1. Graphing Trigonometric Functions

2. Objective (Begin With The End in Mind)
I can determine the amplitude, period, phase shift, and vertical shift of a cosecant and secant trigonometric function from the equation of the function and from the graph of the function.
I can describe the effect of changing A, B, C, or D in the standard form of a cosecant and secant trigonometric equation
I can state the domain and the range of a cosecant and secant function written in standard form
I can sketch the graph of a cosecant and secant function written in standard form by using transformations for at least a two period interval.

3. Graphing Trigonometric Functions
Standard form: y = A tan (Bx - C) + D or y = A cot (Bx - C) + D
Amplitude = “none” (but it does change “slope” of curve (check when “tan x = 1”)
Period = π / |B|
Asymptotes: (tan x): x=(C ± (π/2)) / B
Asymptotes (cot x): x= C / B , (C + π) / B
Vertical shift = D
Domain (tan x): -∞ ≤ x ≤ +∞ , x ≠ n(π/2) , n odd integer
Domain (cot x): -∞ ≤ x ≤ +∞ , x ≠ n(π)
Range: -∞ ≤ y ≤ +∞
Three key points: ( , ) , ( , ) , ( , )
Drawing the curve:
(1) Determine D, the vertical phase shift. This will be the “new” horizontal axis at y = D.
(2) Determine and draw the asymptotes (this identifies period and phase shift).
(3) Divide the period into four equivalent parts - label each interval.
(4) Evaluate the function for each of the three x values resulting from Step 3.
(5) Plot those points found in Step 4 and join them with a curve.
(6) Draw for two cycles

4. Graphing y = csc x y = sin x x = 0, csc (0) = _____ ( , )

5. Graphing y = csc x
What is period? __________ Where are asymptotes? __________

6. Graph y = 2 csc x
What changes?

7. Graph y = csc 2x
What is period? __________

8. Graphing y = sec x y = cos x x = 0, sec (0) = _____ ( , )

9. Graphing y = sec x
What is period? __________ Where are asymptotes? __________

10. Graph y = 2 sec x – 2

11. Graph y = - sec (x/2)

12. Graphing Trigonometric Functions
Standard form: y = A csc (Bx - C) + D or y = A sec (Bx - C) + D
Amplitude = “none” (but it does change “slope” of curves)
Period = 2π / |B|
Asymptotes: (csc x): x= C / B , (C + π) / B
Asymptotes (sec x): x=(C ± (π/2)) / B
Vertical shift = D
Domain (csc x): -∞ ≤ x ≤ +∞ , x ≠ n(π)
Domain (sec x): -∞ ≤ x ≤ +∞ , x ≠ n(π/2) , n odd integer
Range: -∞ ≤ y ≤ -1 and 1 ≤ y ≤ ∞
Six key points: “positive” ( , ) , ( , ) , ( , ) and “negative” ( , ) , ( , ) , ( , )
Drawing the curve:
(1) Determine D, the vertical phase shift. This will be the “new” horizontal axis at y = D.
(2) Determine and draw the asymptotes (this identifies period and phase shift).
(3) Divide period into 8 equivalent parts - label each interval.
(4) Evaluate the function for each of the six x values resulting from Step 3.
(5) Plot those points found in Step 4 and join them with curves.
(6) Draw for two cycles - two “positive” and two “negative”

13. Graph y = csc (2x – π/4)

14. Graph y = sec (4x – π/2)