Graphs of trigonometric functions

IB Mathematics AI HL· 35 min read

1. Key Properties of Base Trigonometric Graphs★★☆☆☆⏱ 10 min

Sine, cosine, and tangent are all periodic functions, meaning their shape repeats at regular intervals. Sine and cosine form continuous, bounded waves, while tangent has vertical asymptotes and an unbounded range.

📘 Definition

Periodic Function

A function $f (x)$ is periodic if there exists a constant $p > 0$ (the period) such that $f (x + p) = f (x)$ for all $x$ in the domain of $f$ .

Example:

$y = sin x$ is periodic with base period $2 π$ .

📐 Worked Example

State the domain, range, and base period for $y = sin x$ , $y = cos x$ , and $y = tan x$ .

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For $y = sin x$ : Sine is defined for all real input angles, and its output always falls between -1 and 1.
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Domain: $x \in R$ , Range: $[- 1, 1]$ , Period: $2 π$
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For $y = cos x$ : Cosine has the same domain and range as sine, with the same base period.
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Domain: $x \in R$ , Range: $[- 1, 1]$ , Period: $2 π$
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For $y = tan x$ : Tangent is $\frac{s i n x}{c o s x}$ , so it is undefined when $cos x = 0$ , and can output any real value. Its base period is half that of sine/cosine.
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Domain: $x \neq = \frac{π}{2} + k π, k \in Z$ , Range: $R$ , Period: $π$

2. Transformations of Sinusoidal Graphs★★★☆☆⏱ 15 min

Any transformed sine or cosine function can be written in the standard form $y = A sin (B (x - C)) + D$ or $y = A cos (B (x - C)) + D$ , where each constant corresponds to one transformation of the base graph.

📘 Definition

Sinusoidal Transformation Parameters

y = A\sin(B(x-C)) + D

$∣ A ∣$ = amplitude, $\frac{2 π}{∣ B ∣}$ = period, $C$ = phase shift (horizontal shift), $D$ = vertical shift (midline).

📐 Worked Example

Identify the amplitude, period, midline, and phase shift of $y = 3 sin (2 x - π) + 1$ .

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Step 1: Rewrite the function to match the standard factored form:
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$y = 3\sin\left(2\left(x - \frac{\pi}{2}\right)\right) + 1$
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Step 2: Extract each parameter from the standard form:
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- Amplitude = $∣ A ∣ = ∣3∣ = 3$
- Period = $\frac{2 π}{∣ B ∣} = \frac{2 π}{2} = π$
- Midline = $y = D = 1$
- Phase shift = $\frac{π}{2}$ units to the right
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Step 3: Verify: Maximum value = $1 + 3 = 4$ , minimum value = $1 - 3 = - 2$ , which matches the calculated parameters.

✓ Quick check

Test your understanding of period calculation:

What is the period of $y = cos (4 x)$ ?
- $4 π$
- $\frac{π}{2}$
- $2 π$
- $\frac{π}{4}$
Reveal answer
1 —
Correct! Period for sinusoids is always $\frac{2 π}{∣ B ∣}$ , so $\frac{2 π}{4} = \frac{π}{2}$ .

3. Sketching Transformed Sinusoidal Graphs★★★☆☆⏱ 15 min

To sketch an accurate sinusoidal graph, always mark key features first: midline, maximum/minimum values, and the starting position from the phase shift, before drawing a smooth wave through the points.

📐 Worked Example

Sketch $y = 2 cos (x + \frac{π}{2}) - 1$ for $0 \leq x \leq 2 π$ .

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Step 1: Rewrite in standard form and extract parameters:
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$y = 2\cos\left(1\left(x - (-\frac{\pi}{2})\right)\right) - 1$
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Amplitude = 2, period = $2 π$ , phase shift = $\frac{π}{2}$ left, midline = $y = - 1$ . Maximum value = $1$ , minimum value = $- 3$ .
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Step 2: Plot key points in the interval $[0, 2 π]$ :
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Key points are: $(0, - 1)$ , $(\frac{π}{2}, - 3)$ (minimum), $(π, - 1)$ , $(\frac{3 π}{2}, 1)$ (maximum), $(2 π, - 1)$ .
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Step 3: Draw a smooth wave connecting the points, completing one full period across the interval.

4. Transformations of the Tangent Graph★★★★☆HL only⏱ 10 min

The tangent graph follows the same transformation rules as sine and cosine, but its base period is $π$ , so the period formula changes. Tangent also has vertical asymptotes at the end of each period, which shift with the graph.

📐 Worked Example

Find the equations of all asymptotes of $y = tan (2 x)$ for $0 \leq x \leq 2 π$ .

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Step 1: Recall that base $y = tan (θ)$ has asymptotes where $θ = \frac{π}{2} + k π$ for all integers $k$ .
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Step 2: Substitute $θ = 2 x$ and solve for $x$ :
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$2x = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{4} + \frac{k\pi}{2}$
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Step 3: Find all $x$ in the interval $[0, 2 π]$ :
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The asymptotes are $x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$ .

5. Common Pitfalls

Wrong move:

Forgetting to factor $B$ out of the argument when finding phase shift, so reading $C$ as the unfactored constant term.

Why:

Phase shift is only correctly defined in the factored standard form $A sin (B (x - C)) + D$ . Unfactored forms give incorrect shift values.

Correct move:

Always rewrite $A sin (B x - C) + D$ as $A sin (B (x - \frac{C}{B})) + D$ before reading off phase shift.

Wrong move:

Calculating the period of a tangent function as $\frac{2 π}{B}$ instead of $\frac{π}{B}$ .

Why:

Base tangent has a period of $π$ , not $2 π$ like sine and cosine, so the period formula is different.

Correct move:

Always use period = $\frac{π}{∣ B ∣}$ for all transformed tangent functions.

Wrong move:

Confusing the direction of phase shift, saying $y = sin (x + C)$ shifts $C$ units to the right.

Why:

The standard form is $sin (x - C)$ , so a positive constant added inside the function is a negative shift.

Correct move:

$y = sin (x + C) = sin (x - (- C))$ , so it shifts $C$ units to the left.

Wrong move:

Using negative amplitude because $A$ is negative.

Why:

Amplitude is a distance from the midline, so it is always non-negative. A negative $A$ reflects the graph, not changes its amplitude.

Correct move:

Amplitude is always $∣ A ∣$ , regardless of the sign of $A$ .

6. Quick Reference Cheatsheet

Function Type	Standard Form	Amplitude	Period	Base Asymptotes
Sine/Cosine	$y = A sin (B (x - C)) + D$	$∣ A ∣$	$\frac{2 π}{∣ B ∣}$	None
Tangent	$y = A tan (B (x - C)) + D$	N/A	$\frac{π}{∣ B ∣}$	$x = \frac{π}{2 B} + \frac{k π}{B} + C$
Shifted Sinusoid	As above	$∣ A ∣$	$\frac{2 π}{∣ B ∣}$	None
Reflected Sinusoid	As above	$∣ A ∣$	$\frac{2 π}{∣ B ∣}$	None

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.

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2024 · 2
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2023 · 1
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Going deeper

IB SyllabusIB AI HL Syllabus Topic 3

What's Next

Graphs of trigonometric functions are the foundation for modeling periodic real-world phenomena like tides, seasonal temperatures, and sound waves, which are common extended response questions in IB AI HL. Mastery of graph properties is also required to solve trigonometric equations accurately, and to interpret parameters of trigonometric models fit to data. Next, you will build on this knowledge to solve equations and apply these graphs to practical problems.