How do you graph y=2sect by first sketching the related sine and cosine graphs?

1 Answer
Jul 10, 2018

Remember that sec t = 1/cost
Then use your knowledge of reciprocal functions to sketch!

Explanation:

To sketch y_1 = 2sect, the only related graph you need think about is the cosine graph.

The identity relating secant and cosine is as follows:
sec t = 1/cost AA {t : cost != 0}
Hence:
2 sec t = 1/(1/2cost)

Which is just the reciprocal of a regular cosine graph dilated by a factor of 1/2 from the t-axis (i.e. an amplitude of 1/2).

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 att 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.