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Graphing Trigonometric Functions. Chapter 4. The sine and cosine curves. Graph y = sinx. The sine and cosine curves. Graph y = cosx. The sine and cosine curves. Graph y = - cosx. The sine and cosine curves. Graph y = - sinx. Amplitude “a”. y = asinx y = acosx
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Graphing Trigonometric Functions Chapter 4
The sine and cosine curves • Graph y = sinx
The sine and cosine curves • Graph y = cosx
The sine and cosine curves • Graph y = -cosx
The sine and cosine curves • Graph y = -sinx
Amplitude “a” y = asinx y = acosx The amplitude will stretch the graph vertically. The value of “a” is half the distance of the max and min.
Amplitude “a” • Graph y = 3cosx
Period of the sine and cosine y = sinbx and y = cosbx The period of the function will shrink or stretch the graph horizontally. The period of a function is The standard period is 2π, this occurs when b = 1.
Period of the sine and cosine • Graph y = sin3x
Period of the sine and cosine • Graph y = cos2x
Amplitude “a” and Period ”b” • Graph y = 3sin4x
Amplitude “a” and Period ”b” • Graph y = -4cosπx
Phase Shifts of sine and cosine y = sinb(x-d) and y = cosb(x-d) The period of the function will have new endpoints when solving the inequality 0 ≤ b(x-d) ≤ 2π. (x – d) is a shift of “d” to the right (x + d) is a shift of “d” to the left
Phase Shifts of sine and cosine • Graph
Phase Shifts of sine and cosine • Graph
Vertical Translations of sine and cosine y = c + sinx and y = c + cosx The “c” will shift the entire graph “c” units up when “c” is positive and “c” units down when “c” is negative
Vertical Translations of sine and cosine • Graph y = 2 + sinx
Vertical Translations of sine and cosine • Graph y = -2 + cos3x
Combinations of Translations • Graph y = -2 – 2sin5x
Combinations of Translations • Graph y = 1 -2cos3(x+π)
Combinations of Translations • Graph
Identifying Features Give the amplitude, period, phase shift, and vertical translation. Amplitude: 2 Period: 2π Phase Shift: π/3 to the left Vertical Translation: none
Identifying Features Give the amplitude, period, phase shift, and vertical translation. Amplitude: 1 Period: 2π/3 Phase Shift: π/6 to the right Vertical Translation: up 1
Identifying Features Give the amplitude, period, phase shift, and vertical translation. Amplitude: 4 Period: π Phase Shift: π to the right Vertical Translation: down 2
Graphs of Secant and Cosecant • Graph y = secx
Graphs of Secant and Cosecant • Graph y = cscx
Graphs of Secant and Cosecant • Graph y = 2csc5x
Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: π Phase Shift: π/6 to the left Vertical Translation: down 1
Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it.
Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: 2π Phase Shift: π/4 to the right Vertical Translation: up 2
Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it.
Over “2-periods” • Graph y = sinx
Over “2-periods” • Graph
Tangent and Cotangent • Sine,Cosine,Secant, and Cosecant have a standard period of 2π. • The tangent and cotangent have a standard period of π. • The standard tangent graph has asymptotes at –π/2 and π/2 • The standard cotangent graph has asymptotes at 0 and π
Tangent and Cotangent • Graph y = tanx
Tangent and Cotangent • Graph y = cotx
Tangent and Cotangent • Graph y = 1 – tan3x
Tangent and Cotangent • Graph y = 2 + 3cot(x – π) Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: π Phase Shift: π to the right Vertical Translation: up 2
Tangent and Cotangent • Graph y = 2 + 3cot(x – π) Find the amplitude, period, phase shift, and vertical translation…then graph it.
Tangent and Cotangent • Graph Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: π/2 Phase Shift: π/8 to the left Vertical Translation: up 1