Transformation of Trigonometric Graphs

In this lesson, we will learn

  • how Trigonometric Graphs can be transformed.
  • the amplitude and vertical shift of Trigonometric Graphs
  • the period and phase shift of Trigonometric Graphs

Related Topics: More Trigonometric Lessons

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function is the maximum displacement on the graph of that function.

In the case of sin and cos functions, this value is the leading coefficient of the function.
If y = A sin x, then the amplitude is .

In the case of tan, cot, sec, and csc, the amplitude would be infinitely large regardless of the value of A. However, for a limited domain, the value of A would determine the maximum height of these functions.

Period of Trigonometric Function

The period of a function is the displacement of x at which the graph of the function begins to repeat.

Consider y = sin x

The value x = 2π is the point at which the graph begins to repeat that of the first quadrant.
The coefficient of x is the constant that determine the period.

The general form is

y = A sin Bx

where   is the amplitude and B determines the period.

For the functions sin, cos, sec and csc, the period is found by


Find the period of the graph y = sin 2x and sketch the graph of y = sin 2x for 0 ≤ 2x ≤ π.

Since B = 2, the period is


Phase Shift of Trigonometric Functions

The general form for the equation of the sine trigonometric function is

y = A sin B(x + p)
where A is the amplitude, the period is calculated by the constant B, and p is the phase shift.
The graph y = sin x may be moved or shifted to the left or to the right. If p is positive, the shift is to the left; if p is negative the shift is to the right.

A similar general form can be obtained for the other trigonometric functions.

Find the amplitude, period and phase shift of


The amplitude is 2, the period is π and the phase shift is  units to the left.



Periodic Functions Definition, Period, Phase Shift, Amplitude, Vertical Shift

Transformation of sin and cos with amplitude and vertical shift


Transformation of sin and cos with period and phase shift

Examples of transforming basic sine and cosine functions

Transformations of Trigonometric Graphs: Amplitude, Period & Phase Shift

Graph trig functions (sine, cosine, and tangent) with all of the transformations.
The videos explained how to the amplitude and period changes and what numbers in the equations.
Part 1: See what a vertical translation, horizontal translation, and a reflection behaves in three separate examples.


Graph trig functions (sine, cosine, and tangent) with all of the transformations.
In this set of videos, we see how the line of equilibrium is affected by a vertical shift, and how the starting point is affected by a horizontal shift (phase). Shifts of graphs up and down are also called translations.
Part 2: An example of how the tangent graph and its asymptotes are affected different transformations. An example that includes every kind of transformation possible, all in one problem, is shown.


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