Graphing Trigonometric Functions

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.

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

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

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

Graph y = 2 csc x What changes?

Graph y = csc 2x What is period? __________

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

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

Graph y = 2 sec x – 2

Graph y = - sec (x/2)

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”

Graph y = csc (2x – π/4)

Graph y = sec (4x – π/2)