Section 5
Section 5.6 Date______________
Phase Shift & Transformations of Trig Functions
We discussed transformations, which included horizontal and vertical shifts, horizontal and vertical stretches/compressions, and reflections, earlier in the year. All of these transformations also apply to trig functions.
Phase shift is a horizontal shift that applies to trig functions.
← Remember: In general, f(x – k) moves all points of f(x) k units _______, and f(x + k) moves all points of f(x) k units ______.
Ex 1 Graph two cycles of the following functions.
|a) y = sin(x - [pic]) |b) y = sec(x + π) |c) y = tan(x + [pic]) |
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Consider y = A sin[ω(x - [pic])]. If we assume that Φ > 0, it has the same graph as ______ except that all points have been shifted [pic] units right.
← Notice: y = A sin[ω(x - [pic])] is equivalent to y = A sin(ωx – Φ).
|For the graphs of y = A sin(ωx – Φ) or y = A cos(ωx – Φ), ω > 0, |
|Amplitude = |A| Period = T = [pic] Phase Shift = [pic] |
|(Phase shift is left if Φ < 0 and right if Φ > 0.) |
Ex 2 Find the amplitude, period and phase shift of the following functions.
|a) y = 4 sin(3x – π) |b) y = -5 cos(½x + π) |c) y = sec(⅞x + [pic]) |d) y = 3 tan(2x - [pic]) |
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|Amplitude:_____ |Amplitude:_____ |Amplitude:_____ |Amplitude:_____ |
|Period:_______ |Period:_______ |Period:_______ |Period:_______ |
|Phase Shift:____ |Phase Shift:____ |Phase Shift:____ |Phase Shift:____ |
Summary of Graphing Trig Functions:
1. Find the amplitude, period, and phase shift of the function. (Period = ___ or __.)
2. Graph the function using amplitude, period and each function’s pattern.
3. Apply the phase shift of the function.
4. Extend the graph in each direction.
5. Apply any vertical transformation.
Ex 3 Graph the following functions. State the amplitude, period, phase shift, domain, and range. Find two zeros, a local maximum, and a local minimum.
|a) y = 3 sin(2x + π) – 1 |b) y = -6 cos(½x – π) + 1 |
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|Amp:____ Period:____ Phase Shift:____ |Amp:____ Period:____ Phase Shift:____ |
|Domain:_______ Range:________ |Domain:_______ Range:________ |
|Zeros:______ Max:____ Min:____ |Zeros:______ Max:____ Min:____ |
|c) y = 2 tan(3x – π) |d) y = cot(4x) – 3 |
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|Amp:____ Period:____ Phase Shift:____ |Amp:____ Period:____ Phase Shift:____ |
|Domain:_______ Range:________ |Domain:_______ Range:________ |
|Zeros:______ Max:____ Min:____ |Zeros:______ Max:____ Min:____ |
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