= 16π . Because the amplitude is 4, the 2 graph of = −4sin ⎛ ⎞ −4 1 will lie between y x ⎜ 8 ⎟ amplitude is π − 1 = 1 and the period is 2π = . ⎝ ⎠ 2 2 2 and 4 on the Because the period is 16π , y-axis. Because the amplitude is 1 , the graph of one cycle will begin at = 0 and end at x 2 [0,16π ] = 16π . We divide the interval into x (2 ) 1 cos will lie between − y x and 1 on 1 = four subintervals, each of length 16π 4π by 2 2 2 4 0, 4π , 8π , 12π , and 16π π the Because the period is , one cycle y-axis. These values of determine the of π x x-coordinates = = will begin at 0 and end at . We divide x x the five key points on the graph. To obtain the [0, ] y- π the interval into four subintervals, each of coordinates of the five key points for π length by finding the following values: = −4sin ⎛ ⎞ 1 , we multiply the y x y- 4 coordinates finding the following values: π π π 0, , , 3π , and ⎝ ⎠ 8 4 2 4 = = −4 of the five key points for sin by . y x A These values of determine the of x x-coordinates the five key points on the graph. To obtain the of the five key points for The five key points are (0, (4π (8π (12π (16π −4) 0) , , , , 0) , , 4) , , 0) We plot these five points and fill in the graph of the curve. We then extend the graph in either
= 16π . Because the amplitude is 4, the 2 graph of y = −4sin ⎛ 1 x ⎞ will lie between −4 ⎜ 8 ⎟ amplitude is − 1 = 1 and the period is 2π = π . ⎝ ⎠ 2 2 2 Because the amplitude is 1 , the graph of 2 and 4 on the y-axis. Because the period is 16π , one cycle will begin at x = 0 and end at x = 16π . We divide the interval [0,16π ] into y 1 cos (2 x ) will lie between − 1 and 1 on four subintervals, each of length 16π = 4π by 2 2 2 4 the y-axis. Because the period is π , one cycle will begin at x = 0 and end at x = π . We divide the interval [0, π ] into four subintervals, each of length π by finding the following values: 4 finding the following values: 0, 4π , 8π , 12π , and 16π These values of x determine the x-coordinates of the five key points on the graph. To obtain the y- coordinates of the five key points for y = −4sin ⎛ 1 x ⎞ , we multiply the 0, π , π , 3π , and π ⎝ 8 ⎠ 4 2 4 of the five key points for y = sin x by A = −4 . These values of x determine the x-coordinates of the five key points on the graph. To obtain the y- coordinates of the five key points for The five key points are (0, 0) , (4π , −4) , (8π , 0) , (12π , 4) , (16π , 0) We plot these five points and fill in the graph of the curve. We then extend the graph in either