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 2 p
b
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
æ pö
y = sin 4 ç x - ÷
è
4ø
Phase Shifts of sine and cosine
• Graph
y = 2cos ( x + p )
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
æ
pö
• Graph y =1+ sin ç 3x - ÷
è
2ø
Identifying Features
æ pö
y = 2 cos ç x + ÷
è
3ø
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
æ
pö
y =1+ sin ç 3x - ÷
è
2ø
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
y = -2 - 4sin2 ( x - p )
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.
æ
pö
y = -1+ sec ç 2x + ÷
è
3ø
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.
æ
pö
y = -1+ sec ç 2x + ÷
è
3ø
Graphs of Secant and Cosecant
Find the amplitude, period, phase shift, and vertical translation…then graph it.
æ pö
y = 2 - csc ç x - ÷
è
4ø
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.
æ pö
y = 2 - csc ç x - ÷
è
4ø
Over “2-periods”
• Graph y = sinx
Over “2-periods”
• Graph
æ pö
y = -cos ç x - ÷
è
4ø
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
Find the amplitude, period, phase shift, and vertical translation…then graph it.
• Graph y = 2 + 3cot(x – π)
Amplitude: not applicable
Period: π
Phase Shift: π to the right
Vertical Translation: up 2
Tangent and Cotangent
Find the amplitude, period, phase shift, and vertical translation…then graph it.
• Graph y = 2 + 3cot(x – π)
Tangent and Cotangent
Find the amplitude, period, phase shift, and vertical translation…then graph it.
æ
pö
• Graph y =1+ tan ç 2x + ÷
è
4ø
Amplitude: not applicable
Period: π/2
Phase Shift: π/8 to the left
Vertical Translation: up 1