Mercyhurst UniversityMath DeptDr Williams Home

Transformations of Trigonometric Functions

Function Options Base Function:
Graphing y = A sin(Bx - C) + D
A: Amplitude
B: Period Multiplier
C: Phase Shift
D: Vertical Shift

Graph Options Click and drag the graph to change the viewport

r = A + B sin(n θ)

Using the Applet

Choose a base trigonometric function, then change its amplitude, period, phase shift, and vertical shift using the inputs. The original base function will be drawn in grey, and the transformation in blue. Use the slider to zoom in or out on the graph, and drag to reposition.

Transforming Trigonometric Functions

The graphs of the six basic trigonometric functions can be transformed by adjusting their amplitude, period, phase shift, and vertical shift.

The amplitude of a sinusoidal trig function (sine or cosine) is it's 'height,' the distance from the average value of the curve to its maximum (or minimum) value. We can change the amplitude of these functions by multiplying the function by a constant A. The resulting amplitude is given by \[ \mbox{ amplitude} = |A| \] The other trig functions (tangent, cotangent, secant, and cosecant) do not have an amplitude, but multiplication by A will affect their steepness. Note that a negative value of A will flip the graph of any function across the \(x\)-axis.

The period of any trig function is the length of one cycle. sin(x), cos(x), sec(x), and csc(x) all have a period of \(2\pi\), while tan(x) and cot(x) have a period of \(\pi\). By altering the value of B (the multiplier of the variable before the function is evaluated), we can change the period of the function according to the formula \[ \mbox{ period} = \frac{\mbox{original period}}{|B|} \] When |B| is larger than one, the new period is smaller than the original, so the function will appear horizontally compressed. When |B| is less than 1, the period is larger than the original, and the function will appear stretched.

Phase Shift
The phase shift of a trigonometric function refers to its horizontal shift to the right. A phase shift results from adding a value to the variable before the evaluating the function. The phase shift of a trigonometric function is calculated using the formula \[ \mbox{phase shift} = \frac{C}{B} \] When C is positive, the graph will appear to shift to the right. When C is negative, the graph will shift to the left.

Vertical Shift
Adding a value D to a trig function will translate its graph vertically. If D is positive, the graph will shift up by a factor of D; if D is negative, the graph will shift down.

Any combination of these transformations can be applied to a function simultaneously, as demonstrated in this applet.

About the Applet

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This applet was last updated February 2018.