · Unit 2: Functions · 45 min read
1. Algebraic Definition of Parity★★☆☆☆⏱ 15 min
The two categories of parity follow simple algebraic conditions:
- Even function: $f(-x) = f(x)$ for all $x$ in the domain
- Odd function: $f(-x) = -f(x)$ for all $x$ in the domain
2. Graphical Interpretation of Parity★★☆☆☆⏱ 10 min
Parity directly corresponds to graphical symmetry, which allows you to quickly identify parity and sketch graphs faster.
Even functions are symmetric **about the y-axis**: reflecting the right half of the graph ($x>0$) over the y-axis gives the full graph.
Odd functions are symmetric **about the origin**: rotating the right half of the graph ($x>0$) 180° around the origin gives the full graph. For any point $(a,b)$ on an odd function, $(-a, -b)$ is also on the function.
3. Periodicity and Fundamental Period★★★☆☆⏱ 20 min
For transformed trigonometric functions of the form $f(x) = A f(Bx + C) + D$:
- The phase shift $C$ and vertical shift $D$ do not affect the period
- The period depends only on the coefficient $B$ of $x$ inside the function
For $\sin(Bx + C)$, $\cos(Bx + C)$: Fundamental period $T = \frac{2\pi}{|B|}$
For $\tan(Bx + C)$, $\cot(Bx + C)$: Fundamental period $T = \frac{\pi}{|B|}$
Exam tip: Always use the absolute value of B, since period is always positive.
Common Pitfalls
Why: Most functions do not satisfy either parity condition.
Why: If the domain is not symmetric about the origin (e.g. $f(x) = x^2$ for $x \geq 0$), parity cannot exist.
Why: Period is a positive quantity, and negative B does not change the magnitude of the period.
Why: Phase shift only shifts the graph horizontally, it does not change how often the graph repeats.
Why: Confusing horizontal stretch with the effect on period: increasing B compresses the graph horizontally, decreasing the period.