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Combined Transformations of Sinusoidal Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

Combined transformations of sinusoidal functions

How do all four transformations work together?

The general form of a sinusoidal function with all transformations combined is
- $f (θ) = a \sin (b (θ + c)) + d or f (θ) = a \cos (b (θ + c)) + d$

The parameters $a, b, c$ and $d$ describe the following characteristics of the graph

Parameter	Characteristic	How to find it from the graph
$\|a\|$	Amplitude	$\frac{max - min}{2}$
$d$	Vertical shift (midline at $y = d$ )	$\frac{max + min}{2}$
$\frac{2 π}{\|b\|}$	Period	Horizontal distance between two consecutive maxima (or minima)
$- c$	Phase shift	Horizontal displacement from the base function's starting position

If $a > 0$ , the function oscillates normally
If $a < 0$ , the graph is reflected over the midline
- peaks and troughs are swapped

How can the amplitude and vertical shift be found from a graph?

Read the maximum and minimum output values from the graph
The amplitude is
- $| a | = \frac{max - min}{2}$

The vertical shift (midline) is
- $d = \frac{max + min}{2}$

For example, if a sinusoidal graph has a maximum of $7$ and a minimum of $1$
- Amplitude: $| a | = \frac{7 - 1}{2} = 3$
- Vertical shift: $d = \frac{7 + 1}{2} = 4$
  - so the midline is $y = 4$

How can the period and the value of b be found from a graph?

The period is the horizontal distance between
- two consecutive maxima
- two consecutive minima
- or any two consecutive corresponding points on the graph
Once the period is known, the value of $| b |$ can be found using:
- $| b | = \frac{2 π}{period}$

For example, if consecutive maxima occur at $x = 1$ and $x = 5$
- Period: $5 - 1 = 4$
- $| b | = \frac{2 π}{4} = \frac{π}{2}$

How can the phase shift be determined?

The phase shift describes the horizontal displacement of the graph
- compared to the base sine or cosine function
For a cosine model ( $a \cos (b (θ + c)) + d$ )
- The base cosine function has its maximum at $θ = 0$
- If the graph's first maximum occurs at $θ = h$
  - then the phase shift is $h$ units to the right
  - which means $c = - h$
    - or equivalently, the shift is $- c = h$
For a sine model ( $a \sin (b (θ + c)) + d$ )
- The base sine function crosses the midline (going upward) at $θ = 0$
- If the graph first crosses the midline going upward at $θ = h$
  - then the phase shift is $h$ units to the right
  - which means $c = - h$
    - or equivalently, the shift is $- c = h$
The phase shift is often the trickiest parameter to determine
- It helps to first decide whether to use a sine or cosine model (a question may decide this for you)
- and then identify the appropriate reference point on the graph

Examiner Tips and Tricks

When reading a graph to determine the phase shift, be careful about the difference between the shift in the $θ$ -value and the value of $c$ in the equation.

If a cosine graph reaches its first maximum at $θ = h$ , then the equation has $(θ - h)$ inside the cosine, which means $c = - h$ .

A common error is to write $c = h$ instead
Always verify your equation by substituting a known point from the graph

What is the systematic process for finding the equation from a graph?

Start by reading the maximum and minimum values from the graph
- Use these to find the amplitude ( $| a |$ ) and vertical shift ( $d$ )
  - Unless a question specifies otherwise, you can assume that $a$ is positive
    - in which case $a$ is simply equal to the amplitude
Next identify two consecutive maxima (or minima)
- Use these to find the period
- then calculate $| b |$
  - Unless a question specifies otherwise, you can assume that $b$ is positive
    - in which case $b$ is simply equal to $\frac{2 π}{period}$
At this point you need to decide whether to use a sine or cosine model
- Cosine is often convenient when a maximum or minimum is clearly visible
- Sine is convenient when a midline crossing (going upward) is clearly visible
- Often a question will tell you whether a sine or cosine model is to be used
Now you can identify the phase shift
- by comparing the graph's reference point
- to where the base function would normally have that feature

Finally write the equation
- and verify by checking that the function matches at least one or two key points on the graph

Examiner Tips and Tricks

Remember that the same graph can be described by either a sine or cosine model. A cosine function is just a phase-shifted sine function, and vice versa

However on the exam, if you are given a specific form to use (e.g. $h (t) = a \sin (b (t + c)) + d$ ), make sure you use it
- even if a different model might feel more natural to you

Also note that the parameters $a$ and $b$ will almost always be positive in exam questions.

It is always possible to write a model using $a > 0$ and $b > 0$
The difference between that and a model with negative values for $a$ and $b$ can always be represented instead as an appropriate phase shift

Worked Example

Sine wave graph with peaks at (-2pi/3,4), (pi/3,4), (4pi/3,4), and troughs at (-pi/6,-2), (5pi/6,-2), (11pi/6,-2); labelled axes x and y.

The figure shows the graph of a trigonometric function $f$ . Which of the following could be an expression for $f (x)$ ?

(A) $3 \cos (2 (x - \frac{π}{3})) + 1$

(B) $3 \cos (2 (x - \frac{π}{6})) + 1$

(D) $3 \sin (2 (x - \frac{π}{6})) + 1$

Answer:

Start by finding the amplitude and vertical shift

The maximum value is $4$ and the minimum value is $- 2$

$| a | = \frac{4 - (- 2)}{2} = 3, d = \frac{4 + (- 2)}{2} = 1$

So the amplitude is $3$ and the midline is at $y = 1$

This is consistent with all four answer options

Next find the period and $b$

Consecutive maxima appear at $x = \frac{π}{3}$ and $x = \frac{4 π}{3}$

$period = \frac{4 π}{3} - \frac{π}{3} = π$

$| b | = \frac{2 π}{π} = 2$

This is also consistent with all four options.

To distinguish between the models, determine the phase shift

The graph reaches a maximum at $x = \frac{π}{3}$
For a cosine model, the maximum of the base function $\cos θ$ occurs at $θ = 0$ , so the maximum has been shifted right to $x = \frac{π}{3}$
This means

$b (x + c) = 0 when x = \frac{π}{3}$

$2 (\frac{π}{3} + c) = 0 \Rightarrow c = - \frac{π}{3}$

This gives option (A) as the correct answer, $3 \cos (2 (x - \frac{π}{3})) + 1$

You can verify this by checking a couple of points

At $x = \frac{π}{3}$ :
$3 \cos (2 \cdot 0) + 1 = 3 (1) + 1 = 4$ ✓

At $x = \frac{5 π}{6}$ :
$3 \cos (2 (\frac{5 π}{6} - \frac{π}{3})) + 1 = 3 \cos (π) + 1 = - 3 + 1 = - 2$ ✓

It's worth looking at why the other options are incorrect

Option (B) uses $c = - \frac{π}{6}$ instead of $c = - \frac{π}{3}$
- This function would have a maximum at $(\frac{π}{6}, 4)$ instead of at $(\frac{π}{3}, 4)$
Option (C) uses a sine function, $3 \sin (2 (x - \frac{π}{3})) + 1$
- That particular model would have a midline crossing at $x = \frac{π}{3}$ , not a maximum point
Option (D) also uses a sine function, $3 \sin (2 (x - \frac{π}{6})) + 1$
- That particular model would have a midline crossing at $x = \frac{π}{6}$ , and a maximum point at $\frac{5 π}{12}$

(A) $3 \cos (2 (x - \frac{π}{3})) + 1$

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Combined Transformations of Sinusoidal Functions (College Board AP® Precalculus): Study Guide

Combined transformations of sinusoidal functions

How do all four transformations work together?

How can the amplitude and vertical shift be found from a graph?

How can the period and the value of b be found from a graph?

How can the phase shift be determined?

Examiner Tips and Tricks

What is the systematic process for finding the equation from a graph?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis