# Graphs of trigonometric functions

> IB Mathematics AI HL · Geometry and trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u3-graphs-of-trigonometric-functions/

This subtopic covers core properties and transformations of graphs of sine, cosine, and tangent functions, which are required to model periodic real-world phenomena and solve common IB AI HL exam problems.

**Prerequisites:** [Basic trigonometric ratios and the unit circle](https://www.owlsprep.com/study/ib-math-ai-hl-u2-trigonometric-ratios-and-identities/); [Vertical and horizontal function transformations](https://www.owlsprep.com/study/ib-math-ai-hl-u1-function-transformations/)

## Learning objectives

- Identify amplitude, period, phase shift, and vertical shift for transformed trigonometric functions
- Sketch graphs of sine, cosine, and tangent functions with transformations
- Recognize key features including midline, turning points, and asymptotes
- Apply graph properties to model periodic phenomena for exam questions

## Key Properties of Base Trigonometric Graphs

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.

**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 $2\pi$.

**Worked example:** State the domain, range, and base period for $y = \sin x$, $y = \cos x$, and $y = \tan x$.

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

> **tip**
>
> IB AI HL almost always uses radians for trigonometric graphs. Always confirm units if the question mentions degrees.

## Transformations of Sinusoidal Graphs

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.

**Sinusoidal Transformation Parameters** — $|A|$ = amplitude, $\frac{2\pi}{|B|}$ = period, $C$ = phase shift (horizontal shift), $D$ = vertical shift (midline).

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

**Worked example:** Identify the amplitude, period, midline, and phase shift of $y = 3\sin(2x - \pi) + 1$.

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

**Check your understanding**

Test your understanding of period calculation:

1. What is the period of $y = \cos(4x)$?

   - $4\pi$
   - $\frac{\pi}{2}$
   - $2\pi$
   - $\frac{\pi}{4}$

   *Answer:* $\frac{\pi}{2}$

   *Why:* Correct! Period for sinusoids is always $\frac{2\pi}{|B|}$, so $\frac{2\pi}{4} = \frac{\pi}{2}$.

## Sketching Transformed Sinusoidal Graphs

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 $y = 2\cos\left(x + \frac{\pi}{2}\right) - 1$ for $0 \leq x \leq 2\pi$.

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

> **tip**
>
> Always label your axes, midline, and key turning points when sketching. You will lose marks if these required features are missing.

## Transformations of the Tangent Graph

The tangent graph follows the same transformation rules as sine and cosine, but its base period is $\pi$, 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 $y = \tan(2x)$ for $0 \leq x \leq 2\pi$.

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

## Common pitfalls

- **Wrong:** Forgetting to factor $B$ out of the argument when finding phase shift, so reading $C$ as the unfactored constant term.
  - Why it fails: Phase shift is only correctly defined in the factored standard form $A\sin(B(x-C)) + D$. Unfactored forms give incorrect shift values.
  - Correct: Always rewrite $A\sin(Bx - C) + D$ as $A\sin(B(x - \frac{C}{B})) + D$ before reading off phase shift.
- **Wrong:** Calculating the period of a tangent function as $\frac{2\pi}{B}$ instead of $\frac{\pi}{B}$.
  - Why it fails: Base tangent has a period of $\pi$, not $2\pi$ like sine and cosine, so the period formula is different.
  - Correct: Always use period = $\frac{\pi}{|B|}$ for all transformed tangent functions.
- **Wrong:** Confusing the direction of phase shift, saying $y = \sin(x + C)$ shifts $C$ units to the right.
  - Why it fails: The standard form is $\sin(x - C)$, so a positive constant added inside the function is a negative shift.
  - Correct: $y = \sin(x + C) = \sin(x - (-C))$, so it shifts $C$ units to the left.
- **Wrong:** Using negative amplitude because $A$ is negative.
  - Why it fails: Amplitude is a distance from the midline, so it is always non-negative. A negative $A$ reflects the graph, not changes its amplitude.
  - Correct: Amplitude is always $|A|$, regardless of the sign of $A$.

## Cheatsheet

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

## 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.

- [Solving Trigonometric Equations](https://www.owlsprep.com/study/ib-math-ai-hl-u3-solving-trigonometric-equations/)
- [Vectors: basics, operations, dot product](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vectors-basics-operations-dot-product/)
- [Vector Equation of a Line, Intersections](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vector-equation-of-a-line/)

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