source : yahoo.com

## What is the amplitude, period, and phase shift of f(x) = −3 cos(4x + π) + 6?

The general form of the cosine function is:

y = A cos(Bx – C) + D

where:

|A| is the amplitude of the function.

The period of the function is: 2π/B

The phase shift of the function is: C/B

A positive phase shift means the graph has moved to the left, while a negative phase shift means the graph has moved to the right.

‘D’ is the amount of vertical displacement, or ‘y’ shift, of the mid point of the function above the ‘x’ axis.

so in this case:

f(x) = −3 cos(4x – π) + 6

A = |-3| = 3

period = 2π/B = 2π/4 = π/2

The phase shift = C/B = -π/4

A) => answer

PDF Find the amplitude, the period, and the phase shift – (There could be several phase shifts. If there is one, we assume it is the one just left of the y-axis. Remember, if both b and c are positive, the phase shift 1) The graph of the equation is shown in the figure. (a) Find the amplitude, period, and phase shift. (b) Write the equation in the form y = a sin……period, vertical translation and phase shift of this: 4cos(x/3-pi/3) = Form:: y = a*cos(b(x-c)+d Your Problem:: y = 4*cos((1/3)(x-pi)+ 0 —- amp = |a| = 4 period = (2pi/b) = (2pi/(. 1/3)) = 6pi vertical trans = d = 0 phase shift = c/b = pi/(1/3) = 3pi – Cheers, Stan H.But, instead of shifting the graphed sine wave three units up, I'll add room underneath my current graph, shift the horizontal axis three units down You can use the Mathway widget below to practice finding the amplitude, period, and phase shift. Try the entered exercise, or type in your own exercise.

SOLUTION: What is the amplitude, period, vertical translation and… – Amplitude and period of sine and cosine functions. Sketching the graph of y= a sin(bx) plus d or y= a cos(bx) plus d.Section 5.5 Graphing Sinusoidal Functions Lecture Notes 3 To get the transformed x-coordinates, we set 4x+3πequal each original x-coordinate and Step 1: Determine the values of a, b, c, and d. We have a= 1/3 b= πc=3πd = 1 Step 2: Determine the amplitude, period, phase shift, and vertical shift…6.2 Trig Functions Amplitude, Period, & Phase Shift 3 ways we can change our graphs Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 2. The range is the set of y values such that 1 y 1. 3…

Graphing Trig Functions: Phase Shift | Purplemath – The Period Is (Type An Exact Answer, Using Numbers In The Expression.) As Needed. Transcribed Image Text from this Question. Determine the amplitude, period, and phase shift of the function. Graph the function y 3 cos (4x – t) The amplitude is (Simplify your answer.)phase-shift-amplitude-period. asked Nov 11, 2014 in PRECALCULUS by anonymous. This Equation is in the form of y = Asin(Bx-c)+d. Where A is the Amplitude, B is the Strtch along x-axis, D is a cos curve with a period8pi, an amplitude2, a right shift pi/2, and a vertical translation5 units up.A. amplitude = 3; period = pi over two; phase shift: x = negative pi over four. 'D' is the amount of vertical displacement, or 'y' shift, of the mid point of the function above the 'x' axis. so in this case: f(x) = −3 cos(4x – π) + 6.

**FFT Tutorial** – >> FFT Tutorial Tony: Here we are again.

Ian: Hi everybody, it's Tony and Ian from Tektronix.

Tony: Nice to see you guys are back.

We're going to do a video FFT tutorial.

Ian: That’s right, Fast Fourier Transform. Tony: So, just what is FFT Ian?

Ian: It’s just a different way of looking at your signal, on oscilloscopes we're used

to looking at voltage changing over time; a FFT example shows you instead what voltages

are present at each frequency. Tony: I get that, so good. So this is kind

of a clearer way of looking at this? Ian: Right, if you can imagine your signal

as a bunch of wire tangled together, the FFT just helps you untangle it and see what it's

made of, see what components are lurking in that signal.

Tony: Like tangled hair? Ian: Exactly… hey, what are you saying?

Tony: No, you’re ok. Ian: Ok, I used conditioner this morning!

C'mon! Ian: So let's say you’re designing a circuit,

you’re going to send some RF energy out an antenna, so you're going to pull out that

antenna, plug-in a BNC cable, connected to your oscilloscope. Now because you’re generating

a sine wave, you expect to see the voltage changing sinusoidal over time and that would

be represented in the FFT by one spike. Tony: Because there's one frequency?

Ian: Exactly, You've got it all ready. Tony: Excellent!

Ian: But imagine that you plug this in and what you actually see is a second spike right

here. Tony: That is excellent.

Ian: And you wouldn't see that right there in the sine wave because it is so low level,

it's such a miniscule second signal that you have to have something like the FFT to bring

it out. Tony: There's no way I could have seen that.

Ian: And on a real oscilloscope, you'd look and you'd find out, well if I’m expecting

900 MHz and I got 1800 MHz that's a harmonic, I’d probably have some harmonic distortion.

There are other causes of… Tony: And this isn't just Photoshop, this

is… Ian: This is real, the arrow is Photoshoped.

Tony: I can't believe that, it's so good. Ian: So, there are some other common causes

of interference. We're just going to let those sit on the slide for a second so that after

the end of the video you can jump back here. Tony: And you thought it was harmonic because

where it was at in the frequency? Ian: Exactly, right if it was somewhere else

maybe it would have been cross-talk from a fox signal.

Tony: That's really handy. Ian: So, let's talk about how it actually

works. There is this beautiful property of mathematics, where every signal can be thought

of as the sum of a bunch of sine waves. Tony: Did you hear him, he said every signal.

Ian: Any signal you want, any shape, name a shape.

Tony: Let’s do a square wave. Ian: Square wave, by coincidence that is what's

on this next card! Ian: Yes Tony! Even a square wave, is the

sum of a bunch of sine waves. Let's see how that works. If we wanted to make a square

wave by adding together a bunch of sine waves, we'd start with a single sine wave.

Tony: Oh, how smooth. Ian: Yes, very smooth, it's almost like we've

done a bunch of failed takes of this video. Tony: No way, this is the first one.

Ian: So, if we add a sine wave to nothing, we get a sine wave.

Tony: OK, 1 to 1, I got it. Ian: 1 to 1, right. And that would cause a

single spike to appear in the frequency domain. Now we're going to add a second sine wave

that's 1/3 as tall and it's been scaled in by 1/3.

Tony: And this is the mathematical part. This is like there are two humps within this big

one. Ian: Yes, exactly, when we add those two together,

that's going to add a little…as you say that's going to add a little hump at the top

and bottom of each of these. And the frequency domain view because it's two sine waves, that's

two peaks. So let's add a third one. Instead of this one being scaled by a 1/3, it's scaled

by a 1/5 and when we add that it's going to add even more wiggles. It's going to make

the sides steeper, it's going to make the tops and bottoms flatter, and you notice we're

kind of closing in on a square wave here. Tony: Then we'd say we're getting squared

up here? Ian: Oh, terrible!

Tony: OK. Ian: We get to a 1/7, a 1/9, a 1/11 and so

on, and so we keep adding peaks until it kind of vanishes down into the noise floor.

Tony: Oh yea, and we recognize that, that goes down like that.

Ian: Now let's hang on to this card; we'll compare it with the real oscilloscope in a

minute. Tony: OK.

Ian: So real world signals have this same property, they might look like a tangle in

the time domain and then a nice, neat shape once you've…

Tony: Which is something you're probably familiar with, lots of tangled signals.

Ian: Yes. Tony: How to break them down.

Ian: So, let's see this on a real oscilloscope. Tony: Yea, let’s take that away.

Ian: So we're going to see what this looks like on a traditional oscilloscope first,

most modern oscilloscopes have a FFT function, usually in the Math System, so we're going

to push Math-FFT, and right away you can see that signal tapering way off down to the noise

floor. But let's zoom in on that a little bit.

Tony: Just like in our drawing right? How it tapers off.

Ian: Yes…So you can see that tapering off there. In fact, I'll move it down a little

bit. Tony: You sure do know your way around an

oscilloscope don't you? Ian: Uh, this one feature I do. So can we

get the card that we drew earlier? Tony: Here you go.

Ian: You can see that looks pretty much like the FFT there.

Tony: I see the angle and I see the angle, genius!

Ian: There's another class of instruments that also do a lot of FFTs and that's spectrum

analyzers, and this oscilloscope, the mixed domain oscilloscope, has a little bit of spectrum

analyzer personality built-in and it can do FFTs as well.

Tony: Mixed domain. Ian: Mixed domain.

Tony: Two things going together. This tool is really handy. Sorry Ian, go ahead.

Ian: So we've got, we hope it is, so we've got one signal going to both the traditional

oscilloscope channel and this dedicated spectrum analyzer channel. I'm going to turn that on

and right away you see a difference. Tony: Whoa.

Ian: And right away you see a difference. Tony: Oh I do, both types of views are right

there. Ian: Exactly, we've got the frequency domain

view in a separate view. And you can see that second peak stands right out. And we've even

marked it and identified that it's spaced from the original 900 MHz by 900 MHz and it's

down about 25 dB. Tony: So you put the marker in here or did

it mark it for you? Ian: It marked it for us, so this tells us

it's probably harmonic distortion. Tony: So this concludes our video FFT tutorial.

Ian: Yes. Tony: There's more right?

Ian: There's more, if you want to learn more about FFTs, you can go to tektronix.com/fft-basics

Tony: There's a lot of stuff there. Ian: Yes, and also stay tuned for the next

video in the series where we do an FFT example of a musical signal in front of a live audience.

Tony: Oh that sounds great. OK. Ian: See you around. Thanks! .

**Trigonometry Identities: cos(2π + x), sin(2π + x), cos(π/2 -x), sin(π/2 -x)** – .

**FFT Tutorial** – >> FFT Tutorial Tony: Here we are again.

Ian: Hi everybody, it's Tony and Ian from Tektronix.

Tony: Nice to see you guys are back.

We're going to do a video FFT tutorial.

Ian: That’s right, Fast Fourier Transform. Tony: So, just what is FFT Ian?

Ian: It’s just a different way of looking at your signal, on oscilloscopes we're used

to looking at voltage changing over time; a FFT example shows you instead what voltages

are present at each frequency. Tony: I get that, so good. So this is kind

of a clearer way of looking at this? Ian: Right, if you can imagine your signal

as a bunch of wire tangled together, the FFT just helps you untangle it and see what it's

made of, see what components are lurking in that signal.

Tony: Like tangled hair? Ian: Exactly… hey, what are you saying?

Tony: No, you’re ok. Ian: Ok, I used conditioner this morning!

C'mon! Ian: So let's say you’re designing a circuit,

you’re going to send some RF energy out an antenna, so you're going to pull out that

antenna, plug-in a BNC cable, connected to your oscilloscope. Now because you’re generating

a sine wave, you expect to see the voltage changing sinusoidal over time and that would

be represented in the FFT by one spike. Tony: Because there's one frequency?

Ian: Exactly, You've got it all ready. Tony: Excellent!

Ian: But imagine that you plug this in and what you actually see is a second spike right

here. Tony: That is excellent.

Ian: And you wouldn't see that right there in the sine wave because it is so low level,

it's such a miniscule second signal that you have to have something like the FFT to bring

it out. Tony: There's no way I could have seen that.

Ian: And on a real oscilloscope, you'd look and you'd find out, well if I’m expecting

900 MHz and I got 1800 MHz that's a harmonic, I’d probably have some harmonic distortion.

There are other causes of… Tony: And this isn't just Photoshop, this

is… Ian: This is real, the arrow is Photoshoped.

Tony: I can't believe that, it's so good. Ian: So, there are some other common causes

of interference. We're just going to let those sit on the slide for a second so that after

the end of the video you can jump back here. Tony: And you thought it was harmonic because

where it was at in the frequency? Ian: Exactly, right if it was somewhere else

maybe it would have been cross-talk from a fox signal.

Tony: That's really handy. Ian: So, let's talk about how it actually

works. There is this beautiful property of mathematics, where every signal can be thought

of as the sum of a bunch of sine waves. Tony: Did you hear him, he said every signal.

Ian: Any signal you want, any shape, name a shape.

Tony: Let’s do a square wave. Ian: Square wave, by coincidence that is what's

on this next card! Ian: Yes Tony! Even a square wave, is the

sum of a bunch of sine waves. Let's see how that works. If we wanted to make a square

wave by adding together a bunch of sine waves, we'd start with a single sine wave.

Tony: Oh, how smooth. Ian: Yes, very smooth, it's almost like we've

done a bunch of failed takes of this video. Tony: No way, this is the first one.

Ian: So, if we add a sine wave to nothing, we get a sine wave.

Tony: OK, 1 to 1, I got it. Ian: 1 to 1, right. And that would cause a

single spike to appear in the frequency domain. Now we're going to add a second sine wave

that's 1/3 as tall and it's been scaled in by 1/3.

Tony: And this is the mathematical part. This is like there are two humps within this big

one. Ian: Yes, exactly, when we add those two together,

that's going to add a little…as you say that's going to add a little hump at the top

and bottom of each of these. And the frequency domain view because it's two sine waves, that's

two peaks. So let's add a third one. Instead of this one being scaled by a 1/3, it's scaled

by a 1/5 and when we add that it's going to add even more wiggles. It's going to make

the sides steeper, it's going to make the tops and bottoms flatter, and you notice we're

kind of closing in on a square wave here. Tony: Then we'd say we're getting squared

up here? Ian: Oh, terrible!

Tony: OK. Ian: We get to a 1/7, a 1/9, a 1/11 and so

on, and so we keep adding peaks until it kind of vanishes down into the noise floor.

Tony: Oh yea, and we recognize that, that goes down like that.

Ian: Now let's hang on to this card; we'll compare it with the real oscilloscope in a

minute. Tony: OK.

Ian: So real world signals have this same property, they might look like a tangle in

the time domain and then a nice, neat shape once you've…

Tony: Which is something you're probably familiar with, lots of tangled signals.

Ian: Yes. Tony: How to break them down.

Ian: So, let's see this on a real oscilloscope. Tony: Yea, let’s take that away.

Ian: So we're going to see what this looks like on a traditional oscilloscope first,

most modern oscilloscopes have a FFT function, usually in the Math System, so we're going

to push Math-FFT, and right away you can see that signal tapering way off down to the noise

floor. But let's zoom in on that a little bit.

Tony: Just like in our drawing right? How it tapers off.

Ian: Yes…So you can see that tapering off there. In fact, I'll move it down a little

bit. Tony: You sure do know your way around an

oscilloscope don't you? Ian: Uh, this one feature I do. So can we

get the card that we drew earlier? Tony: Here you go.

Ian: You can see that looks pretty much like the FFT there.

Tony: I see the angle and I see the angle, genius!

Ian: There's another class of instruments that also do a lot of FFTs and that's spectrum

analyzers, and this oscilloscope, the mixed domain oscilloscope, has a little bit of spectrum

analyzer personality built-in and it can do FFTs as well.

Tony: Mixed domain. Ian: Mixed domain.

Tony: Two things going together. This tool is really handy. Sorry Ian, go ahead.

Ian: So we've got, we hope it is, so we've got one signal going to both the traditional

oscilloscope channel and this dedicated spectrum analyzer channel. I'm going to turn that on

and right away you see a difference. Tony: Whoa.

Ian: And right away you see a difference. Tony: Oh I do, both types of views are right

there. Ian: Exactly, we've got the frequency domain

view in a separate view. And you can see that second peak stands right out. And we've even

marked it and identified that it's spaced from the original 900 MHz by 900 MHz and it's

down about 25 dB. Tony: So you put the marker in here or did

it mark it for you? Ian: It marked it for us, so this tells us

it's probably harmonic distortion. Tony: So this concludes our video FFT tutorial.

Ian: Yes. Tony: There's more right?

Ian: There's more, if you want to learn more about FFTs, you can go to tektronix.com/fft-basics

Tony: There's a lot of stuff there. Ian: Yes, and also stay tuned for the next

video in the series where we do an FFT example of a musical signal in front of a live audience.

Tony: Oh that sounds great. OK. Ian: See you around. Thanks! .

**Trigonometry Identities: cos(2π + x), sin(2π + x), cos(π/2 -x), sin(π/2 -x)** – .