Secant Graph : We know that sec x = 1 /cos x. The cosine function has period 2π so the secant function has also period 2π.
As secant is a reciprocal of cosine function so for some values this function is discontinuous.
We know that sec x ≤ -1 or sec ≥ 1.
First, we graph y = cos x and then y = sec x immediately below it. Compare the y-values in each of the 2 graphs and assure yourself they are the reciprocal of each other.
Phase shift in secant-graph
The secant function is y= sec(bx -c) - d
Period = 2π/b and phase shift = c/b
vertical shift = -d
Example 1 :
y = sec(πx -1)
b = π
period =2π/b = 2π/π = 2
phase shift=c/b=1/π= 1/3.14 = 0.3184
So the graph will shift to right side by 0.3184 units.
If we draw a secant curve with no shifts then the (x,y)coordinates are (0,1),(1,-1) and (2,1)
After shifting the coordinates will be (0 +0.3184,1),(1.3184,-1) and (2.3184,1)
Example 2 :
y = – sec(x + π/2) + 3
Guide function: y = – cos(x + π/2) + 3
a = –1 ; b = 1 ; c = – π/2 and d = 3
period = 2π/1 = 2π
phase shift = x + π/2 = 0
⇒ x = -π/2
vertical shift =3
Now you can easily graph this.
1) y = -5 sec(πx/4 - π/2)-3
2) y = 2 sec(x)
3) y = sec(x - π)
4) y = -sec(x) + 1
5) What is the period of y = −4sec(πx)?
6) Find the period and phase shift of y = -4sec(πx/4 + π/4)
7) Find the period and phase shift of y = -4sec(πx/8 + π/4)
8) What is the period of y = 2sec(x/8)?
9) Find the period and phase shift of y = sec(3x) + 4
10) What is the period of y = sec(-2x)?