AP®MathsCollege BoardPrecalculusRevision NotesTrigonometric & Polar FunctionsSinusoidal Functions & ModelingTranslations & Dilations of Sinusoidal Functions

Translations & Dilations of Sinusoidal Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

Transforming sinusoidal functions

What is the general form of a sinusoidal function?

Functions that can be written in the form
- $f (θ) = a \sin (b (θ + c)) + d or g (θ) = a \cos (b (θ + c)) + d$
  - where $a$ , $b$ , $c$ , and $d$ are real numbers
  - with $a \neq 0$ and $b \neq 0$
- are sinusoidal functions

These are all transformations of the base sine and cosine functions
The four parameters each control a different aspect of the transformation:
- $a$ controls the vertical dilation (amplitude)
- $b$ controls the horizontal dilation (period)
- $c$ controls the horizontal translation (phase shift)
- $d$ controls the vertical translation (vertical shift)

Why do the same transformations apply to both sine and cosine?

The cosine function is itself a phase shift of the sine function
- $\cos θ = \sin (θ + \frac{π}{2})$

This means any transformed cosine function can also be written as a transformed sine function (and vice versa)
- Therefore, although the descriptions below are all described as transformations of the sine function
- the effects of the transformations are the same if applied to cosine

Translations of sinusoidal functions

What is a vertical translation of a sinusoidal function?

The transformation $g (θ) = \sin θ + d$
- produces a vertical translation of the graph of $f (θ) = \sin θ$ by $d$ units

The entire graph shifts up if $d > 0$ or down if $d < 0$
The midline of the graph shifts from $y = 0$ to $y = d$
- The function now oscillates symmetrically about $y = d$ rather than about the $x$ -axis
The maximum value becomes $1 + d$
- and the minimum value becomes $- 1 + d$
The amplitude and period are not affected by a vertical translation

What is a horizontal translation (phase shift) of a sinusoidal function?

The transformation $g (θ) = \sin (θ + c)$ produces a horizontal translation of the graph of $f (θ) = \sin θ$ by $- c$ units
- This horizontal translation is called a phase shift

If $c > 0$ , the graph shifts to the left by $c$ units
- If $c < 0$ , the graph shifts to the right by $| c |$ units

The amplitude, midline, and period are not affected by a phase shift
- only the horizontal position of the graph changes
- This means the location of maximum points, minimum points and midline crossing points will be shifted

Examiner Tips and Tricks

Be careful with the sign of a horizontal phase shift.

In the expression $\sin (θ + c)$ , the graph shifts by $- c$ units
- so a positive value of $c$ shifts the graph to the left
- and a negative value of $c$ shifts the graph to the right

This is the same convention used for horizontal translations of any function, but it is a common source of errors.

Dilations of sinusoidal functions

What is a vertical dilation of a sinusoidal function?

The transformation $g (θ) = a \sin θ$ produces a vertical dilation of the graph of $f (θ) = \sin θ$
- The graph is dilated vertically by a factor of $| a |$

The amplitude of the transformed function is $| a |$
- The maximum value becomes $| a |$
- and the minimum value becomes $- | a |$
  - assuming there is no additional vertical translation
If $a < 0$ , the graph is also reflected over the $x$ -axis
- This flips the graph upside down
  - peaks become troughs and vice versa
The period and midline are not affected by a vertical dilation

What is a horizontal dilation of a sinusoidal function?

The transformation $g (θ) = \sin (b θ)$ produces a horizontal dilation of the graph of $f (θ) = \sin θ$
- The graph is dilated horizontally by a factor of $\frac{1}{| b |}$

The period of the transformed function changes by a factor of $\frac{1}{| b |}$
- So the new period is $\frac{2 π}{| b |}$
- If $| b | > 1$ , the period is shorter
  - the graph is compressed horizontally (more cycles fit in the same width)
- If $| b | < 1$ , the period is longer
  - the graph is stretched horizontally (fewer cycles fit in the same width)
- E.g., $g (θ) = \sin (4 θ)$ has a period of $\frac{2 π}{4} = \frac{π}{2}$
  - meaning four complete cycles occur over an interval of $2 π$

If $b < 0$ , the graph is also reflected over the $y$ -axis
- However, because of the symmetry properties of sine and cosine
- this reflection can often be expressed as a phase shift instead
The amplitude and midline are not affected by a horizontal dilation

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