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Transformations of Sinusoidal Functions

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Wize High School Grade 12 Pre-Calculus Textbook > Trigonometric (Sinusoidal) Graphs

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A sinusoidal function has the following properties:
  • Ymax=−1Y_{\text{max}}=-1
  • Yavg=1Y_{\text{avg}}=1
  • Ymin=3Y_{\text{min}}=3
  • Period of π3\dfrac{\pi}{3}
  • A phase shift of π9\dfrac{\pi}{9}.
Determine a sin⁡x\sin{x} function to describe the above.
More Transformations of Sinusoidal Functions Questions:
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
If (π4,3)\Bigg(\displaystyle\frac{\pi}{4}, 3\Bigg) is on the function y=2cos⁡2(θ−π3)−3y=2\cos{2\Big(\theta-\displaystyle\frac{\pi}{3}\Big)}-3, what point must be on the function y=cos⁡θy=\cos{\theta}?
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
Let y=cos⁡θy=\cos{\theta} undergo the following transformations:
A vertical compression by a factor of 12\displaystyle\frac{1}{2}
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
The following is a table of values describing a sinusoidal relationship between xx and 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}
Equivalent Sinusoidal Functions
Let y=3cos⁡4(x+π16)y=3\cos4\Big(x+\frac{\pi}{16}\Big).
Write an equivalent expression using sin⁡θ\sin\theta with the smallest possible phase shift and a>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)}
More Transformations of Sinusoidal Functions Questions:
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
If (π4,3)\Bigg(\displaystyle\frac{\pi}{4}, 3\Bigg) is on the function y=2cos⁡2(θ−π3)−3y=2\cos{2\Big(\theta-\displaystyle\frac{\pi}{3}\Big)}-3, what point must be on the function y=cos⁡θy=\cos{\theta}?
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
Let y=cos⁡θy=\cos{\theta} undergo the following transformations:
A vertical compression by a factor of 12\displaystyle\frac{1}{2}
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
The following is a table of values describing a sinusoidal relationship between xx and 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}
Equivalent Sinusoidal Functions
Let y=3cos⁡4(x+π16)y=3\cos4\Big(x+\frac{\pi}{16}\Big).
Write an equivalent expression using sin⁡θ\sin\theta with the smallest possible phase shift and a>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)}
More Graphs of Sinusoidal Functions Questions:
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
Let y=cos⁡θy=\cos{\theta} undergo the following transformations:
A vertical compression by a factor of 12\displaystyle\frac{1}{2}
More Graphs of Sinusoidal Functions Questions:
Transformations of Sinusoidal Functions
Practice: Transformations of Sinusoidal Functions
Let y=cos⁡θy=\cos{\theta} undergo the following transformations:
A vertical compression by a factor of 12\displaystyle\frac{1}{2}