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

by M. Bourne

### Phase shift Interactives...

Later on this page:

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

y=asin(bx+c)

and

y=acos(bx+c)

Both ** b** and

**in these graphs affect the**

*c***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`.

### Example 1

### Need Graph Paper?

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`

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## 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 is`30°`.

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`

Copyright © www.intmath.com

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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