How to find the reciprocal of a graph, such as y=cos(x)?

To find the reciprocal of any graph, first consider the key points, in this case: (0,1), (90,0), (180,-1), (270,0) and (360,1). Since we are finding the reciprocal in the y-axis, we can find the reciprocal of the key points as values on the x-axis are unchanged and the reciprocal of 1 is 1, the reciprocal of -1 is -1 and the reciprocal of 0 is invalid so there is an asymptote. Using this information the key points on the reciprocal graph, y=sec(x), can be plotted along with its asymptotes. Once these have been done, the majority of the curve can be sketched, but it is important to check the range of the graph. The range of the graph of y=cos(x) is -1 ≤ x ≤ 1, applying our knowledge of reciprocals means that the range of the new graph must be x ≤ -1 and 1 ≤ x. This means that the curve will never touch the line y=0 and that it will only touch and never cross the lines y=1 and y=-1. Since y=cos(x) is an even function, y=sec(x) is also an even function so can be reflected in the y-axis to find the curve in the 2^nd and 3^rd quadrants.