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 cosiney = 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 cosiney = 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

• Graph y = -2 – 2sin5x

Combinations of Translations

• Graph y = 1 -2cos3(x+π)

Combinations of Translations

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

• Graph y = secx

Graphs of Secant and Cosecant

• Graph y = cscx

Graphs of Secant and Cosecant

• Graph y = 2csc5x

Graphs of Secant and Cosecant

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)

Amplitude: not applicable

Period: π

Phase Shift: π to the right

Vertical Translation: up 2

Find the amplitude, period, phase shift, and vertical translation…then graph it.

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

Period: π/2

Phase Shift: π/8 to the left

Vertical Translation: up 1

Find the amplitude, period, phase shift, and vertical translation…then graph it.

Amplitude: not applicable