IB-Why does 2 π/b=period?

[Leah5467]

We are taught to see a graph and then make an equation based on y=asin(b(x-c))+d to represent the graph. I don't really get why b is in the equation and...what b is,honestly....Please help! Thank you so much! saved a life?

 

[Dr.Peterson]

Have you learned about transformations of functions, in particular the horizontal stretch or compression? That is what b does.

In particular, as your title says, it determines the period (by stretching or shrinking the graph horizontally). Taking the simplest case, [MATH]y = \sin(bx)[/MATH], notice that the start of one cycle of the sine is when the argument [MATH]bx = 0[/MATH], and the end is when [MATH]bx = 2\pi[/MATH]. Those are, respectively, [MATH]x=0[/MATH] and [MATH]x=\frac{2\pi}{b}[/MATH]. That is why the period is [MATH]\frac{2\pi}{b}[/MATH].

 

[Leah5467]

Thank you for helping! Then...If 2π/b=12,does it mean within one 2π,it rolls b times? And what does 12 mean in this equation then?

 

[Dr.Peterson]

It means the period is 12, so each cycle is 12 units long. What you say is sort of right: b is the number of cycles per 2pi. In this case, though, b = pi/6, and it would be a little awkward to talk about the cycle repeating 0.524 times in 6.28 units, but it's not wrong to say that.

 

[Leah5467]

i see. thank you!

 

[Steven G]

Let y=asin(bx+c)
One period of the graph starts when bx+c=0 or x= -c/b
This period of the graph finishes when bx+c=2pi or x = (2pi - c)/b
The period is then (2pi -c)/b - -c/b = 2pi/b
That is why the period of a sine (cosine, csc and sec) is 2pi/b

 

[Leah5467]

thank you!