GRAPHS OF THE TANGENT AND COTANGENT FUNCTIONS

We are interested in the graph of y = f(x) = tan x Start with a "t" chart and let's choose values from our unit circle and find the tangent values. Tangent has a period of  so it will repeat every . y would mean there is a vertical asymptote here x y = tan x x

y = tan x Let's choose more values. Since we went from we have one complete period x y = tan x y x would mean there is a vertical asymptote here

Let's see what the graph would look like for y = tan x for 3 complete periods. The vertical lines are not part of the graph but are the asymptotes. If you use your graphing calculator it will probably put those in as well as showing the graph.

Transformations apply as usual. Let’s try one. up 2 reflect over x-axis right /4

Since the period of tangent is , the period of tan Bx is: The period would be /2 y = tan x y = tan 2x

What about the graph of y = f(x) = cot x? This would be the reciprocal of tangent so let's take our tangent values and "flip" them over. y x tan x y = cot x x

y = cot x Let's choose more values. x tan x y = cot x y x We need to see more than one period to get a good picture of this.

Let's look at the tangent graph again to compare these. y = cot x y = tan x Again the vertical lines are not part of the graph but are the asymptotes. Notice vertical asymptotes of one are zeros of the other.

This can be expressed algebraically in the form: So from a basic cotangent graph we have A = -1 which will cause the graph to be reflected about the x axis and we have B = /2 which will change the period of the graph from  to  divided by /2 which is 2.

reflects about x-axis changes the period to 2 Since cotangent is cosine over sine that there are asymptotes at values where and there are x intercepts at values where Now let’s think of the shape of the cotangent graph without the reflection and draw that in. And finally, apply the reflection about the x-axis for the final graph.