Function : y=cot(2x+pi)
1)determine the interval for the principal cycle.
(Type this answer in interval notation)
2)Then for the principal cycle, determine the equations of the vertical asymptotes.
3)find the coordinates of the center point.
4) Find the coordinates of the halfway points.
( Direction - if needed type your points as ordered pair)
THANKS TUTOR!! Please show work, for I have to solve many more.
read the previous two posts and see what you can glean from them. I suspect a name change going on here.
Chichi/Ana/Lio/Mike/Nancy -- please use the same name for your posts.
Ms. Sue, I'm Nancy and I've only posted one previous question that has nothing to do with questions like other posts, which you're clearly blaming me to be catfishing. I just happen to be doing my summer h/w and stumbled upon the same questions others have. I clicked on other users post and saw the format they used. So, I used the same format. The format from my h/w is the combination of all my questions in one sentence. I didn't want any tutor to get confused, so I copied other user format. Thanks. If you want, you can click on my name to see the previous question I posted named "HELP NEEDED FOR CALC".
Hmm! -- All of you are posting from the same internet address in New York City.
To determine the interval for the principal cycle of the function y = cot(2x + π), we need to find the period of the function. The period is the distance between each full cycle of the function.
1) Determining the interval for the principal cycle:
The general formula for the period of the cotangent function is given by π/b, where b is the coefficient of x in the argument of cot(x). In this case, b = 2.
So, the period of the function y = cot(2x + π) is given by π/2.
The interval for the principal cycle is from -π/2 to π/2. Therefore, the answer, in interval notation, is (-π/2, π/2).
2) Determining the equations of the vertical asymptotes:
The vertical asymptotes occur when the tangent function becomes undefined, which happens when the denominator of the cotangent function equals zero.
In this case, the denominator of the cotangent function is 2x + π. Setting it equal to zero and solving for x, we get:
2x + π = 0
2x = -π
x = -π/2
So, the equation of the vertical asymptote is x = -π/2.
3) Finding the coordinates of the center point:
The center point refers to the point where the cotangent function is at its average value. In this case, the average value of the cotangent function is 0, which occurs when the cotangent function crosses the x-axis.
To determine the x-coordinate of this point, we set the argument of the cotangent function equal to 0:
2x + π = 0
2x = -π
x = -π/2
Therefore, the coordinates of the center point are (-π/2, 0).
4) Finding the coordinates of the halfway points:
The halfway points refer to the points where the cotangent function reaches its maximum and minimum values. These occur halfway between the vertical asymptotes.
Since the vertical asymptotes are at x = -π/2 and x = π/2, we can find the halfway points by taking the average of these values.
Halfway point 1: (x-coordinate)
-π/2 + 0 = -π/4
Halfway point 2: (x-coordinate)
0 + π/2 = π/4
The y-coordinate of both halfway points is infinity as the function approaches the vertical asymptotes.
Therefore, the coordinates of the halfway points are (-π/4, ∞) and (π/4, ∞).
I hope this helps! Let me know if you have any further questions.