How do you graph y=2sect by first sketching the related sine and cosine graphs?
1 Answer
Remember that
Then use your knowledge of reciprocal functions to sketch!
Explanation:
To sketch
The identity relating secant and cosine is as follows:
Hence:
Which is just the reciprocal of a regular cosine graph dilated by a factor of
Things to remember for sketching reciprocal functions:
-
f(t) = 1/f(t) wheneverf(t) = 1
This will never occur with the secant graph! becausey_2 = 1/2cost never equals one. -
Whenever
f'(t) = 0 ,d/dt(1/f(t)) will also be zero.
This means that the maximum and minimum values of both graphs will occur at the samet -values -
ran y_2 = [-1/2,1/2] , so the range of the reciprocal will be given byran y_1 = (-oo,-2] uu [2, oo) . -
Maximum and minimum points will occur at
t in {-2pi, -pi, 0, pi, 2pi ...}
(Check this with the derivative:
d/dt(2sect) = 2 sect tant = (2sint)/cos^2t
d/dt(1/2cost) = -sint/2
Of course these can only be zero whensint = 0 . -
Whenever
y_1 is decreasing,y_2 is increasing, and vice-versa.
I'm not sure what else to say... I hope this helps! Use a graphing calculator to do some of these things if you need.