# Aim: How do we sketch y = tan x Aim: What are the graphs of tangent function and reciprocal functions? Do Now: If y = tan x, fill in the table and then sketch the graph. x 0 /6 y 0 0.58 /4 /3 4/9 17/36 /2 1 1.73 5.67 u.d. HW: Handout

11.43 M at h Composer 1. 1. 5 http: / / www. mathcomposer. com 5 4 y y = tan x 3 2 1 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 -1 -2 -3 -4 -5 x 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 The sine and cosine curves are continuous while the graph of y = tan x is not or discontinuous at 3

etc and the graph repeats every x ,x ,x 2 2 2 , Summary of the properties of f(x) = tan x Domain : {x x real numbers, and x ( 2 k 1) , k is integer} Range: { y real numbers} Period: 2 within which lies 1 complete curve Range: there is no range existed. Vertical asymptotes: at odd multiples of /2 M at h Composer 1. 1. 5 ht t p: // www. mat hcomposer. com 5

y y = 2 tan x 4 3 2 1 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 -1 Period is x 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 -2 -3 -4 -5 M at h Composer 1. 1. 5 ht tp: / / www. mat hcomposer . com 5 y 4

y = 2 tan 2x 3 2 1 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 -1 -2 -3 -4 -5 x 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Period is /2 M ath Composer 1. 1. 5 htt p: / / www. mat hcomposer. com 5 y y = tan 0.5x 4 3

2 1 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 -1 x 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Period is 2 -2 -3 -4 -5 M at h Composer 1. 1. 5 ht tp: / / www. mat hcomposer . com 5 y 4 y = tan 0.25x 3 2 1

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 -1 -2 -3 -4 -5 x 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Period is 4 The general formula to find the period of tangent function is B a) Graph: y = 2sin x and y = tan x on the same axes, over x b) How many values are the solutions of tan x = 2sin x? M ath Composer 1. 1. 5 htt p: / / www. mathcomposer. com 5 y = tan x

y 4 3 2 1 . -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 -1 -2 -3 -4 -5 5 points y = 2sin x x 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 M at h Composer 1. 1. 5 ht t p: // www. mat hcomposer. com 5 y

y = cot x 4 3 2 1 -2.0 -1.5 -1.0 -0.5 x 0.5 1.0 1.5 2.0 -1 -2 -3 -4 -5 For integral values of n, the vertical lines on the graph at x = n are asymptotes Mat h Composer 1. 1. 5 ht t p: // www. mat hcomposer. com 5

y 4 3 2 1 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 y = sec x x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -1 -2 -3 -4 -5 For integral values of n, every /2 + n , sec x is undefined. The vertical lines on the graph at x = /2 + n are asymptotes. M at h Composer 1. 1. 5 ht tp: // www. mat hcom poser. com 5

y 4 3 2 1 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 y = csc x x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -1 -2 -3 -4 -5 For integral values of n, every n , csc x is undefined The vertical lines on the graph at x = n are asymptotes. What are the maximum and minimum values of the function a) y = 3 sin 2x ? b) y = 3 cos 2x ? c) y = 3 tan 2x ? a) Max. = 3, Min. = -3

b) Max. = 3, Min. = -3 c) Max. is infinity, Min. is negative infinity If f(x) = 2tan 2x, what is the value of 1. f ( ) ? 4 2. f ( ) ? 3 )? 3. f ( 2