Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
November 11, 2014 Section 4.6 Graphs of Other Trigonometric Functions Objective: Know how to sketch the graphs of trigonometric functions (by hand!). . all.. c e R sin (-x) = cos (-x) = even or odd odd sec (-x) = even csc (-x) = odd even tan (-x) = odd cot (-x) = odd Let's take a closer look at tangent. Complete the table. x -π/2 -1.57 -1.5 -π/4 0 π/4 1.5 1.57 π/2 tan x Asymptotes sin x tan x = cos x Where does cos x = 0? Answers:-π/2, π/2, 3π/2,...so vertical asymptotes: x= y = tan x Period: π Domain: R, x = π/2 +nπ Range: (-∞, ∞) Vert. asymptote: x = π/2 +nπ Symmetry: origin (odd) *Tip: Use "dot mode" for the TI-83 if you do not want asymptotes displayed. *Note: the key points are the asymptotes at the ends of the period and the intercept in the middle. November 11, 2014 Try these: a) Graph y = 2 tan 2x Graph 2π 3π 2 π π 2 π 2 π 3π 2 2π 2π 3π 2 π π 2 π 2 π 3π 2 2π b) Graph y = -tan ( x ) 2 Graph Now, let's take a closer look at cotangent. cot x = cos x Where does sin x = 0? sin x y = cot x Answers:0, π, 2π,...so vertical asympt Period: π Domain: R, x = nπ Range: (-∞, ∞) Vert. asymptote: x = nπ Symmetry: origin (odd) November 11, 2014 Ex: Graph y = -2 cot 2x Graph 2π General equations: y = d + a tan (bx-c) 3π 2 π π 2 π 2 π 3π 2 2π y = d + a cot (bx-c) Amplitude ( a ) - not defined If a < 0, then the graph is a reflection in the x-axis. Vertical shift (d) Phase shift (c/b) - horizontal shift (according to period) Period: π/b - distance between asymptotes *Should sketch 2 additional cycles to left and right. Asymptotes: Solve bx - c = -π/2 bx - c = π/2 Solve bx - c = 0 bx - c = π x-intercepts: Solve sin(bx-c) = 0 Solve cos(bx-c) = 0 or find midpoint between asymptotes. November 11, 2014 Graphs of Reciprocal Functions 1 y = csc x = sin x Period: 2π Domain: R, x = nπ Range: (-∞, 1] ∪[1, ∞) Vert. asymptote: x = nπ Symmetry: origin (odd) *"hills" and "valleys" 1 y = sec x = cos x Period: 2π Domain: R, x = π/2 +nπ Range: (-∞, -1] ∪ [1, ∞) Vert. asymptote: x = π/2 +nπ Symmetry: y-axis (even) To graph sec or csc functions... Make a sketch of its reciprocal fct. x-int. = asymptotes of sec x or csc x max = rel. min and min. = rel. max. November 11, 2014 Try these: a) Graph y = 2 csc ( x ) 2 Graph 4π 3π 2π π π 2π 3π 4π 2π 3π 2 π π 2 π 2 π 3π 2 2π b) Graph y = sec (x + π) Graph Damped Trig. Graphs Graph the following on your calculator: a) f(x) = x sin x b) f(x) = (x2+1)-1 sin x Graph a) damping factor Graph b) To graph by hand... 1. Sketch y = ±(damping factor). They become the guides. 2. Sketch the graph of the trig. function "between" these guides. 3. The graph touches the guides at points that would have been rel. max. and min-values of the trig. fct. The x-intercepts stay the same.