# Sinusoidal functions: amplitude, period, translation

> IB Mathematics: Applications and Interpretation SL · IB MATH AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u3-sinusoidal-functions-amplitude-period-translation/

This subtopic explains how amplitude, period, and horizontal/vertical transformations change the graph and equation of sine and cosine functions. You will learn to connect parameters to graphical features for exam questions.

**Prerequisites:** [Basic sine and cosine functions](https://www.owlsprep.com/study/ib-math-ai-sl-u3-basic-trigonometric-functions/)

## Learning objectives

- Identify amplitude, period, and translations of sinusoids from graphs and equations
- Write the equation of a transformed sinusoidal function given its key features
- Connect each parameter of the general form to its graphical meaning

## General Form, Amplitude and Vertical Translation

**General form of a sinusoidal function** — A transformed version of the parent sine or cosine function, where each constant controls one specific transformation of the base graph.

*Notation:* f(x) = A\sin(B(x - D)) + C \quad \text{or} \quad f(x) = A\cos(B(x - D)) + C

The parameter $A$ controls the amplitude (vertical stretch) and any vertical reflection of the wave. The parameter $C$ controls the vertical translation, which sets the midline of the wave at $y=C$. Amplitude is always positive, equal to $|A|$.

**Worked example:** For the function $y = -3\sin(x) + 4$, identify the amplitude and midline.

1. Compare the given function to the general form to identify parameters:
2. $$A = -3, \quad C = 4$$
3. Amplitude is the absolute value of $A$, since it is a distance:
4. $$\text{Amplitude} = |A| = |-3| = 3$$
5. The midline of the function is given by $y=C$:
6. $$\text{Midline}: y = 4$$

> **tip**
>
> The maximum value of any sinusoid is $C + |A|$, and the minimum is $C - |A|$. This is useful for checking your results.

## Period and Horizontal Stretch

The parameter $B$ controls the horizontal stretch or compression of the parent wave, which changes the length of the period. The parent $\sin(x)$ and $\cos(x)$ have a period of $2\pi$ when $B=1$.

**Period of a sinusoid** — The horizontal length of one full repeating cycle, calculated from the parameter $B$ as:

*Example:* \text{Period} = \frac{2\pi}{|B|}

**Worked example:** Calculate the period of the function $y = \cos(4x) - 1$.

1. Identify $B$ from the function: $B = 4$
2. Substitute into the period formula:
3. $$\text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{4} = \frac{\pi}{2}$$
4. A larger value of $B$ gives a shorter period, which matches our result.

**Check your understanding**

Check your understanding:

1. What is the period of $y = \sin\left(\frac{1}{3}x\right)$?

   - $\frac{2\pi}{3}$
   - $3\pi$
   - $6\pi$
   - $\frac{\pi}{3}$

   *Why:* Remember the formula: $\text{Period} = \frac{2\pi}{B}$. For $B = 1/3$, this gives $2\pi \div (1/3) = 6\pi$.

## Horizontal Translation (Phase Shift)

The parameter $D$ controls the horizontal shift of the graph. A positive $D$ shifts the graph $D$ units to the right, and a negative $D$ shifts it $D$ units to the left. **You must factor $B$ out of the $x$-term to find the correct value of $D$.**

**Worked example:** Identify the horizontal translation of $y = \sin(2x - \pi)$.

1. First, factor $B=2$ out of the argument of the sine function to match the general form:
2. $$y = \sin\left(2\left(x - \frac{\pi}{2}\right)\right)$$
3. From the factored form, we read $D = \frac{\pi}{2}$.
4. This means the parent graph $y = \sin(2x)$ is shifted $\frac{\pi}{2}$ units to the right.

> **warning**
>
> If you forget to factor $B$, you will incorrectly read $D = \pi$, which is double the actual shift. This is a very common exam mistake.

## Writing Equations from Graphs

A common IB exam question asks you to write the equation of a sinusoidal function from its graph. Follow this reliable order of steps to find all parameters:

1. Find $C$ (midline): average the maximum and minimum $y$-values
2. Find $A$ (amplitude): subtract the midline from the maximum $y$-value
3. Find the period: measure distance between two consecutive peaks, then calculate $B = 2\pi / \text{period}$
4. Find $D$ (horizontal shift): compare the position of a key point to the parent function

**Worked example:** A sinusoidal graph has a maximum at $(0, 7)$, next maximum at $(10, 7)$, and a minimum value of $1$. Write a cosine function for this graph.

1. 1. Calculate $C$ (midline):
2. $$C = \frac{\text{max} + \text{min}}{2} = \frac{7 + 1}{2} = 4$$
3. 2. Calculate $A$ (amplitude):
4. $$A = \text{max} - C = 7 - 4 = 3$$
5. 3. Calculate period and $B$: distance between maxima is $10-0=10$, so period $=10$:
6. $$B = \frac{2\pi}{\text{period}} = \frac{2\pi}{10} = \frac{\pi}{5}$$
7. 4. Calculate $D$: parent cosine has a maximum at $x=0$, so this graph has no horizontal shift: $D=0$
8. Final equation:
9. $$y = 3\cos\left(\frac{\pi}{5}x\right) + 4$$

## Common pitfalls

- **Wrong:** Forgetting to factor $B$ before reading the horizontal shift $D$
  - Why it fails: The general form is $B(x-D)$, so $D$ is only correct after factoring, not from the unfactored argument
  - Correct: Always factor the coefficient of $x$ out of the sine/cosine argument before identifying $D$
- **Wrong:** Using $B$ as the period instead of calculating $2\pi/B$
  - Why it fails: Students mix up the inverse relationship between $B$ and period
  - Correct: Remember: $\text{period} = 2\pi/|B|$, larger $B$ gives a shorter period
- **Wrong:** Calling the maximum value the amplitude, ignoring the vertical translation
  - Why it fails: Amplitude is measured from the midline, not from the x-axis
  - Correct: Amplitude is half the distance between maximum and minimum, always equal to $|A|$, regardless of $C$
- **Wrong:** Reversing the direction of horizontal shift: positive $D$ is left shift
  - Why it fails: The notation $f(x-D)$ is counter-intuitive for many students
  - Correct: Remember: $f(x-D)$ shifts right $D$ units, $f(x+D) = f(x - (-D))$ shifts left $D$ units
- **Wrong:** Reporting a negative amplitude when $A$ is negative
  - Why it fails: Negative $A$ means reflection over the midline, but amplitude is a distance so it is always positive
  - Correct: Always take the absolute value of $A$ when asked for amplitude in an exam question

## Cheatsheet

| Parameter | What it controls | Calculation from graph |
| --- | --- | --- |
| $A$ | Amplitude (vertical stretch) | $A = \text{max } y - C = \|A\| > 0$ |
| $B$ | Period scaling | $B = 2\pi / \text{period}$ |
| $C$ | Midline (vertical translation) | $C = (\text{max } y + \text{min } y)/2$ |
| $D$ | Horizontal shift (phase shift) | Factor $B$: shift right $D$, left $-D$ |

## What's next

Mastering amplitude, period, and translations of sinusoidal functions is the critical foundation for all work with periodic models in IB AI SL. This subtopic appears in every exam, in both calculator and non-calculator papers, so fluency with these parameters is essential for high marks. You will next apply these skills to model real-world periodic phenomena like seasonal temperatures, tidal heights, and oscillating motion, then solve problems involving sinusoidal functions in context.

- [Voronoi diagrams and closest site problems](https://www.owlsprep.com/study/ib-math-ai-sl-u3-voronoi-diagrams-and-closest-site/)
- [Triangle area formula for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-triangle-area-formula-for-non/)
- [Statistics and Probability](https://www.owlsprep.com/study/ib-math-ai-sl-u4-overview/)

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