Consider the graph of y = sec2x - 1

Question

a) How has this graph been transformed from the graph of y = secx? State the period of y = sec2x - 1 

b) Determine the domain and range of y = sec2x - 1

Answer 1

Given a parent function, y=f(x), it is convenient to represent the generalized parent function as follows.

 

(y-k)=Af[B(x-h)]

 

Here,

 

Step 1:  B represents compressing the parent function along the x axis toward the origin by a factor of B.  The previous x values become Bx.

 

Step 2:  After accomplishing Step 1, h represents a translation to the right by h units.  You can think of moving the function to the right by h units.

 

Step 3:  After accomplishing Steps 1 and 2, A represents compressing the function along the y axis toward the origin by a factor of A.  The previous y values become Ay.

 

Step 4:  After accomplishing Steps 1, 2, and 3, k represents a translation upward by k units.  You can think of moving the function upward by k units.

 

Part (a)

Since B=2, there is a compression by a factor of 2 along the x axis toward the origin.

Since h=0, there is no translation along the x axis.

Since A=1, there is no compression along the y axis.

Since k=-1, the function is translated downward by 1 unit.

 

Since cos(x) has a period of 2π, so does sec(x).  A typical cycle is from x=0 to x=2π

 

For sec(2x), a typical cycle is from 2x=0 to 2x=2π, i.e., from x=0 to x=π

 

Therefore, the period is π.

 

The domain of sec(x) is all reals except for x=(π/2)±mπ

 

The domain of sec(2x) is all reals except for 2x=(π/2)±mπ, i.e., x=(π/4)±(m/2)π

 

The range of sec(2x) is the same as for sec(x),  (-∞,-1]U[1,∞).

 

Check my work for errors.