will begin at = 0 and end at = 8π . We x x [0, ] divide the interval 8π into four (2 ) 1 cos , we multiply the y x y-coordinates subintervals, 2 each of length 8π = of the five key points for = cos by y x by finding the following 2π 4 values: = − 1 .The five key points are A 2 0, 2π , 4π , 6π , and 8π ,0 π ⎛ , 1 ⎞ , ⎛ 3π ⎛ − ⎞ ⎛ π 1 , ,0 , π ⎛ − , 1 ⎞ ⎞ These values of determine the of x x-coordinates ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0, 2 4 2 2 4 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ the five key points on the graph. To obtain the y- of the five key points for We plot these five points and fill in the graph of = ⎛ ⎞ 2 cos 1 , we multiply the y x y-coordinates the curve. We then extend the graph in either direction to obtain the graph shown below. ⎜ ⎟ 4 ⎝ ⎠ of the five key points for = cos by y x = 2 .The five key points are A (0, (2π (4π (6π (8π 2) , , 0) , , −2) , , 0) , , 2) We plot these five points and fill in the graph of the curve. We then extend the graph in either direction to obtain the graph shown below. y ( 2 , 0) (8 , 2) 2 (0, 2) , 0) ( 8 , 2) x (2 , 0) 8 4 (ω ) 54. = sin , = −4sin ⎛ 1 ⎞ y A x y x 4 8 to ⎜ 8 ⎟ we find = −4 ω = and 1 . Therefore, A 2 ⎝ ( 4 , 2) (4 , 2) the 8 ⎠ (2x ) 1 cos y Comparing amplitude is −4 = 4 and the period is 53. to (ω = cos 2π y A x ) , and the ω =− A = 2 we find . 1/ 8 1 2
will begin at x = 0 and end at x = 8π . We divide the interval [0, 8π ] into four subintervals, y 1 cos (2 x ) , we multiply the y-coordinates 2 each of length 8π = 2π by finding the following of the five key points for y = cos x by 4 values: A = − 1 .The five key points are 2 0, 2π , 4π , 6π , and 8π ⎛ − 1 ⎞ , ⎛ π ,0 , ⎛ π , 1 ⎞ , ⎛ 3π ⎞ ,0 ⎛ π , − 1 ⎞ These values of x determine the x-coordinates of ⎜ 0, 2 ⎟ ⎜ 4 ⎟ ⎜ 2 2 ⎟ ⎜ 4 ⎟ ⎜ 2 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ the five key points on the graph. To obtain the y- of the five key points for y = 2 cos ⎛ 1 x ⎞ , we multiply the y-coordinates We plot these five points and fill in the graph of the curve. We then extend the graph in either direction to obtain the graph shown below. ⎜ 4 ⎟ ⎝ ⎠ of the five key points for y = cos x by A = 2 .The five key points are (0, 2) , (2π , 0) , (4π , −2) , (6π , 0) , (8π , 2) We plot these five points and fill in the graph of the curve. We then extend the graph in either direction to obtain the graph shown below. y ( 2 , 0) 2 (0, 2) (8 , 2) ( 8 , 2) (2 , 0) , 0) x 8 4 4 8 54. y = −4sin ⎛ 1 x ⎞ to y = A sin (ω x ) , ⎜ 8 ⎟ 2 ( 4 , 2) (4 , 2) we find ⎝ ⎠ A = −4 and ω = 1 . Therefore, the 8 53. Comparing y 1 cos (2x ) to y = A cos (ω x ) , amplitude is 2π −4 = 4 and the period is we find 1 A =− 2 and ω = 2 . the 1/ 8