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

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1 Answer

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:

enter image source here