Combined Transformations of Sinusoidal Functions (College Board AP® Precalculus): Revision Note

Combined transformations of sinusoidal functions

How do all four transformations work together?

• The general form of a sinusoidal function with all transformations combined is

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

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

• If $\mathit{a}\mathbf{>}\mathbf{0}$, the function oscillates normally

• If $\mathit{a}\mathbf{<}\mathbf{0}$, 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

  • $\left|a\right|=\frac{\text{max}-\text{min}}{2}$

• The vertical shift (midline) is

  • $d=\frac{\text{max}+\text{min}}{2}$

• For example, if a sinusoidal graph has a maximum of $7$ and a minimum of $1$

  • Amplitude: $\left|a\right|=\frac{7-1}{2}=3$

  • 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 $\mathbf{\left|}\mathit{b}\mathbf{\right|}$ can be found using:

  • $\left|b\right|=\frac{2\pi }{\text{period}}$

• For example, if consecutive maxima occur at $x=1$ and $x=5$

  • Period: $5-1=4$

  • $\left|b\right|=\frac{2\pi }{4}=\frac{\pi }{2}$

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 ($a\cos \left(b\right(\theta +c\left)\right)+d$)

  • The base cosine function has its maximum at $\theta =0$

  • If the graph's first maximum occurs at $\theta =h$

    • 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 ($a\sin \left(b\right(\theta +c\left)\right)+d$)

  • The base sine function crosses the midline (going upward) at $\theta =0$

  • If the graph first crosses the midline going upward at $\theta =h$

    • 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

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 ($\left|a\right|$) 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 $\left|b\right|$

    • Unless a question specifies otherwise, you can assume that $b$ is positive

      • in which case $b$ is simply equal to $\frac{2\pi }{\text{period}}$

• 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