Question

How do you graph `y = 2sin(4x) + 1`, by using the period, domain, range and intercepts?

Answer 1

a) In the function `y = asin(b(x+ c)) + d`, the period is given by `(2pi)/b`.

Call the period P.

`P = (2pi)/4`

`P = pi/2`

`:.`The period is `pi/2` units.

b) In the function `y = asin(b(x + c)) + d`, the amplitude is `a` and the vertical displacement is `d`.

The maximum is given by `a + d`.

The minimum is given by `a - d`.

Accordingly, the maximum is at `y= 3` and the minimum is at `y = 1`. Therefore, the range is `{y|1 ≤ y ≤ 3, y in RR}`.

c) We have most of the information needed to graph from the above questions, but I recommend you find the intercepts (the y intercept and a general rule for the x-intercepts.

y intercept

`y = 2sin(4 xx 0) + 1`

`y = 2sin(0) + 1`

`y = 0 + 1`

`y = 1`

x intercepts

`0 = 2sin(4x) + 1`

`-1/2 = sin(4x)`

`arcsin(-1/2) = 4x`

`(7pi)/6, (11pi)/6= 4x`

`x = (7pi)/24 and (11pi)/24`

Due to the periodicity of the sine function, `x = (7pi)/24 + 2pin` and `(11pi)/24 + 2pin` where `n` is an integer.

We have all the information necessary to graph. My grapher won't let me label points, but you can look above and find a good many (using x-intercepts, periodicity, maximums and minimums).

Hopefully this helps!