# Graphs of trigonometric functions

> IB Mathematics: Analysis and Approaches SL · Geometry & Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u3-graphs-of-trigonometric-functions/

This sub-topic covers key features of sine, cosine and tangent graphs, and how transformations alter their shape, position and key points. You will learn to sketch graphs and derive equations from given graphs, a core exam skill.

**Prerequisites:** [Basic trigonometric values and identities](https://www.owlsprep.com/study/ib-math-aa-sl-u3-trigonometric-identities/); [General function transformations](https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-transformations/)

## Learning objectives

- Identify amplitude, period, vertical shift and phase shift of transformed trigonometric graphs
- Sketch graphs of sine, cosine and tangent functions from their equations
- Derive equations of trigonometric functions from given graphs
- Recognize key features of base and transformed trigonometric graphs for exam questions

## Key Features of Base Trigonometric Graphs

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.

**Period** — The smallest horizontal length after which the graph repeats its shape

*Notation:* For $f(x) = A \sin(Bx + C) + D$, period $= \frac{2\pi}{|B|}$ (sin/cos), $\frac{\pi}{|B|}$ (tan)

*Example:* $y = \sin 2x$ has period $\pi$

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

**Worked example:** Sketch $y = \cos x$ for $0 \leq x \leq 2\pi$, label all intercepts and turning points.

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

## Vertical Transformations: Amplitude and Vertical Shift

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

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

*Notation:* Amplitude $= |A|$

*Example:* $y = 3 \sin x$ has amplitude 3, range $[-3, 3]$

> **tip**
>
> A negative $A$ reflects the graph over the x-axis, but does not change amplitude, which is always positive.

**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 $-2$.

## Horizontal Transformations: Period and Phase Shift

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.

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

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

*Example:* $y = \sin(2x - \pi) = \sin(2(x - \frac{\pi}{2}))$, phase shift is $\frac{\pi}{2}$ right

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

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

## Finding Equations of Trigonometric Graphs

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:

1. Find $D$ (vertical shift): midpoint of maximum and minimum values
2. Find $A$ (amplitude): maximum value minus $D$
3. Find period from the graph, calculate $B = \frac{2\pi}{\text{period}}$ (sin/cos) or $\frac{\pi}{\text{period}}$ (tan)
4. 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 $x=4\pi$, 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 $4\pi$, so $B = \frac{2\pi}{4\pi} = \frac{1}{2}$.
4. The graph crosses the midline at $x=0$ increasing, matching base $\sin x$, so phase shift is 0, $C=0$.
5. Final equation: $y = 3 \sin\left(\frac{1}{2}x\right) + 2$

## Common pitfalls

- **Wrong:** Taking $C$ as phase shift without factoring $B$ from the $x$ term
  - Why it fails: Phase shift is scaled by $B$, so using the raw $C$ value gives an incorrect shift
  - Correct: Always factor $B$ out of $Bx + C$ to get $B(x + \frac{C}{B})$, so phase shift is $-\frac{C}{B}$
- **Wrong:** Using $2\pi$ to calculate period for tangent functions
  - Why it fails: Base tangent has period $\pi$, not $2\pi$, so the formula is different from sine and cosine
  - Correct: Use period $= \frac{\pi}{|B|}$ for all tangent functions, period $= \frac{2\pi}{|B|}$ for sine/cosine
- **Wrong:** Writing a negative amplitude when $A$ is negative
  - Why it fails: Amplitude is a measure of distance, so it is always non-negative
  - Correct: Take the absolute value of $A$ for amplitude, note the reflection over the x-axis separately if required
- **Wrong:** Mixing up sine and cosine when writing equations from graphs
  - Why it fails: Base sine and cosine have different starting points at $x=0$, so using the wrong base gives an incorrect phase shift
  - Correct: Use sine if the graph crosses the midline at $x=0$, use cosine if it has a maximum/minimum at $x=0$

## Cheatsheet

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

## 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.

- [Solving Trigonometric Equations](https://www.owlsprep.com/study/ib-math-aa-sl-u3-solving-trigonometric-equations/)
- [Pythagorean and double angle trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-sl-u3-pythagorean-and-double-angle-trigonometric/)

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