Graphing transformations of trigonometric functions

Graphing transformations of trigonometric functions

After learning all the graphs of basic trigonometric functions, in this lesson, we are going to go a little bit further on how the graphs will be transformed as the functions change. The general form for the equation of trig functions is y = f [B(x + c)] + D, where f refers the trig function; A refers to the amplitude/steepness; B represents the period of the graph; C refers to phase shift (left or right) and D represents vertical shift (up or down). We will learn how to graph the trig function for multiple periods; state the vertical displacement, phase shift, period and amplitude; and also find the domain and range of the transformed functions.
Basic Concepts: Combining transformations of functions, Sine graph: y = sin x, Tangent graph: y = tan x, Secant graph: y = sec x
Related Concepts: Ferris wheel trig problems, Tides and water depth trig problems, Spring (simple harmonic motion) trig problems

Lessons

  • 1.
    For each trigonometric function:
    (i) Graph the trigonometric function for one period.
    (ii) State the vertical displacement, phase shift, period, and amplitude.
    (iii) State the domain and the range.
    a)
    y=2sin⁡π4(x+3)+1y = 2\sin \frac{\pi }{4}(x + 3) + 1

    b)
    y=3sec⁡(π2x−π)−1y = 3\sec (\frac{\pi }{2}x - \pi ) - 1

    c)
    y=−2sin⁡(4x+4π)−3y = - 2\sin (4x + 4\pi ) - 3

  • 2.
    For the trigonometric function: y=−tan⁡(x3−π6)y = - \tan \left( {\;\frac{x}{3} - \frac{\pi }{6}\;} \right)
    i) Graph the trigonometric function for two periods.
    ii) State the domain and the range.
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10.
Trigonometry
10.1
Converting between degrees and radians
10.2
Radian measure and arc length
10.3
Angle in standard position
10.4
Coterminal angles
10.5
Reference angle
10.6
Find the exact value of trigonometric ratios
10.7
ASTC rule in trigonometry (All Students Take Calculus)
10.8
Unit circle
10.9
Trigonometric ratios for angles in radians
10.10
Solving first degree trigonometric equations
10.11
Determining non-permissible values for trig expressions
10.12
Use sine ratio to calculate angles and side (Sin = oh \frac{o}{h} )
10.13
Use cosine ratio to calculate angles and side (Cos = ah \frac{a}{h} )
10.14
Use tangent ratio to calculate angles and side (Tan = oa \frac{o}{a} )
10.15
Combination of SohCahToa questions
10.16
Law of sines
10.17
Law of cosines
10.18
Sine graph: y = sin x
10.19
Cosine graph: y = cos x
10.20
Tangent graph: y = tan x
10.21
Cotangent graph: y = cot x
10.22
Secant graph: y = sec x
10.23
Cosecant graph: y = csc x
10.24
Graphing transformations of trigonometric functions
10.25
Determining trigonometric functions given their graphs
10.26
Quotient identities and reciprocal identities
10.27
Pythagorean identities
10.28
Sum and difference identities
10.29
Double-angle identities
10.30
Word problems relating ladder in trigonometry
10.31
Word problems relating guy wire in trigonometry
10.32
Other word problems relating angles in trigonometry
Precalculus
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