The graph of the function ???? = cos ( 1 2 ???? ) y=cos( 2 1 x) will have the following characteristics: Shape: The graph will be a wave-like curve typical of cosine functions, oscillating between 1 and -1. Period: The period of the function is 4 ???? 4. This means that the pattern of the wave will repeat every 4 ???? 4 units along the x-axis. Key Points: At ???? = 0 x=0, ???? = 1 y=1 (maximum point). At ???? = ???? x=, ???? = 0 y=0 (crosses the x-axis). At ???? = 2 ???? x=2, ???? = 1 y=1 (minimum point). At ???? = 3 ???? x=3, ???? = 0 y=0 (crosses the x-axis again). At ???? = 4 ???? x=4, ???? = 1 y=1 (returns to the maximum point). X-Axis and Y-Axis: The x-axis will be labeled with values such as 0 , ???? , 2 ???? , 3 ???? , 4 ???? 0,,2,3,4, and you can also include negative values if you extend the graph in both directions. The y-axis will range from -1 to 1. Graph Appearance: The graph will start at the maximum point (1), decrease to 0 at ???? = ???? x=, reach the minimum (-1) at ???? = 2 ???? x=2, return to 0 at ???? = 3 ???? x=3, and finally return to the maximum (1) at ???? = 4 ???? x=4. The wave will smoothly transition between these points. Visual Representation If you were to sketch it, it would look something like this: will the wave be going up or down