MHB How do I find A,B,C, and D in a sinusoidal function?

AI Thread Summary
To find the values A, B, C, and D in a sinusoidal function, the equation y=Asin(2π/B(θ-C))+D is used. The amplitude is determined by |A|, the period by T=2π/B, the phase shift is C, and the mean is simply D. For the example problems, the first function y=sin(2x-π)+1 yields A=1, B=2, C=π/2, and D=1, while the second function y=6sin(πx)-1 gives A=6, B=1, C=0, and D=-1. Understanding these parameters allows for accurate analysis of sinusoidal functions.
Randi
Messages
1
Reaction score
0
I really need someone to break it down for me. I think I understand A and D, but I am confused on B and C. I have some example problems. But first, the equation my pre-calculus teacher has given us is y=Asin(2π/B(θ-C))+D. But I am still having a lot of trouble.

Find amplitude, period, a phase shift, and the mean of the following sinusodial functions.
a.) y=sin(2x-π)+1
b.) y=6sin(πx)-1

The answers to a.) are 1. π. π/2. 1. and the answers to b.) are 6. 2. 0. -1.

I just don't understand how these answers were found. PLEASE HELP.
 
Mathematics news on Phys.org
If we use the form:

$$y=A\sin\left(B(x-C)\right)+D$$

The amplitude is defined as $|A|$.

The period is:

$$T=\frac{2\pi}{B}$$

The phase shift is $C$. This comes from the horizontal shift of a function.

Finally, let's look at the mean. We may begin with:

$$-1\le\sin(\theta)\le1$$

Multiply through by $0\le A$ (if $A$ is negative, we would just reverse the inequality):

$$-A\le A\sin(\theta)\le A$$

Add through by $D$:

$$D-A\le A\sin(\theta)+D\le D+A$$

And so the mean will be the average of the boundaries:

$$\frac{(D-A)+(D+A)}{2}=\frac{2D}{2}=D$$

Now, for the first problem, we may write it as:

$$y=1\cdot\sin\left(2\left(x-\frac{\pi}{2}\right)\right)+1$$

So, we identify:

$$A=1,\,B=2,\,C=\frac{\pi}{2},\,D=1$$

And so the values are:

Amplitude: $$|1|=1$$

Period: $$T=\frac{2\pi}{2}=\pi$$

Phase shift: $$C=\frac{\pi}{2}$$

Mean: $$D=1$$

Now, see if you can do the second one. :D
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Replies
9
Views
4K
Replies
1
Views
1K
Replies
4
Views
5K
Replies
1
Views
1K
Replies
17
Views
3K
Replies
2
Views
2K
Back
Top