Question

I have a few questions about this equation>> y=3sin(6(x+2))+4
I know that the amplitude is 3
How would I find what the period is?
What is the horizontal shift? (left or right? and by how much?)
How would I find the midline?

Answer 1

Think on y=sin(x)
Amplitude: 1
Period : 2π
Midline: y = 0
Horizontal Shift; None

You recognized that y=3sin(x) results in an amplitude of 3.
Amplitude: 3
Period : 2π
Midline: y = 0
Horizontal Shift; None

The midline shift is the only other easy part. That 4 is just kind of hanging out there. Once you see it, you are not likely to forget it. Thus y=3sin(x)+4 results in
Amplitude: 3
Period : 2π
Midline: y = 4
Horizontal Shift; None

Are we making sense, so far?

Answer 2

Yes, so far :)

Answer 3

Okay, let's add another piece. Multiplying the argument be a value greater than +1 just makes things happen faster. Just look at these x-values

1, 2, 3, 4, 5

What happens if we multiply them by 6?

6, 12, 18, 24, 30

See how we are clipping along quite a bit faster? If we are moving faster, we SHORTEN the period.

This gives us a clue about that 6 in the argument.

Thus \(y=3\sin(6x)+4\) results in
Amplitude: 3
Period : \(2\pi/6 = \pi/3\)
Midline: y = 4
Horizontal Shift; None

Do you believe?

Answer 4

Okay, so whatever is being multiplied to x is then made as a denominator to 2pi to find the period?<<since it is making it move faster<<makes the period shorter?

Answer 5

Yes. Multiply by 5 and the new period is \(2\pi/5\). Multiply by 1/3 and the new period is \(2\pi/(1/3) = 6\pi\)

Okay, the last little piece. This is sometimes counter-intuitive. Think about the difference between x and x-h. If we want x = 3, we just use x = 3 in the first version, but we have to use x = 3+h in the second version. Do you see how we have to INCREASE the value we put in, in order to reproduce it in the changed version? Thus, x-3 is a shift RIGHT and x+3 is a shift LEFT. That's not quite how we usually think about positive and negative.

Thus \(y=3\sin(6(x+2))+4\) results in
Amplitude: 3
Period : \(2\pi/6 = \pi/3\)
Midline: y = 4
Horizontal Shift; -2

Make sure you factor that argument.

6x+12 = 6(x+2)

Some folks will mistake this for a shift of -12! that would be quite incorrect.

Answer 6

Thank you sooo much!