Graphs of trigonometric functions

IB Mathematics: Analysis and Approaches SL· Unit 3: Geometry & Trigonometry· 10 min read

1. Key Features of Base Trigonometric Graphs★★☆☆☆⏱ 15 min

The three core trigonometric functions have distinct, periodic graphs that repeat at fixed intervals. Sine and cosine are continuous waves with a bounded range, while tangent has vertical asymptotes and a shorter period.

📘 Definition

Period

For $f (x) = A sin (B x + C) + D$ , period $= \frac{2 π}{∣ B ∣}$ (sin/cos), $\frac{π}{∣ B ∣}$ (tan)

The smallest horizontal length after which the graph repeats its shape

Example:

$y = sin 2 x$ has period $π$

Function	Range	Period	Key Features
$sin x$	$[- 1, 1]$	$2 π$	Crosses origin, crosses midline at start of cycle
$cos x$	$[- 1, 1]$	$2 π$	Y-intercept at $(0, 1)$ , maximum at start of cycle
$tan x$	All real $x$	$π$	Vertical asymptotes every $π$ units

📐 Worked Example

Sketch $y = cos x$ for $0 \leq x \leq 2 π$ , label all intercepts and turning points.

1
Identify base features: period $2 π$ , amplitude 1, y-intercept at $(0, 1)$
2
Find x-intercepts when $cos x = 0$ : at $x = \frac{π}{2}$ and $x = \frac{3 π}{2}$
3
Find turning points: maximum at $(0, 1)$ and $(2 π, 1)$ , minimum at $(π, - 1)$
4
Draw a smooth continuous wave connecting all labelled key points.

2. Vertical Transformations: Amplitude and Vertical Shift★★☆☆☆⏱ 20 min

For the general form $y = A sin (B x + C) + D$ , $A$ controls amplitude (vertical stretch/compression) and $D$ controls the vertical shift of the midline.

📘 Definition

Amplitude

Amplitude $= ∣ A ∣$

Half the distance between the maximum and minimum value of a sinusoidal function

Example:

$y = 3 sin x$ has amplitude 3, range $[- 3, 3]$

📐 Worked Example

Find the range and maximum value of $y = 2 sin x - 4$ .

1
Identify $A = 2$ , $D = - 4$ . Amplitude is $∣2∣ = 2$ .
2
Base $sin x$ has range $[- 1, 1]$ , so $2 sin x$ has range $[- 2, 2]$ .
3
Shift all values down by 4: subtract 4 from both bounds to get range $[- 6, - 2]$ .
4
The maximum value is the upper bound of the range, which is $- 2$ .

3. Horizontal Transformations: Period and Phase Shift★★★☆☆⏱ 25 min

Horizontal transformations change the length of one cycle (period) and the horizontal position (phase shift) of the graph. $B$ controls period, while $C$ controls phase shift.

📘 Definition

Phase Shift

For $y = A sin (B x + C) + D$ , phase shift $= - \frac{C}{B}$

The horizontal shift of a graph compared to the corresponding base function. To find it correctly, always factor $B$ out of the $x$ term.

Example:

$y = sin (2 x - π) = sin (2 (x - \frac{π}{2}))$ , phase shift is $\frac{π}{2}$ right

📐 Worked Example

Find the period of $y = tan (3 x)$ , and state the position of the first positive vertical asymptote.

1
For tangent functions, period is $\frac{π}{∣ B ∣}$ . Here $B = 3$ , so period $= \frac{π}{3}$ .
2
The base function $y = tan x$ has its first positive asymptote at $x = \frac{π}{2}$ .
3
Horizontal scaling by a factor of $\frac{1}{3}$ shifts the asymptote to $\frac{π}{2} \times \frac{1}{3} = \frac{π}{6}$ .
4
The first positive asymptote is at $x = \frac{π}{6}$ .

4. Finding Equations of Trigonometric Graphs★★★★☆⏱ 20 min

A common exam question asks you to derive the equation of a trigonometric function from its graph. Follow this ordered process to find $A, B, C, D$ correctly:

Find $D$ (vertical shift): midpoint of maximum and minimum values
Find $A$ (amplitude): maximum value minus $D$
Find period from the graph, calculate $B = \frac{2 π}{period}$ (sin/cos) or $\frac{π}{period}$ (tan)
Find $C$ (phase shift) by substituting a known point on the graph

📐 Worked Example

A sinusoidal graph has maximum $y = 5$ , minimum $y = - 1$ , one full cycle from $x = 0$ to $x = 4 π$ , and crosses the midline at $(0, 2)$ increasing. Find its equation.

1
Calculate $D$ : midpoint of 5 and -1 is $\frac{5 + ( - 1 )}{2} = 2$ , so $D = 2$ .
2
Calculate $A$ : $5 - 2 = 3$ , so $A = 3$ .
3
Period is $4 π$ , so $B = \frac{2 π}{4 π} = \frac{1}{2}$ .
4
The graph crosses the midline at $x = 0$ increasing, matching base $sin x$ , so phase shift is 0, $C = 0$ .
5
Final equation: $y = 3 sin (\frac{1}{2} x) + 2$

5. Common Pitfalls

Wrong move:

Taking $C$ as phase shift without factoring $B$ from the $x$ term

Why:

Phase shift is scaled by $B$ , so using the raw $C$ value gives an incorrect shift

Correct move:

Always factor $B$ out of $B x + C$ to get $B (x + \frac{C}{B})$ , so phase shift is $- \frac{C}{B}$

Wrong move:

Using $2 π$ to calculate period for tangent functions

Why:

Base tangent has period $π$ , not $2 π$ , so the formula is different from sine and cosine

Correct move:

Use period $= \frac{π}{∣ B ∣}$ for all tangent functions, period $= \frac{2 π}{∣ B ∣}$ for sine/cosine

Wrong move:

Writing a negative amplitude when $A$ is negative

Why:

Amplitude is a measure of distance, so it is always non-negative

Correct move:

Take the absolute value of $A$ for amplitude, note the reflection over the x-axis separately if required

Wrong move:

Mixing up sine and cosine when writing equations from graphs

Why:

Base sine and cosine have different starting points at $x = 0$ , so using the wrong base gives an incorrect phase shift

Correct move:

Use sine if the graph crosses the midline at $x = 0$ , use cosine if it has a maximum/minimum at $x = 0$

6. Quick Reference Cheatsheet

Function Form	Amplitude	Period	Phase Shift	Vertical Shift
$y = A sin (B x + C) + D$	$∣ A ∣$	$\frac{2 π}{∣ B ∣}$	$- \frac{C}{B}$	$D$
$y = A cos (B x + C) + D$	$∣ A ∣$	$\frac{2 π}{∣ B ∣}$	$- \frac{C}{B}$	$D$
$y = A tan (B x + C) + D$	N/A	$\frac{π}{∣ B ∣}$	$- \frac{C}{B}$	$D$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 1
Find equation from sinusoidal graph
2022 · 1
Sketch transformed tangent graph
2021 · 2
Identify key features of cosine graph

What's Next

Mastering graphs of trigonometric functions is the foundation for solving trigonometric equations and modelling periodic real-world phenomena such as tide heights, seasonal temperatures, and wave motion, which are common extended response questions in IB AA SL. This skill also connects to calculus, where you will differentiate and integrate trigonometric functions, and you will need to recall their graph shapes to find critical points and intercepts. Understanding the periodic nature of these graphs is key to avoiding mistakes when finding all solutions to trigonometric equations over a given domain.