how to find the period of a sine function

Assignment 1: Exploring Sine Curves

by Kristina Dunbar, UGA

In this consignment, nosotros will be investigating the graph of the equation

y = a sin (bx + c)

using different values for a, b, and c.

In the above equation,

• a is the amplitude of the sine curve
• b is the menses of the sine bend
• c is the phase shift of the sine curve

What is the amplitude of a sine curve?

The aamplitude of a sine bend is its peak.

What is the menstruum of the sine bend?

The period of the sine curve is the length of one cycle of the curve.  The natural flow of the sine curve is 2π.  So, a coefficient of b=one is equivalent to a period of iiπ.  To go the period of the sine curve for any coefficient b, just separate 2π by the coefficient b to get the new period of the curve.

The coefficient b and the flow of the sine curve have an inverse relationship, so as b gets smaller, the length of ane bicycle of the curve gets bigger.  Likewise, as you increase b, the flow will decrease.

What is the phase shift of a sine curve?

The phase shift of a sine bend is how much the curve shifts from zero.  If the phase shift is cipher, the curve starts at the origin, but it can move left or correct depending on the stage shift.  A negative phase shift indicates a movement to the right, and a positive phase shift indicates movement to the left.

Let's look at the graph y = sin ten.

As you look at the graph, recall that the numerical value of π is approximately iii.1416, then 2π is approximately vi.2832.

In the to a higher place graph

• The amplitude a is 1.  This means that the elevation of the graph volition be i, and the tiptop of the showtime "hump" is 1.

• The period b has a coefficient of one, so the period is (2π)/1, or simply 2π.

• The phase shift c is zero, and then the curve starts at the origin.

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Permit'south examine the sine curve with unlike amplitudes.

We've already seen the case where the amplitude is 1; it's in the to a higher place graph.  What about other amplitudes?

y = two sin x

  y = 5 sin x

y = -i sin x

What's different nigh the in a higher place graph?  Information technology has a coefficient of a = -one.  What does that mean?  Nosotros see that the highest point of the curve is however i, only the first hump is at -1 instead of i.  We've essentially flipped the graph over.

Now let'due south look at several different sine graphs together.

 

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Let's examine the sine curve with unlike periods.

We've already seen the case where the b coefficient is 1; it's in the above graph.  What well-nigh other periods?

Retrieve, the b coefficient and the period of the curve have an inverse relationship.

y = sin (2x)

The coefficient b in the above graph is 2, so the period of the sine curve inverse by a factor of 1/2, making the new period π, or about 3.14.

y = sin (.5x)

For the above graph, the coefficient b = ane/2, and then the catamenia of the sine curve will be twice as long as it usually is, or iv π.

y = sin (3x)

Notice that the new period is i/iii of the original period, of 2 π/3, which is approximately ii.09.

Now let'south look at several different sine graphs together, with dissimilar periods.

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Let'southward examine the sine curve with a stage shift.

Normally, the sine curve does not accept a phase shift, so the variable c is 0.  This ways that the sine bend starts at the origin, as shown in the starting time graph at the top of this page.

What about when c is not equal to null?

y = sin (x + π)

In the to a higher place graph of y = sin (x + π), the graph has been shifted past a unit of measurement of π to the left.

In fact, a positive phase shift c actually indicates a shift to the left.  Allow'due south look at some more examples:

y = sin (ten + 1)

The sine bend shifted 1 unit of measurement to the left.

y = sin (10 + π/2)

The bend shifted π/2 units to the left.  Recall that π/ii is approximately ane.57.

What if the variable c is negative?

y = sin (ten - 1)

The bend shifted 1 unit to the right.

y = sin (x - π/2)

Let's look at a few phase shifts together:

Notation:  A phase shift of π will look exactly the same every bit a phase shift of -π.

y = sin (x + π)

y = sin (x - π)

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In the in a higher place exercises, nosotros have explored what happens to the sine curve when we vary the coefficients a, b, and c individually.  What if you varied more than one at a time?

y = 2 sin (2x)

a = 2    b=2    c=0

The amplitude is 2 and the period is two π/2, or π.  At that place is no phase shift.

y = ii sin (2x -one)

a = 2    b = 2    c = -1

The amplitude is 2 and the period is two π/two, or π.  The entire curve is shifted ane unit to the right.

y = 3 sin (2x + ii)

a = 3    b = 2    c = 2

The amplitude is 3, as we would expect.  The flow of the graph is two π/two, or π.  We would await the phase shift to be two units to the left, just we see that that is not the example.  Why?  Because the phase shift is in relation to the period.  The period of the graph is 1/2 its original size, and therefore the phase shift will likewise be 1/2 of the c coefficient, or 1.  This is shown in the graph above.

y = .5 sin (.5x -iii)

a = .five    b = .five    c = -iii

The aamplitude is .five, which we see clearly in the graph.  The coefficient b is .5, so the flow of the sine curve is twice what it usually is, or 4 π (approximately 12.57).  Because the menses of the bend is twice what it ordinarily is, the phase shift volition be twice the c coefficient, or 6 units to the right.

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Would you like to piece of work some do problems?  Click hither.

Source: http://jwilson.coe.uga.edu/EMAT6680/Dunbar/Assignment1/sine_curves_KD.html

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