Warm-up: HW: Graph Tangent and Cotangent Functions Complete the table from θ = −/2 to θ = /2 (one  interval), with the given 5 input angle values. x sinx cosx y = tanx −/2 −/4 /4 /2 −1 und −1 1 1 1 und HW: Graph Tangent and Cotangent Functions

Objective: Graph tangent and cotangent functions.

Tangent Function Recall that when cos θ = 0, tan θ is undefined. This occurs at  intervals, offset by /2: { … − /2, /2, 3/2, 5/2, … } From the warm-up we can create a t-table for y = tanx x y = tanx −/2 und −/4 −1 /4 1 /2 x sinx cosx y = tanx −/2 −/4 /4 /2 −1 und −1 1 1 1 und

We are interested in the graph of y = tan x vertical asymptote x y = tan x x

Graph of the Tangent Function y x Properties of y = tan x 1. Domain : all real x 2. Range: (–, +) 3. Period: /ω 4. Vertical asymptotes: period:

Transformations apply as usual. Let’s try one. up 2 reflect about x-axis right /4

Finding Domain and Range: Domain for y = tanx: all real x For 2. Range: (–, +)

Finding the Period: Since the period of tangent is , the period of tan x is: The period would be /2 y = tan x y = tan 2x y x y x period: period:

Graphing Variations of y = tan x ¼ and ¾ distance from asymptote to asymptote Properties of y = tan x 1. Domain : all real x 2. Range: (–, +) 3. Period: /ω 4. Vertical asymptotes:

Example: Graphing a Tangent Function ¼ and ¾ distance from asymptote to asymptote Graph y = 3 tan 2x Step 1: Find two asymptotes. An interval containing one period is or /ω = /2

x = 0 is midway between and The graph passes through (0, 0). Graph y = 3 tan 2x Step 2: Identify an x-intercept, midway between the consecutive asymptotes. x = 0 is midway between and The graph passes through (0, 0). ¼ and ¾ distance from asymptote to asymptote

The graph passes through and ¼ and ¾ distance from asymptote to asymptote Graph y = 3 tan 2x Step 3: Find points on the graph 1/4 and 3/4 of the way between the consecutive asymptotes. These points have y-coordinates of –A and A. The graph passes through and

y = 3 tan 2x Finding Domain and Range: Properties of y = tan x 1. Domain : all real x 2. Range: (–, +) 3. Period: /ω 4. Vertical asymptotes: Properties for y = 3tan2x Range: (–, +) Vertical asymptotes: Domain:

Cotangent Function Recall that . when sin θ = 0, cot θ is undefined. This occurs at  intervals, starting at 0: { … −, 0, , 2, … } Let’s create a t-table from θ = 0 to θ =  (one  interval), with 5 input angle values. θ cot θ Und /4 1 /2 3/4 −1  θ sin θ cos θ cot θ /4 /2 3/4  1 Und 1 1 −1 –1 Und

Graph of Cotangent Function: Periodic Vertical asymptotes where sin θ = 0 cot θ θ cot θ und /4 1 /2 3/4 −1  −3/2 -  − /2 /2  3/2

Graph of the Cotangent Function To graph y = cot x, use the identity . At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y x Properties of y = cot x vertical asymptotes 1. Domain : all real x 2. Range: (–, +) 3. Period: /ω 4. Vertical asymptotes:

Graphing Variations of y = cot x Properties of y = cot x 1. Domain : all real x ¼ and ¾ distance from asymptote to asymptote 2. Range: (–, +) 3. Period: /ω 4. Vertical asymptotes:

Example: Graphing a Cotangent Function   y x  

x =  is midway between x = 0 and x = 2. Graph Step 2 Identify an x-intercept midway between the consecutive asymptotes. x =  is midway between x = 0 and x = 2. The graph passes through (, 0). y x

The graph passes through and Step 3 Find points on the graph 1/4 and 3/4 of the way between consecutive asymptotes. These points have y-coordinates of A and –A. The graph passes through and y x

Finding Domain and Range: Properties for Properties of y = cot x Domain: 1. Domain : all real x Range: (–, +) 2. Range: (–, +) y x 3. Period: /ω 4. Vertical asymptotes: Vertical asymptotes:

¼ and ¾ distance from asymptote to asymptote  

Sneedlegrit:   HW: Graph Tangent and Cotangent Functions