In this lesson, you’ll learn what is meant by amplitude, frequency, period, phase shift, and vertical shift, and how to find them from the function itself, or a graph.

Waves are important – they’re everywhere! In nature, they’re found in water waves, tsunamis, earthquakes, sound, and light. Everywhere you look, waves are proof of the world happening.

We have found ways to use many forms of waves, including electricity, x-rays, microwaves, speakers, and sending information across the world at mind-boggling speeds, just to name a few!

In mathematics, waves are usually shown as a function called a **sinusoidal function**.

At the end of the lesson, there are some Practice Questions, so you can test what you’ve learned.

What Makes a Periodic Function

Amplitude

Period

Frequency

Phase/Horizontal Shift

Vertical Shift

Sinusoidal Functions

Finding the Amplitude

Finding the Amplitude

Finding the Period

Finding the Frequency

Finding the Vertical Shift

Finding the Phase/Horizontal Shift

Practice Questions

To Sum Up (Pun Intended!)

## What Makes a Periodic Function: Amplitude, Period, Phase Shift

A **periodic function** repeats after a certain time or distance and, if left alone, would never end. The measurement between repeats is the **period**, or **wavelength**.

A periodic function that comes along the most is the sine function.

This is what it looks like on a graph.

These functions have 5 main attributes, which are also called transformations.

### 1. Amplitude

The **amplitude** of a wave is the greatest displacement from the rest position. The bigger the amplitude, the taller the wave.

In sound waves, a bigger amplitude means a louder sound.

In light waves – a kind of electromagnetic radiation – a bigger amplitude means a brighter light.

Radio is often broadcasted using **AM** or Amplitude Modulation transmissions, meaning information can be picked up by a receiver tuned to the correct amplitude of the waves.

### 2. Period

The period of a wave is the time or distance between two oscillations – the time taken for a point on an oscillating object to return to where it started. It can be called the **time period** and has the symbol, T.

The word *period* is used for waves measured with time. If the measurement is distance, it is called the **wavelength**.

The period of a wave is closely related to its frequency.

### 3. Frequency

You know when you get an SMS, or someone’s calling? Your phone *vibrates* to give you a warning, right? Your phone is doing something called **oscillating**.

The frequency of the oscillation tells you how many times the phone “vibrates” back-and-forth every second.

A higher frequency means more oscillations per second, so on a graph, the waves look closer together.

A higher frequency means more oscillations per second, making the waves look closer together.

The higher the frequency, the higher pitched the sound. You can experiment with the relationship between pitch and frequency.

Frequency modulation (FM) is a type of transmission for radio except, instead of changing the amplitude to send information, the frequency is changed instead.

There are other ways of changing the way a wave behaves though.

### 4. Phase Shift

**Phase shift**, measures how far left or right, or *horizontally*, the wave has been *shift*ed from the normal sine function. It is for this reason that it’s sometimes called **horizontal shift**.

Just like data can be transmitted on different channels by changing the frequency or amplitude, as mentioned for radio, sometimes the horizontal shift is what has changed.

This is called Phase Shift Keying, or PSK. It is used for lots of things, Wi-Fi and Bluetooth being two of the most widely used.

### 5. Vertical Shift

The final piece of the puzzle, the **vertical shift** works the same way, but measures how far up or down the *y*-axis, or vertically, the wave has been shifted from the normal sine function.

## Sinusoidal Functions

**Sinusoidal functions** are periodic, but also use a trigonometric function like sine or cosine. *Sin*-usoidal.

You met one of them earlier, sin (*t*). In this case, the *t* shows time.

Here’s what it looks like on a graph again:

This graph only shows part of the function, between about -π and 6π but, much like Doctor Strange’s time loop, it actually goes on forever, repeating the same pattern.

You will often see other letters for the argument of the sine function instead of *t*. Usually *x* shows distance, and *θ* shows angles.

In this lesson only *t* will be used – it is still important to remember that other variables can be used.

## How to Find Attributes of a Sinusoidal Function

These attributes are the ones you just looked at:

They’re easy to find when only one of them is different from the standard sine wave, but when you combine several transformations of different types, it’s easy to get lost if you don’t know what you’re looking for.

In this section, you’ll be given the tools to make sense of it all.

### How to Find the Amplitude

*t*– φ)) + C

The amplitude, A is the number that multiplies the sine function.

Think of it this way: a sound will be twice as loud if you doubled its amplitude. Multiplying the whole function by 2 is doubling the amplitude.

On a graph, multiplying the whole sine function by some number, A, looks like stretching or squashing the sine graph in the *y*-direction

The amplitude, A, is found by taking **half** the vertical distance between the peaks and the troughs.

The **peaks** are the highest points of each wave, and the **troughs** the lowest points. When writing a function for a wave using sin(*t*), the sine function is multiplied by the amplitude.

You *can* have a negative value for A, but that doesn’t mean the amplitude is negative! It just means that the graph has been flipped in the *t*-axis.

The amplitude must be a positive number, and so is the absolute value, magnitude, or size of this coefficient, |A|.

Here’s an example of how to find the amplitude.

In this graph, the peak is at *y* = 2.5, and the trough is at *y* = -0.5, so the vertical distance between them is 2.5 – (-0.5) = 3 units.

The amplitude is half this distance, so 1.5 units.

This means the equation for this function would look like this:

*t*– φ)) + C

Don’t worry about the values of B, φ and C yet – we’ll get to them one at a time!

### How to Find the Period

So, what if you don’t have the equation, just a graph?

The period of a wave can found by measuring the distance between two consecutive peaks or troughs.

Measure the distance between *two peaks* OR *two troughs* that are next to each other.

Remember, the peaks are the highest point on the function and the troughs are the lowest.

When you look at the default sine function (without any transformations), you can see the function comes back to the same point every time you look 2π further along the *t*-axis, so the period of the basic sine function is 2π.

The period, T of a wave is found by measuring the distance between two consecutive peaks or troughs.

If a periodic function has a period of 2, then

*t*+ 2)

= f(

*t*+ 4)

= f(

*t*+ 6)

This means that adding one period, T, to the value of *t* doesn’t change the outcome.

You could actually pick any point and measure the distance between it and the same point on the next wavefront, but peaks or troughs are usually the easiest to use.

Here’s an example of how to find the period.

This is the same wave as in the previous example. One peak occurs at *t*=0s, and the next peak occurs at *t*=2s. The difference between these times 2, so the period is 2 seconds.

Now, what if you don’t have a graph, only an equation?

That’s where the frequency comes in!

### How to Find the Frequency

*t*– φ))+ C

The frequency of a wave, f, isn’t as easy to picture as some of the other attributes, but it can be calculated from the time period, T, or vice versa, using these:

^{1}⁄

_{T}

T =

^{1}⁄

_{f}

So, to find the frequency of a wave first find the period like we showed you above, then take its reciprocal to find the frequency.

It’s helpful to know that the frequency refers to how many times the wave repeats itself in a certain time.

If the time period is measured in seconds, then the frequency is measured in Hertz (Hz).

The longer the period, the less often the wave will repeat itself, so it will have a lower frequency. The shorter the period, the higher the frequency.

But we’re trying to find B, not T or f?!

When dealing with radians **angular frequency**, B, is used instead of frequency. It is related to the time period by the formula

OR

B =

^{2π}⁄

_{T}

In the last section we already found the period, which was 2 seconds.

Using the formula for angular frequency, you can calculate that the frequency is:

^{2π}⁄

_{(2 s)}= π rad/s

Now we know the amplitude and the angular frequency, so the sine function so far looks like this as an equation:

*t*– φ)) + C

When it comes to writing the equation of a wave as a transformation of sin(*t*), the input for the sine function gets multiplied by the angular frequency *after* any horizontal shifting.

### How to Find the Vertical Shift

*t*– φ)) + C

Vertical shift is the constant C added on at the end of the general sinusoidal function.

The rest position of the sine function is the line *y* = 0.

When you introduce a vertical shift of C, the rest position of this function moves from the line *y*=0 to the line *y* = C.

If the vertical shift is -5, then the rest position moves from the line *y*=0 to the line *y*=-5.

A sine function is always symmetrical around its rest position, so the amplitude will help find the rest position if you don’t have an equation to work with.

The amplitude is *half* the total height of the wave from peak to trough.

If you already know the amplitude A, it’s probably simpler to measure the vertical distance, d, from the *t*-axis to the troughs, then the vertical shift is

If the rest position is above the *t*-axis, the vertical shift is positive. If the rest position is below the *t*-axis, the vertical shift is negative.

Here’s an example of how to find the vertical shift.

This is the same example we used before, and found that the amplitude A = 1.5 units.

The vertical distance from the *t*-axis to the troughs is d=-0.5, so the vertical shift is

= 1.5 + (-0.5)

= 1

So far we have found this function’s amplitude, period, frequency and vertical shift. It can be written as:

*t*– φ)) + 1

Now, on to the final piece of the puzzle.

### How to Find the Phase Shift

*t*– φ) + C

Remember from earlier: The phase shift of a wave, φ, measures how far the wave has been moved **horizontally** from the default sine wave.

To see how far a wave has been shifted, you need to find the closest time to *t*=0, when the wave is both at the rest position and has a positive slope.

If there is no vertical shift this is simple since the rest position is just the *t*-axis, the line *y*=0.

If there *is* some vertical shift, you’ll need to calculate that first – have a look at the section above to find the rest position using the peak-trough positions and the amplitude – then you can draw the line *y*=C.

C is the vertical shift you just calculated. The line *y*=C is the rest position for a vertically shifted wave.

Now that you have the rest position, let’s get back to the phase shift.

To choose the right point on the function to measure from, use this guidance. The function at that point must:

- Be one of the two closest point to the
*y*-axis - Have a positive slope. That means as you move along the
*t*-axis, the*y*-value increases.

If the point you will use is to the right of the y-axis, φ is positive, and if the point is to the left of the y-axis, φ is negative.

It doesn’t matter how close your point is to *t*=0, but the slope must still be positive – the function is periodic, so adding or subtracting whole number multiples of the period to φ doesn’t change the resulting function – this method makes things that little bit simpler.

The distance from the *y*-axis to the point that satisfies the guidance above is the phase shift, φ.

Here’s an example of how to find the phase shift.

The function is not symmetrical around the *t*-axis, so there is some vertical shift going on.

We know from before that the time period T=2s, the angular frequency B=π, the vertical shift C=1, and the line *y*=1 is the rest position.

Let’s look at the graph. The closest point to *t*=0 which passes through this line with a positive slope has been highlighted and is at *t*=0.25 s. This is 0.25 seconds to the right, so the phase shift is φ=0.25 seconds!

*t*– 0.25)) + 1

Even though the sign in front of φ is negative in the general equation, a positive value of φ will still cause a shift in the positive direction.

Think about graph transformations – shifting the input backward appears to make the graph move forwards.

## Practice Questions

Now have a go at these practice questions. If you can get through them all, you’ll be much more confident solving problems about period functions. If you can’t, leave a comment on what you’re struggling with and one of us will be happy to help!

And remember: if you want to keep something fresh in your brain, you need to practice it regularly!

A sinusoidal wave takes 4π seconds to complete an oscillation.

b) The amplitude of this wave is 3 units, and there is no horizontal or vertical shift. Write the equation for the wave.

a)The angular frequency, B is

^{2π}⁄

_{T}

=

^{2π}⁄

_{4π}

= 2 rad/s

b) *y* = 3 sin(2*t*)

The equation of a wave is given by

*y*= 3 sin(

*t*) – 1

b) What is the period of the wave? (Hint: We’re using radians)

c) Sketch the graph in the range -π≤t≤π.

a) 3 units

b) 2π seconds, since the and the angular frequency B=1 and the period T=^{2π}⁄_{B}

c) The sketch should look like this, with a trough at (-^{π}⁄_{2},-4) and a peak at (^{π}⁄_{2},2). The rest position, and where it crosses the *y*-axis, should be *y*=1.

This graph has the equation

*y*= sin(P

*t*+ Q)

Find the values of the constants P and Q

(Hint: P would normally multiply both *t* AND the phase shift constant φ)

T=2, so B=^{2π}⁄_{2}=π. Also, φ=1 since the rest position is y=0.

*y*= sin(π(

*t*– φ))

= sin(π(

*t*– 1))

= sin(π

*t*– π)

This means that P=π and Q=-π.

This graph shows a wave.

b) Find the frequency.

c) Find the horizontal shift.

d) Finally, write down the equation of the wave.

a) A = ^{1.6}⁄_{2} = 0.8 units

Since the troughs have *y*=0, then C=0.8 as well.

b) The clearest way to see this is between the troughs at (0.375,0) and (1.375,0). The time between them is 1 second, but there are 3 oscillations, so T=^{1}⁄_{3} seconds.

Use that to find B.

^{1}⁄

_{3}

= 6π

c) Drawing the line *y*=0.8 there is positive-slope intersection at *t*=0.125s, and so φ=0.125.

d) y = 0.8 sin(6π (*t* – 0.125)) + 0.8

Sketch the graph with the equation

*y*= 2 sin(

^{16}⁄

_{π}(t + 1)) – 1

In the range 0≤t≤2.

5) Your sketch should look like this:

## To Sum Up (Pun Intended!)

In this article, you learned not only about all the different attributes of periodic and sinusoidal functions but how to find them all too!

A summary of them all:

**Amplitude**: the ‘height’ of the wave, equal to half the vertical distance between the peaks and the troughs.**Period**: the time between oscillations, found as the distance between two consecutive peaks or troughs.**Frequency**: the number of oscillations per second, related to the period by the formula f = 1/T**Angular Frequency**: the number of revolutions per minute, measured usually in radians. B = 2π/T

**Phase Shift**: how much the wave has been moved horizontally from the default position, found by looking at the points at the rest position.**Vertical Shift**: how much the wave has been shifted vertically, given by the height of the midpoint between the peaks and the troughs.

If you haven’t done so already, have a go at some of the practice questions to test your knowledge!

If you have any questions on anything covered here, let us know in the comments. Also, please let us know how you got on with the practice questions, it helps us make sure they’re right for you.