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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24.
Graphing Trigonometric Functions
24.1
Sine graph: y = sin x
24.2
Cosine graph: y = cos x
24.3
Tangent graph: y = tan x
24.4
Cotangent graph: y = cot x
24.5
Secant graph: y = sec x
24.6
Cosecant graph: y = csc x
24.7
Graphing transformations of trigonometric functions
24.8
Determining trigonometric functions given their graphs
GCE O-Level A Maths
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