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)# whenever#f(t) = 1#
This will never occur with the secant graph! because#y_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 same#t# -values -
#ran y_2 = [-1/2,1/2]# , so the range of the reciprocal will be given by#ran 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 when#sint = 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.