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4-5 Graphing Other Trigonometric Functions Locate the vertical asymptotes, and sketch the graph of each function. 2. SOLUTION: is the graph of y = tan x shifted The graph of y = a tan (bx + c), so a = 1, b = 1, and c = units to the left. The period is or . . Use the tangent asymptote equations to find the location of the asymptotes. Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int y = tan x for one period on . y = tan (x + 4) (0, 0) Intermediate Point Vertical Asymptote (0, 1) Sketch the curve through the indicated key points for the function. Then repeat the pattern. eSolutions Manual - Powered by Cognero Page 1 4-5 Graphing Other Trigonometric Functions 4. y = –3 tan SOLUTION: The graph of y = –3 tan axis. The period is is the graph of y = tan x expanded vertically, expanded horizontally, and reflected in the x- or 3 . y = a tan (bx + c), so a = –3, b = , and c = 0. Use the tangent asymptote equations to find the location of two consecutive vertical asymptotes. Create a table listing the coordinates of key points for y = tan 2x for one period on Function . y = tan x Vertical Asymptote Intermediate Point eSolutions Manual - Powered by Cognero Page 2 4-5 Graphing Other Trigonometric Functions x-int (0, 0) (0, 0) Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 6. y = –tan 3x SOLUTION: The graph of y = – tan 3x is the graph of y = tan x compressed horizontally and reflected in the x-axis. The period is or . y = a tan (bx + c), so a = –1, b = 3, and c = 0. Use the tangent asymptote equations to find the location of two consecutive vertical asymptotes. Create a table listing the coordinates of key points for y = tan 2x for one period on Function y = tan x eSolutions Manual - Powered by Cognero . y = 2 tan x Page 3 4-5 Graphing Other Trigonometric Functions Vertical Asymptote Intermediate Point x-int (0, 0) (0, 0) Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 8. SOLUTION: The graph of is the graph of y = cot x expanded horizontally. The period is or 2 . y = a cot (bx + c), so a = 1, b = , and c = 0. Use the tangent asymptote equations to find the location of the asymptotes. Find the location of two consecutive vertical asymptotes. Create a table listing the coordinates of key points for eSolutions Manual - Powered by Cognero for one period on . Page 4 4-5 Graphing Other Trigonometric Functions Function y = cot x Vertical Asymptote Intermediate Point x-int x=0 x=0 Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 10. SOLUTION: is the graph of y = csc x compressed horizontally and shifted units to the left. The The graph of period is or . y = a csc (bx + c), so a = 1, b = 4, and c = asymptotes. eSolutions Manual - Powered by Cognero . Use the asymptote equations to find the location of two vertical Page 5 4-5 Graphing Other Trigonometric Functions Create a table listing the coordinates of key points for Function for one period on . y = csc x Vertical Asymptote Intermediate Point x-int Intermediate Point Vertical Asymptote x = –π x= x=0 x= x=π x= Sketch the curve through the indicated key points for the function. Then repeat the pattern. 12. y = –2 csc 3x SOLUTION: is the graph of y = csc x expanded vertically, compressed horizontally, and reflected in the The graph of x-axis. The period is or . y = a csc (bx + c), so a = –2, b = 3, and c = 0. Use the asymptote equations to find the location of two vertical asymptotes. eSolutions Manual - Powered by Cognero Page 6 4-5 Graphing Other Trigonometric Functions Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int y = csc x y = –2 csc 3x x=0 x=0 for one period on . Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 14. SOLUTION: The graph of eSolutions Manual - Powered by Cognero is the graph of y = sec x expanded horizontally and shifted π units to the left. The Page 7 4-5 Graphing Other Trigonometric Functions period is or 10 . y = a sec (bx + c), so a = 1, b = , and c = . Use the asymptote equations to find the location of two vertical asymptotes. Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int for one period on . y = sec x (0, 1) Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. eSolutions Manual - Powered by Cognero Page 8 4-5 Graphing Other Trigonometric Functions 16. y = –sec SOLUTION: The graph of is the graph of y = sec x expanded horizontally and reflected in the x-axis. The period is or 16 . y = a sec (bx + c), so a = –1, b = , and c = 0. Use the asymptote equations to find the location of two vertical asymptotes. Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int for one period on [−4 , 12 ]. y = sec x (0, 1) eSolutions Manual - Powered by Cognero (0, –1) Page 9 4-5 Graphing Other Trigonometric Functions Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. Locate the vertical asymptotes, and sketch the graph of each function. 29. y = sec x + 3 SOLUTION: The graph of y = sec x + 3 is the graph of y = sec x shifted 3 units up. The period is or 2 . y = a sec (bx + c), so a = 1, b = 1, and c = 3. Use the asymptote equations to find the location of two vertical asymptotes. Create a table listing the coordinates of key points for y = sec x + 3 for one period on Function y = sec x eSolutions Manual - Powered by Cognero . y = sec x +3 Page 10 4-5 Graphing Other Trigonometric Functions Vertical Asymptote Intermediate Point x-int (0, 1) (0, 4) Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 30. SOLUTION: The graph of is the graph of y = sec x shifted to the right and 4 units up. The period is or 2 . y = a sec (bx + c), so a = 1, b = 1, and c = . Use the asymptote equations to find the location of two vertical asymptotes. eSolutions Manual - Powered by Cognero Page 11 4-5 Graphing Other Trigonometric Functions Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int for one period on . y = sec x x =0 (0, 1) Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 31. y = csc −2 SOLUTION: The graph of is the graph of y = csc x expanded horizontally and shifted 2 units down. The period is or 6π. y = a tan (bx + c), so a = 1, b = , and c = 2. Use the asymptote equations to find the location of two vertical asymptotes. eSolutions Manual - Powered by Cognero Page 12 4-5 Graphing Other Trigonometric Functions Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int for one period on . y = csc x x=0 x=0 Intermediate Point Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern. 32. SOLUTION: The graph of is the graph of y = csc x compressed horizontally, shifted shifted 3 units up. The period is . y = a csc (bx + c), so a = 1, b = 3, and c = asymptotes. eSolutions Manual - Powered by Cognero units to the left, and . Use the asymptote equations to find the location of two vertical Page 13 4-5 Graphing Other Trigonometric Functions Create a table listing the coordinates of key points for Function for one period on . y = csc x Vertical Asympt Interm Point x-int x=0 Interm Point Vertical Asympt Sketch the curve through the indicated key points for the function. Then repeat the pattern. 33. y = cot (2x + π) − 3 SOLUTION: The graph of eSolutions Manual - Powered by Cognero is the graph of y = cot x compressed horizontally, shifted units to the left, and Page 14 4-5 Graphing Other Trigonometric Functions shifted 3 units down. The period is . y = a cot (bx + c), so a = 1, b = 2, and c = . Use the asymptote equations to find the location of two consecutive vertical asymptotes. Create a table listing the coordinates of key points for Function Vertical Asymptote Intermediate Point x-int for one period on . y = cot x x=0 Intermediate Point Vertical Asymptote x=0 Sketch the curve through the indicated key points for the function. Then repeat the pattern. 34. eSolutions Manual - Powered by Cognero Page 15 4-5 Graphing Other Trigonometric Functions SOLUTION: is the graph of y = cot x expanded horizontally, shifted π units to the left, and shifted The graph of 1 unit down. The period is or 2π. y = a cot (bx + c), so a = 1, b = , and c = . Use the asymptote equations to find the location of two consecutive vertical asymptotes. Create a table listing the coordinates of key points for Function Vertical Asympt Interm Point x-int for one period on . y = cot x x=0 Interm Point Vertical Asympt Sketch the curve through the indicated key points for the function. Then repeat the pattern. eSolutions Manual - Powered by Cognero Page 16 4-5 Graphing Other Trigonometric Functions eSolutions Manual - Powered by Cognero Page 17