# 3. Graphs of y = a sin(bx + c) and y = a cos(bx + c)

by M. Bourne

### Phase shift Interactives...

Phase shift interactive

Amplitude, period and phase shift interactive

In this section, we meet the following 2 graph types:

y = a sin(bx + c)

and

y = a cos(bx + c)

Both b and c in these graphs affect the phase shift (or displacement), given by:

text(Phase shift)=(-c)/b

The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive.

There is nothing magic about this formula. We are just solving the expression in brackets for zero; bx + c = 0.

Sketch the curve

y = sin(2x + 1)

### Example 2

Sketch

y=12cos(2x-pi/8)

## Phase shift Interactive

In the following interactive, drag the Point P left or right, which displaces the curve. Observe the "c" and "displacement" values and how they change when you move the curve.

The example you see is y=sin(pi t). This has period given by (2 pi)/b = (2π)/π = 2.

You can also see the cosine case by choosing it at the top.

Choose graph type:

Sine
Cosine

Guide curve:

On
Off

Graph: y=sin(pi t - 0)

Displacement = (-c)/b = 0/pi = 0

## Phase Angle or Phase Shift?

Phase angle is not always defined the same as phase shift.

The phase angle for the sine curve y = a sin(bx + c) is usually taken to be the value of c and the phase shift is usually given by -c/b, as we saw above.

Reminder: In the last section, we saw how to express sine curves in terms of frequency.

Example: Electronics engineers separate the terms "phase angle" and "phase shift", and they use a mix of radians and degrees. We may have a current expressed as follows:

I = 50 sin (2π(100)t + 30°)

This means the amplitude is 50\ "A", the frequency is 100\ "Hz" and the phase angle is30°.

See an application of phase angle at An Application to AC Circuits in the complex numbers chapter.

Also, see a discussion on this issue at Phase shift or phase angle? in the math blog.

To keep things simple for now, we will mostly use the term phase shift in this chapter.

## Amplitude, period and phase shift graph applet

The following interactive will help you to do the graph sketching exercises below.

Use the sliders under the graph to vary each of the amplitude, period and phase shift of the graph.

The x-axis has an integer scale (it's radians, of course), and multiples of pi are indicated with a red stroke.

You can also change the function to cosine. Hopefully you can see the concepts work the same for both sine and cosine curves.

Choose graph type:

Sine
Cosine

Guide curve:

On
Off

Graph: y=a sin(bx + c)=sin(x)

Period = (2pi)/b = 6.28/1 = 6.28

Displacement = (-c)/b = 0/1 = 0

NOTE: The graph may look a little jagged as you drag the sliders. When you let go, the graph will be smooth again.

### Exercises

Sketch the graph of the following.

1. y = sin(2x + π/6)

2. y = 3\ sin(x + π/4)

3. y = 2\ cos(x - π/8)

4. y = -cos(2x - π)

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