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Transformations of Sinusoidal Functions

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Transformations of Sinusoidal Functions

Let y=af(b(x−h))+ky=af(b(x-h))+ky=af(b(x−h))+k where f(x)f(x)f(x) is a trigonometric function. Then, 


HorizontalVerticalb<0‾:a<0‾:Reflection about theReflection about they-axisx-axis∣b∣>1‾:∣a∣>1‾:Compresses the period  1b  unitsExpansion of a units∣b∣<1‾:∣a∣<1‾:Expands the period  1b  unitsCompression of a unitsh>0‾:k>0‾:Phase shift h units rightDisplacement k units uph<0‾:k<0‾:Phase shift h units leftDisplacement k units down\begin{array}{l c c l}
\textbf{Horizontal}&&&\textbf{Vertical}\\\\
\underline{b<0}:&&&\underline{a<0}:\\
\text{Reflection about the}&&&\text{Reflection about the}\\
\text{\textit y-axis}&&&\text{\textit x-axis}\\\\\\
\underline{|b|>1}:&&&\underline{|a|>1}:\\
\text{Compresses the period}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{Expansion of \textit a units}\\\\\\
\underline{|b|<1}:&&&\underline{|a|<1}:\\
\text{Expands the period}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{Compression of \textit a units}\\\\\\
\underline{h>0}:&&&\underline{k>0}:\\
\text{Phase shift \textit h units right}&&&\text{Displacement \textit k
 units up}\\\\\\
\underline{h<0}:&&&\underline{k<0}:\\
\text{Phase shift \textit h units left}&&&\text{Displacement \textit k units down}
\end{array}Horizontalb<0​:Reflection about they-axis∣b∣>1​:Compresses the period  b1​  units∣b∣<1​:Expands the period  b1​  unitsh>0​:Phase shift h units righth<0​:Phase shift h units left​​​Verticala<0​:Reflection about thex-axis∣a∣>1​:Expansion of a units∣a∣<1​:Compression of a unitsk>0​:Displacement k units upk<0​:Displacement k units down​


Wize Tip
The period can be calculated as: P=360°∣b∣P=\dfrac{360\degree}{|b|}P=∣b∣360°​

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How to Graph the Transformation of a Sine or Cosine Function
Step 1. 
Find the period P=360°∣b∣\boxed{P=\displaystyle\frac{360\degree}{|b|}}P=∣b∣360°​​
Label the rest of the graph by splitting each period into 4 (label the x-axis with steps of P4\dfrac{P}{4}4P​).

Step 2.
Find the amplitude aaa 


Step 3. 
Find the vertical displacement kkk


Step 4.
Identify:

 Max:y=a+kAxis:y=kMin:y=−a+k\begin{array}{rl}
\text{Max}:&y=a+k\\\\
\text{Axis}:&y=k\\\\
\text{Min}:&y=-a+k
\end{array}Max:Axis:Min:​y=a+ky=ky=−a+k​


Step 5.  
Identify the phase shift (horizontal translation) hhh to determine how many units to the left or right the function must move.


Step 6.
Graph.


Wize Tip
For y=sin⁡xy=\sin xy=sinx, use the pattern:
Start in the middle ➡ go up ➡ back to the middle ➡ go down ➡ back to the middle

For y=cos⁡xy=\cos xy=cosx, use the pattern:
Start at the top ➡ back to the middle ➡ go down ➡ back to the middle ➡ go up 

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Example: Transformations of Sinusoidal Functions
Graph y=2sin⁡(2(θ−45°))+1y=2\sin(2(\theta-45\degree))+1y=2sin(2(θ−45°))+1.

Step 1. Find the period. 

P=360°2=180°P=\displaystyle\frac{360\degree}{2}=180\degreeP=2360°​=180°

Leave space to label the period.
Label the rest of the graph by splitting each period into 4 steps: label the x-axis with steps of 180°4=45°\dfrac{180\degree}{4}=45\degree4180°​=45°.

Step 2. Find the amplitude.

y=2sin⁡(2(θ−45°))+1y=\colorbox{yellow}2\sin(2(\theta-45\degree)) +1y=2​sin(2(θ−45°))+1
So a=2a=2a=2.

Step 3. Find the vertical translation.

y=2sin⁡(2(θ−45°))+1y=2\sin(2(\theta-45\degree)) \colorbox{yellow}{+1}y=2sin(2(θ−45°))+1​
So k=1k=1k=1.
This also gives us the equation of the axis: y=1y=1y=1

PAGE BREAK
Step 4. Identify the Max, Axis, Min\colorThree{\text{Max},~\text{Axis},~\text{Min}}Max, Axis, Min.

Min=−1\text{Min}=-1Min=−1

Axis=1\text{Axis}=1Axis=1

Max=3\text{Max}=3Max=3

Step 5.  Identify the phase shift to determine how many units left or right the function moves.

The phase shift is 45°45\degree45° units right.

PAGE BREAK
Step 6.

Since we are graphing sin⁡θ\sin{\theta}sinθ , we begin on the Axis\color{red}\text{Axis}Axis at (45°,1)(45\degree,1)(45°,1)

To the right, plot the Max\color{red}\text{Max}Max at (90°,3)(90\degree,3)(90°,3)

To the right, plot on the Axis\color{red}\text{Axis}Axis at (135°,1)(135\degree,1)(135°,1)

To the right, plot the Min\color{red}\text{Min}Min of (180°,−1)(180\degree,-1)(180°,−1)

To the right, plot the Axis\color{red}\text{Axis}Axis of (225°,1)(225\degree,1)(225°,1)

Continue on with this pattern in both directions.

Example: Transformations of Sinusoidal Functions
Match each transformed sinusoidal function with the property that is changed by the transformation.

A.

Axis

B.

Amplitude

C.

Period

D.

No change in axis, amplitude, or period

y=cos⁡x−4y=\cos x-4y=cosx−4

y=8sin⁡xy=8\sin xy=8sinx

y=cos⁡(12x)y=\cos(\frac{1}{2}x)y=cos(21​x)

y=sin⁡(x−20°)y=\sin(x-20\degree)y=sin(x−20°)

I don't know

Practice: Transformations of Sinusoidal Functions
The following is a table of values describing a sinusoidal relationship between xxx and y.y.y.

x15°45°75°105°135°y0−5050\begin{array}{|c|c|c|c|c|c|}\hline\\
\textbf{x}& 
15\degree & 45\degree& 75\degree& 105\degree &135\degree\\\\\hline\\
\textbf{y}&0&-5&0&5&0\\\\\hline
\end{array}xy​15°0​45°−5​75°0​105°5​135°0​​

What function best describes the tables of values? 

y=−5sin⁡(3(x−15°))y=-5\sin{\Bigg(3\Big(x-15\degree\Big)\Bigg)}y=−5sin(3(x−15°)) 

y=5sin⁡(3(x−15°))y=5\sin{\Bigg(3\Big(x-15\degree\Big)\Bigg)}y=5sin(3(x−15°)) 

y=−5sin⁡(3(x+15°))y=-5\sin{\Bigg(3\Big(x+15\degree\Big)\Bigg)}y=−5sin(3(x+15°)) 

 y=5sin⁡(3(x+15°))y=5\sin{\Bigg(3\Big(x+15\degree\Big)\Bigg)}y=5sin(3(x+15°)) 

I don't know

Practice: Transformations of Sinusoidal Functions
Let y=cos⁡θy=\cos{\theta}y=cosθ undergo the following transformations:
A vertical compression by a factor of 12\displaystyle\frac{1}{2}21​
A horizontal compression by a factor of 13\displaystyle\frac{1}{3}31​
A phase shift/horizontal translation of 60°60\degree60° left
Which of the following graphs best displays the graph of the transformed function?