Sinusoidal functions: amplitude, period, translation

IB Mathematics: Applications and Interpretation SL· 5 min read

1. General Form, Amplitude and Vertical Translation★★☆☆☆⏱ 15 min

📘 Definition

General form of a sinusoidal function

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

A transformed version of the parent sine or cosine function, where each constant controls one specific transformation of the base graph.

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
$\text{Midline}: y = 4$

2. Period and Horizontal Stretch★★★☆☆⏱ 15 min

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 $2 π$ when $B = 1$ .

📘 Definition

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 $y = cos (4 x) - 1$ .

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.

✓ Quick check

Check your understanding:

What is the period of $y = sin (\frac{1}{3} x)$ ?
- $\frac{2 π}{3}$
- $3 π$
- $6 π$
- $\frac{π}{3}$
Reveal answer
6\pi —
Remember the formula: $Period = \frac{2 π}{B}$ . For $B = 1/3$ , this gives $2 π \div (1/3) = 6 π$ .

3. Horizontal Translation (Phase Shift)★★★☆☆⏱ 20 min

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 $y = sin (2 x - π)$ .

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 $D = \frac{π}{2}$ .
4
This means the parent graph $y = sin (2 x)$ is shifted $\frac{π}{2}$ units to the right.

4. Writing Equations from Graphs★★★★☆⏱ 20 min

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:

Find $C$ (midline): average the maximum and minimum $y$ -values
Find $A$ (amplitude): subtract the midline from the maximum $y$ -value
Find the period: measure distance between two consecutive peaks, then calculate $B = 2 π / period$
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
1. Calculate $A$ (amplitude):
4
$A = \text{max} - C = 7 - 4 = 3$
5
1. 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
1. Calculate $D$ : parent cosine has a maximum at $x = 0$ , so this graph has no horizontal shift: $D = 0$
8
Final equation:
9
$y = 3\cos\left(\frac{\pi}{5}x\right) + 4$

5. Common Pitfalls

Wrong move:

Forgetting to factor $B$ before reading the horizontal shift $D$

Why:

The general form is $B (x - D)$ , so $D$ is only correct after factoring, not from the unfactored argument

Correct move:

Always factor the coefficient of $x$ out of the sine/cosine argument before identifying $D$

Wrong move:

Using $B$ as the period instead of calculating $2 π / B$

Why:

Students mix up the inverse relationship between $B$ and period

Correct move:

Remember: $period = 2 π /∣ B ∣$ , larger $B$ gives a shorter period

Wrong move:

Calling the maximum value the amplitude, ignoring the vertical translation

Why:

Amplitude is measured from the midline, not from the x-axis

Correct move:

Amplitude is half the distance between maximum and minimum, always equal to $∣ A ∣$ , regardless of $C$

Wrong move:

Reversing the direction of horizontal shift: positive $D$ is left shift

Why:

The notation $f (x - D)$ is counter-intuitive for many students

Correct move:

Remember: $f (x - D)$ shifts right $D$ units, $f (x + D) = f (x - (- D))$ shifts left $D$ units

Wrong move:

Reporting a negative amplitude when $A$ is negative

Why:

Negative $A$ means reflection over the midline, but amplitude is a distance so it is always positive

Correct move:

Always take the absolute value of $A$ when asked for amplitude in an exam question

6. Quick Reference Cheatsheet

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

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.

2022 · Paper 1
Find amplitude and period of function
2023 · Paper 2
Write equation from sinusoidal graph

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.