How do you find the period of a sine function?

For y = a sin(bx + c) + d, the period is 2pi/|b|. For y = a tan(bx + c) + d, the period is pi/|b|. The coefficient b compresses or stretches the graph horizontally. Larger |b| means shorter period (more oscillations).

What is amplitude in a trig function?

Amplitude is the vertical distance from the midline to the peak. For y = a sin(bx+c) + d, amplitude = |a|. It represents half the height of the wave. For tangent, amplitude is undefined since the function has no maximum or minimum.

How does the phase shift affect the graph?

The phase shift is -c/b for y = a sin(bx+c) + d. It shifts the graph horizontally. A positive phase shift (c negative) moves right, a negative phase shift (c positive) moves left. The graph shape remains unchanged.

What is the frequency of a trig function?

Frequency = 1/period = |b|/(2pi). It indicates how many cycles occur per unit interval. Higher frequency means more oscillations. Combined with amplitude, it fully describes a simple harmonic wave.

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Home›Trigonometry Tools›Trigonometric Graph Period Calculator

Trigonometric Graph Period Calculator

Enter function parameters to compute period, amplitude, midline, and phase shift

Function Type

Amplitude (a)

b (frequency)

c (horizontal shift)

d (vertical shift)

y = a * sin(bx + c) + d

Trig Graph Properties

Period: T = 2pi/|b| (sin/cos), T = pi/|b| (tan)

Amplitude: A = |a| (sin/cos only)

Phase Shift: PS = -c/b

Midline: y = d

Frequency: f = 1/T = |b|/(2pi)

The parameters a, b, c, and d fully describe a sinusoidal function. Each parameter affects the graph in a specific way: a controls height, b controls width, c controls horizontal position, and d controls vertical position.

⚠ For tangent functions, amplitude is undefined (unbounded). Period for tan is pi/|b|, not 2pi/|b|.

What Affects a Trig Graph?

Each parameter in y = a sin(bx+c) + d transforms the graph in a predictable way. Understanding these effects is essential for graphing trigonometric functions and analyzing wave behavior.

Amplitude |a|

Vertical stretch/compression. Negative a reflects across the midline. Only applies to sin and cos.

Period |b|

Horizontal stretch/compression. Larger b = shorter period = more oscillations. Period = 2pi/|b|.

Phase Shift

Horizontal translation = -c/b. Positive shift moves right. The starting point of one cycle shifts.

Vertical Shift d

Moves the entire graph up/down. The midline becomes y = d. All max/min values shift by d.

Teaching Example: y = 3 sin(2x + 1) + 0. Amplitude = 3, Period = 2pi/2 = pi, Phase shift = -1/2 = -0.5 (left 0.5 units), Midline y = 0. Frequency = 2/(2pi) = 0.318 cycles per unit.

Applications

Sound Waves AC Circuits Signal Processing Physics SHM Graphing Engineering

FAQs about Trig Graph Periods

How to find period of sine?▼

For y = a sin(bx+c)+d: period = 2pi/|b|. For y = 3 sin(2x): period = 2pi/2 = pi. For y = sin(x): period = 2pi.

What is the period of tan?▼

For y = a tan(bx+c)+d: period = pi/|b|, not 2pi/|b|. Tangent repeats twice as fast as sine/cosine.

How does b affect the graph?▼

b controls horizontal stretch. Large b = compressed graph (more cycles in same space). Small b = stretched graph (fewer cycles).

What is the midline of a trig function?▼

The midline is y = d. It is the horizontal line that the function oscillates around. Maximum = d + |a|, Minimum = d - |a|.

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