Question 968429
the normal period of cot(x) would be pi.
this is because the graph repeats every pi units on the x-axis.
the graph of cot(x) is shown below:


<img src = "http://theo.x10hosting.com/2015/050501.jpg" alt="$$$" </>


the period of cot(4x) would be pi/4.
this is because the graph repeats every pi/4 units on the x-axis.
the graph of cot(4x) is shown below:


<img src = "http://theo.x10hosting.com/2015/050502.jpg" alt="$$$" </>


period for sine and cosine and secant and cosecant function = 2pi / frequency


frequency for sine and cosine and secant and cosecant function = 2pi / period.


period for tangent and cotangent function = pi / frequency.


frequency for tangent and cotangent function = pi / period.


in your probolem, the frequency of the cotangent function is 4.


this makes the period pi/4.


if you were given the period as pi/4, then the freq8uency would be calculated as pi / (pi/4) which would then become 4.


the general formula for repeating trigonometric functions is


y = a*sin(b*(x-c))+d


a is the amplitude.
b is the frequency
c is the horizontal displacement.
d is the vertical displacement.


your equation would be shown as:


y = a*cot(b*(x-c)+d


in your equation:
a = 1
b = 4
c = 0
d = 7


your amplitude is 1.
your frequency is 4
your horizontal displacement is 0.
your vertical displacement is 7.


vertical displacement means that your center line for the graph will be 7 units above the x-axis.


here's the graph of y = cot(4x).


{{{graph(400,400,-2,2,-15,15,1/tan(4x),0)}}}


here's the graph of y = cot(4x) + 7


{{{graph(400,400,-2,2,-15,15,1/tan(4x)+7,7)}}}


you can see that the centerline for the grpah has been vertically displaced by 7 units in a positive direction.


here's a couple of references for you to ponder.


<a href = "http://www.regentsprep.org/regents/math/algtrig/att7/sincostan.htm" target = "_blank">http://www.regentsprep.org/regents/math/algtrig/att7/sincostan.htm</a>


<a href = "http://www.regentsprep.org/regents/math/algtrig/att7/othergraphs.htm" target = "_blank">http://www.regentsprep.org/regents/math/algtrig/att7/othergraphs.htm</a>


note that the normal period for sine and cosine is shown as 2pi and the normal period for tangent and cotangent is shown as pi.