Translations & Dilations of Sinusoidal Functions (College Board AP® Precalculus): Study Guide

Transforming sinusoidal functions

What is the general form of a sinusoidal function?

• Functions that can be written in the form

  • $f\left(\theta \right)=a\sin \left(b\right(\theta +c\left)\right)+d\text{or}g\left(\theta \right)=a\cos \left(b\right(\theta +c\left)\right)+d$

    • where $a$, $b$, $c$, and $d$ are real numbers

    • with $a\ne 0$ and $b\ne 0$

  • 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

  • $\cos \theta =\sin \left(\theta +\frac{\pi }{2}\right)$

• 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 $g\left(\theta \right)=\sin \theta +d$

  • produces a vertical translation of the graph of $f\left(\theta \right)=\sin \theta$ 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 $g\left(\theta \right)=\sin (\theta +c)$ produces a horizontal translation of the graph of $f\left(\theta \right)=\sin \theta$ 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 $\left|c\right|$ 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

Dilations of sinusoidal functions

What is a vertical dilation of a sinusoidal function?

• The transformation $g\left(\theta \right)=a\sin \theta$ produces a vertical dilation of the graph of $f\left(\theta \right)=\sin \theta$

  • The graph is dilated vertically by a factor of $\left|a\right|$

• The amplitude of the transformed function is $\left|a\right|$

  • The maximum value becomes $\left|a\right|$

  • and the minimum value becomes $-\left|a\right|$

    • 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 $g\left(\theta \right)=\sin \left(b\theta \right)$ produces a horizontal dilation of the graph of $f\left(\theta \right)=\sin \theta$

  • The graph is dilated horizontally by a factor of $\frac{1}{\left|b\right|}$

• The period of the transformed function changes by a factor of $\frac{1}{\left|b\right|}$

  • So the new period is $\frac{2\pi }{\left|b\right|}$

  • If $\left|b\right|>1$, the period is shorter

    • the graph is compressed horizontally (more cycles fit in the same width)

  • If $\left|b\right|<1$, the period is longer

    • the graph is stretched horizontally (fewer cycles fit in the same width)

  • E.g., $g\left(\theta \right)=\sin \left(4\theta \right)$ has a period of $\frac{2\pi }{4}=\frac{\pi }{2}$

    • meaning four complete cycles occur over an interval of $2\pi$

• 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