2.7 TAN, COT, SEC and CSC Graphs HOMEWORK: 2.7: 21-24, 29-32, 35, 36, 39, 40 2.8: 19, 20, 22-25 y = tan x Key Points: (-π/4, -1), (0,0), (π/4, 1) Key asymptotes: x = π/2 + kπ p. 1 of 19 2.7 TAN, COT, SEC and CSC Graphs y = cot x Key points: (π/4, 1), (π/2, 0), (3π/4, -1) Key Asymptotes: x = 0 + kπ p. 2 of 19 2.7 TAN, COT, SEC and CSC Graphs y = sec x and y = csc x To get these graphs, we will just use the identities: sec x = 1/ cos x and csc x = 1 / sin x y = sec x p. 3 of 19 2.7 TAN, COT, SEC and CSC Graphs y= csc x To graph y Asec(x ) or y Acsc(x ) p. 4 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Graph y = 6 tan(2x – π/4) + 1 and clearly label all important features. State the equations of the asymptotes and the period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 5 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Graph y = 3 cot(5x) and clearly label all important features. State the equations of all asymptotes and the period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 6 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Graph y = 2sec(3x – π/4). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 7 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Graph y = -4csc(2x – π/3). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 8 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Graph y =1 - 3sec(5x + π/2). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 9 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Given y = - csc(x/2 + π/3). State the equations of the asymptotes and period in one cycle. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 10 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Graph y = - 5tan(3x + π/2). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 11 of 19 2.7 TAN, COT, SEC and CSC Graphs Example: Given y = 1-5cot(4x + π/2). State the equations of the asymptotes and period. Period _________ Phase shift ___________ Asymptotes in one cycle ____________________ General equation of asymptotes _________________ Range ____________ p. 12 of 19 2.7 TAN, COT, SEC and CSC Graphs y x y x y x y x The above graphs are of the equation y A tan(x) B or y Acot(x) B and 0 Determine which equation and state if A and B are positive or negative p. 13 of 19 2.7 TAN, COT, SEC and CSC Graphs y x y x y x y x The above graphs are of the equation y A tan(x ) B or y Acot(x ) B and 0 & A>0. Determine which equation and if is 0 + k*period or not 0 + k*period p. 14 of 19 2.7 TAN, COT, SEC and CSC Graphs Strategy for graphing trig functions y Atrig(x ) B Step 1: Identify A, B, & and base graph {Note: Remember minus sign in equation!} Step 2: Find Amp., period and phase shift h Amp A (sine, cosine only) ** period 2 if base sin, cos, sec, or csc. ** period if base tan or cot p.s. h Step 3: Find base points 3 , 1) ; (2 , 0) 2 2 3 Cosine : (0, 1); ( , 0) ; ( , 1) ; ( , 0) ; (2 ,1) 2 2 Secant: Graph y Acos(x ) and draw asymptotes at zeros and Sine: (0, 0); ( ,1) ; ( , 0) ; ( cups toward asymptotes. If B, shift vertically. Cosecant: Graph y Asin(x ) and draw asymptotes at zeros and cups toward asymptotes. If B, shift vertically ,), ( ,1), (0,0) , ( ,1), ( ,) 2 4 4 2 3 ,1), ( ,) Cotangent: (0,) , ( ,1), ( , 0) , ( 4 2 4 Remember, if y value is, then there is an asymptote at x = the x coordinate! Tangent: ( p. 15 of 19 2.7 TAN, COT, SEC and CSC Graphs Step 4: Transform the key points {Watch order!} Horiz stretch: x coord or x 1 Horiz shift: x coord p.s. x Vert stretch: y coord A {NOT just } {NOT amp!} Vert shift: y coord B Step 5: Plot resulting points and/or asymptotes and sketch graph. Step 6: Draw additional cycle if needed. {Add new period to x values or use scale} Step 7: Verify amp, period, phase shift and asymptotes etc. are correct for your graph. Check on your calculator. p. 16 of 19 2.7 TAN, COT, SEC and CSC Graphs Strategy to find asymptotes Method 1: Transform the original asymptotes of the function. For tangent, see where ( ,) and ( ,) went 2 2 For cotangent, see where (0,) and ( ,) went For secant, findthe zeros of y Acos(x ) the zerosof y Asin(x ) For cosecant, find find the primary asymptotes in one cycle and For general asymptotes, add k* new period Method 2: Set x = to old asymp. of base function + k* period. y Asec(x ) B , 3 x 2k x 2k and solve for x. set and 2 2 y Atan(x ) B x k x k and solve for x. set and 2 2 y Acsc(x ) B set x 0 2k and x 2k and solve for x. y Acot(x ) B set x 0 k and x k and solve for x. p. 17 of 19 2.7 TAN, COT, SEC and CSC Graphs True or False. If the answer is false, correct it 2 T F The period of y 3tan(5x ) 1 is 3 5 T F The phase shift of y 3tan(5x 3 T F The amplitude of of y 2sec(5x ) 1 is 3 15 ) 1 is 2 T F Two consecutive asymptotes of y 2 cot(5x 2 and 15 15 T F Two consecutive asymptotes of y 2 cot(5x x 15 and x 2 15 T F Two consecutive asymptotes of y 2 tan(5x x 15 and x 2 15 3 3 3 ) 1 are ) 1 are ) 1 are T F The general equation of the asymp. of y 2sec(5x are x p. 18 of 19 20 2k and x 4 2k 4 ) 1 2.7 TAN, COT, SEC and CSC Graphs T F The general equation of the asymptotes of y 2sec(5x are x 2k 2k and x 20 5 4 5 T F The general equation of the asymptotes of y 2 tan(5x are x 4 2k 3 2k and x 20 5 20 5 4 0, then the graph of y A tan(x ) B T F If 0 & A< y x could look like T F If 0 & A< 0, then the graph of y Acot(x ) B y could look like x 2 T F The period of y 3csc(5x ) 1 is 3 5 p. 19 of 19 ) 1 ) 1