30 lines
1.3 KiB
Markdown
30 lines
1.3 KiB
Markdown
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Given the above graph:
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- At the origin, $sin(x) = 0$ and $cos(x) = 1$
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- A full wavelength takes $2\pi$
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# Manipulation
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| Formula | Movement |
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| ---------------- | ---------------------------------- |
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| $y = cos(x) - 1$ | Vertical shift down by 1 |
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| $y = 2cos(x)$ | Vertical stretch by a factor of 2 |
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| $y = -cos(x)$ | Flip over x axis |
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| $y = cos(2x)$ | Horizontal shrink by a factor of 2 |
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# Periodic Functions
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A function is considered periodic if it repeats itself at even intervals, where each interval is a complete cycle, referred to as a *period*.
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# Sinusoidal Functions
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A function that has the same shape as a sine or cosine wave is known as a sinusoidal function.
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Given a function in the form:
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$$ A * sin(B*x - C) + D $$
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| Variable | Meaning |
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| -------- | --------------------------------------------------------------------------------------------- |
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| $A$ | The amplitude of a function can be found by taking $\|A\|$. The sign flips it over the x axis |
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| $B$ | Horizontal, or phase |
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$$ y = A * \sin(B(x-\frac{C}{B})) $$ |