Question

It is 1:08 AM and I am trying to solve some problems but I am stuck :) Can I get some help please? Thanks!

Answer 1

See below.

Explanation:

If we express the cosine function in the following way:

`y=acos(bx+c)+d`

Then:

`\ \ \bba \ \ \ ="the amplitude"`

`bb((2pi)/|b|) \ \="the period"`

`bb((-c)/b)= "the phase shift"`

`\ \ \ \bbd \ \ \="the vertical shift"`

a)

For given function we have:

`b=(2pi)/|3|`

We know from above that:

`"period"=(2pi)/|b|`

Hence:

`(2pi)/((2pi)/3)=color(blue)(3)`

b)

From above:

`"amplitude" =a=color(blue)(1/4)`

c)

The midline is a line that runs between the maximum and minimum values of the function.

These max/min values are the same as the amplitude, remember amplitude is given as an absolute value:

`|a|`

We can see from the function written in the form:

`y=acos(bx+c)+d`

that there is no vertical shift. i.e `d=0`. This means the midline is just the `x` axis. This is the same as for `cos(theta)`

`color(blue)(y=0)`

The graph confirms our findings: