Transformations of Sinusoidal Functions

7 Activities

Wize High School Algebra II Textbook (Common Core) > Trigonometric (Sinusoidal) Graphs

Transformations of Sinusoidal Functions

7 Activities

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

xπ12π45π127π123π4y0−5050\begin{array}{|c|c|c|c|c|c|}\hline\\
\textbf{x}&\displaystyle\frac{\pi}{12}&\displaystyle\frac{\pi}{4}&\displaystyle\frac{5\pi}{12}&\displaystyle\frac{7\pi}{12}&\displaystyle\frac{3\pi}{4}\\\\\hline\\
\textbf{y}&0&-5&0&5&0\\\\\hline
\end{array}xy​12π​0​4π​−5​125π​0​127π​5​43π​0​​

What function best describes the tables of values? 

y=−5sin⁡(3(x−π12))y=-5\sin{\Bigg(3\Big(x-\displaystyle\frac{\pi}{12}\Big)\Bigg)}y=−5sin(3(x−12π​)) 

y=5sin⁡(3(x−π12))y=5\sin{\Bigg(3\Big(x-\displaystyle\frac{\pi}{12}\Big)\Bigg)}y=5sin(3(x−12π​)) 

y=−5sin⁡(3(x+π12))y=-5\sin{\Bigg(3\Big(x+\displaystyle\frac{\pi}{12}\Big)\Bigg)}y=−5sin(3(x+12π​)) 

 y=5sin⁡(3(x+π12))y=5\sin{\Bigg(3\Big(x+\displaystyle\frac{\pi}{12}\Big)\Bigg)}y=5sin(3(x+12π​)) 

I don't know

Transformations of Sinusoidal Functions

Practice: Transformations of Sinusoidal Functions
If (π4,3)\Bigg(\displaystyle\frac{\pi}{4}, 3\Bigg)(4π​,3) is on the function y=2cos⁡2(θ−π3)−3y=2\cos{2\Big(\theta-\displaystyle\frac{\pi}{3}\Big)}-3y=2cos2(θ−3π​)−3, what point must be on the function y=cos⁡θy=\cos{\theta}y=cosθ?

Transformations of Sinusoidal Functions

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​

Equivalent Sinusoidal Functions

Let y=3cos⁡4(x+π16)y=3\cos4\Big(x+\frac{\pi}{16}\Big)y=3cos4(x+16π​). 
Write an equivalent expression using sin⁡θ\sin\thetasinθ with the smallest possible phase shift and a>0a>0a>0.

Transformations of Sinusoidal Functions

Sketch a graph of the function y=2cos⁡2(θ−π8)y=2\cos{2\Big(\theta-\dfrac{\pi}{8}\Big)}y=2cos2(θ−8π​)

Transformations of Sinusoidal Functions

A sinusoidal function has the following properties:
Ymax=−1Y_{\text{max}}=-1Ymax​=−1
Yavg=1Y_{\text{avg}}=1Yavg​=1

Transformations of Sinusoidal Functions

Practice: Transformations of Sinusoidal Functions
If (π4,3)\Bigg(\displaystyle\frac{\pi}{4}, 3\Bigg)(4π​,3) is on the function y=2cos⁡2(θ−π3)−3y=2\cos{2\Big(\theta-\displaystyle\frac{\pi}{3}\Big)}-3y=2cos2(θ−3π​)−3, what point must be on the function y=cos⁡θy=\cos{\theta}y=cosθ?

Transformations of Sinusoidal Functions

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​

Equivalent Sinusoidal Functions

Let y=3cos⁡4(x+π16)y=3\cos4\Big(x+\frac{\pi}{16}\Big)y=3cos4(x+16π​). 
Write an equivalent expression using sin⁡θ\sin\thetasinθ with the smallest possible phase shift and a>0a>0a>0.

Transformations of Sinusoidal Functions

Sketch a graph of the function y=2cos⁡2(θ−π8)y=2\cos{2\Big(\theta-\dfrac{\pi}{8}\Big)}y=2cos2(θ−8π​)

Transformations of Sinusoidal Functions

A sinusoidal function has the following properties:
Ymax=−1Y_{\text{max}}=-1Ymax​=−1
Yavg=1Y_{\text{avg}}=1Yavg​=1

Transformations of Sinusoidal Functions

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