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Unit 5 – Graphs of the other Trigonometric Functions Tangent and Cotangent Warm Up 1-9 on the trig graphs review page done MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. In this section, you will learn to: • Sketch the graphs of tangent and cotangent functions (Thursday/Friday) • Sketch the graphs of secant and cosecant functions (Monday/Tuesday) MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the tangent curve: •The graph of y tan x : y 5 4 3 2 1 x -360 -270 -180 -90 90 -1 -2 -3 -4 -5 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the tangent curve: • The graph of y tan x : Period : Domain : all real numbers except x 2 n Range : , Asymptotes : x 2 n MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the cotangent curve: •The graph of y cot x : y 5 4 3 2 1 x -360 -270 -180 -90 90 -1 -2 -3 -4 -5 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the cotangent curve: • The graph of y cot x : Period : Domain : all real numbers except x n Range : , Asymptotes : x n MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Graphical effects of constants a, b, c and d in y a tan bx c d and y a cot bx c d functions : Question: Is there an amplitude for a tangent or a cotangent function? Why or why not? No amplitude, since the two curves extend infinitely in both directions. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Period of Tangent and Cotangent Functions: The period of tangent and cotangent functions y a tan bx c d and y a cot bx c d is b . THIS IS DIFFERENT 1 a) Period of y 2 tan x 1: 4 2 b 1 2 2 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Asymptotes of the tangent graph function: Where do you think you need to set the left and right endpoints for a tangent graph below? MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Asymptotes of the tangent graph function: Asymptotes for a tangent function can be found by solving for the two equations: bx c 2 and bx c 2 Find the asymptotes of y 4 tan 3x 1: 4 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Find the asymptotes of y 4 tan 3 x 1: 4 bx c 3x 4 3x x 2 2 and bx c and 3x 4 2 4 3 and 3x 4 12 and x 4 2 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Find the asymptotes of y 4 tan 3 x 1: 4 Solve for the period and verify that the distance between the asymptotes is equal to the period: 3 4 Distance 4 12 12 12 12 3 Period b 3 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Asymptotes of the cotangent graph function: Where do you think you need to set the left and right endpoints for a cotangent graph below? MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Asymptotes of the cotangent graph function: Asymptotes for a cotangent function can be found by solving for the two equations: bx c 0 and bx c Find the asymptotes of y 4cot 3x 1: 4 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Find the asymptotes of y 4cot 3x 1: 4 bx c 0 and bx c 3x 4 3x x 4 12 0 and 3x 4 5 and 3x 4 and 5 x 12 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Find the asymptotes of y 4cot 3x 1: 4 Solve for the period and verify that the distance between the asymptotes is equal to the period: 5 4 Distance 12 12 12 3 Period b 3 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Reflection: • If a is positive, then there is no reflection about the x-axis. • If a is negative, then there is a reflection about the x-axis. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Reflection : y a tan bx c d y tan x y tan x y y 5 5 4 4 3 3 2 2 1 1 x -90 -45 45 90 x -90 -45 45 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Reflection : y a tan bx c d y tan x and y tan x y 5 4 3 2 1 x -90 -45 45 -1 -2 -3 -4 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Reflection : y a cot bx c d y cot x y cot x y y 5 5 4 4 3 3 2 2 1 x 45 -1 -2 90 135 1 x 180 45 -1 -2 -3 -3 -4 -4 -5 -5 90 135 180 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Reflection : y a cot bx c d y cot x and y cot x y 5 4 3 2 1 x 45 -1 -2 -3 -4 -5 90 135 180 Effects of a on the tangent and cotangent graphs: 1 y tan x 2 y 2 tan x y y 5 5 4 4 3 3 2 2 1 1 x x -90 -45 45 90 -90 -45 45 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Effects of a on the tangent and cotangent graphs: a) If a > 1, then the graph rises faster. b) If 0< a < 1, then the graph rises slower. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Effects of c on the tangent and cotangent graphs: y tan x 4 y tan x 4 y y 5 5 4 4 3 3 2 2 1 1 x x -90 -45 45 -1 -2 -3 -4 -5 90 135 -135 -90 -45 45 -1 -2 -3 -4 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Horizontal Translation or Phase Shift: The constant c determines the phase shift of the graph. Phase Shift = - c/b (or –c/b) a) If c is positive, then the shift is toward the left. b) If c is negative, then the shift is toward the right. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Effects of d on the tangent and cotangent graphs: y tan x 2 y tan x 2 y y 5 5 4 4 3 3 2 2 1 1 x x -90 -45 45 90 -90 -45 45 -1 -1 -2 -2 -3 -3 -4 -5 -4 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Vertical Translation: The constant d determines the vertical translation of the graph. a) If d is positive, then the vertical shift is upward. b) If d is negative, then the vertical shift is downward. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph. EXAMPLE : y 2cot 2 x 3 4 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. EXAMPLE : y 2cot 2 x 3 4 a) Amplitude: b) Period: none b 2 c) Horizontal Translation: d) Vertical Shift: e) Reflection: to the right 8 3 units downward about the x-axis MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Problem : y 2cot 2 x 3 4 e) Endpoints: 0 and 2 x 4 4 5 2x 2x 4 4 5 x x 8 8 5 Verify distance with the period: 8 8 2 2x MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Graph of y 2cot 2 x 3 4 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the secant curve: • The graph of y sec x : y 5 4 3 2 1 x 45 -1 -2 -3 -4 -5 90 135 180 225 270 315 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the secant curve: The graph of y sec x : Period : 2 Domain : all real numbers except x 2 n Range : , 1 and 1, Asymptotes : x 2 n MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the cosecant curve: • The graph of y csc x : y 5 4 3 2 1 x 45 -1 -2 -3 -4 -5 90 135 180 225 270 315 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. The graph of the cosecant curve: The graph of y csc x : Period : 2 Domain : all real numbers except x n Range : , 1 and 1, Asymptotes : x n MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Graphical effects of constants a, b, c and d in y a sec bx c d and y a csc bx c d functions : a) Is there an amplitude for a secant or a cosecant function? Why or why not? MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Graphical effects of constants a, b, c and d in y a sec bx c d and y a csc bx c d functions : b) Period is 2 . b c) The horizontal translation, vertical translation, and reflection all stay the same. d) It is best to sketch the cosecant and secant graph by first graphing the reciprocal functions of sine and cosine. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph. EXAMPLE : y 2csc 2 x 3 4 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. EXAMPLE : y 2csc 2 x 3 4 a) Amplitude: 2 2 b) Period: b 2 c) Horizontal Translation: d) Vertical Shift: e) Reflection: none to the right 8 3 units downward about the x-axis MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Problem : y 2csc 2 x 3 4 e) Endpoints: 2x 0 and 2 x 4 2x 4 x 8 2 4 9 2x 4 9 x 8 9 Verify distance with the period: 8 8 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Graph of y 2csc 2 x 3 4 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a sin bx c d y sin x y sin x y 2 y 2 1 1 x -360 -270 -180 -90 90 180 270 360 x -360 -270 -180 -90 90 -1 -1 -2 -2 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a sin bx c d y sin x and y sin x y 2 1 x -360 -270 -180 -90 90 -1 -2 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a cos bx c d y cos x y cos x y y 2 2 1 1 x -360 -270 -180 -90 90 -1 -2 180 270 360 x -360 -270 -180 -90 90 -1 -2 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a cos bx c d y cos x and y cos x y 2 1 x -360 -270 -180 -90 90 -1 -2 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a tan bx c d y tan x y tan x y y 5 5 4 4 3 3 2 2 1 1 x -90 -45 45 -1 -2 -3 -4 90 x -90 -45 45 -1 -2 -3 -4 -5 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a tan bx c d y tan x and y tan x y 5 4 3 2 1 x -90 -45 45 -1 -2 -3 -4 -5 90 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a cot bx c d y cot x y cot x y y 5 5 4 4 3 3 2 2 1 1 x 90 180 x -180 -90 -1 -1 -2 -2 -3 -3 -4 -5 -4 -5 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a cot bx c d y cot x and y cot x y 5 4 3 2 1 x -180 -90 90 -1 -2 -3 -4 -5 180 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a sec bx c d y sec x y sec x y 5 y 5 4 4 3 3 2 2 1 1 x x 90 180 270 360 -360 -270 -180 -90 -1 -1 -2 -2 -3 -3 -4 -5 -4 -5 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a sec bx c d y sec x and y sec x y 5 4 3 2 1 x -360 -270 -180 -90 90 -1 -2 -3 -4 -5 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a csc bx c d y csc x y csc x y y 5 5 4 4 3 3 2 2 1 1 x 90 180 270 360 x -360 -270 -180 -90 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y-axis Reflection : y a csc bx c d y csc x and y csc x y 5 4 3 2 1 x -360 -270 -180 -90 90 -1 -2 -3 -4 -5 180 270 360 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Reflection: • If a is positive, then there is no reflection about the xaxis. • If a is negative, then there is a reflection about the xaxis. • If b is positive, then there is no reflection about the yaxis. • If b is negative, then there is a reflection about the yaxis. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Example : y 2sin 2 x 1 y 5 4 3 2 1 x -180 -90 -1 -2 -3 -4 -5 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. Example : y 2sin 2 x 1 y 5 4 3 2 1 x -180 -90 -1 -2 -3 -4 -5 MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. y 2sin 2 x 1 y 2sin 2 x 1 y y 5 5 4 4 3 3 2 2 1 1 x -180 -90 x -180 -90 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 HOMEWORK • Graphing Trig Functions – Kuta • Trig graphs review page 1 and 2