1. Next  Back Tangent and Cotangent Graphs Reading and Drawing Tangent and Cotangent Graphs Some slides in this presentation contain animation. Slides will be more meaningful if you allow each slide to finish its presentation before moving to the next one.

2. Next  Back This is the graph for y = tan x. This is the graph for y = cot x.

3. Next  Back One definition for tangent is. Notice that the denominator is cos x. This indicates a relationship between a tangent graph and a cosine graph. This is the graph for y = cos x.

4. Next  Back To see how the cosine and tangent graphs are related, look at what happens when the graph for y = tan x is superimposed over y = cos x.

5. Next  Back In the diagram below, y = cos x is drawn in gray while y = tan x is drawn in black. Notice that the tangent graph has VERTICAL asymptotes (indicated by broken lines) everywhere the cosine graph touches the x-axis.

6. Next  Back One definition for cotangent is. Notice that the denominator is sin x. This indicates a relationship between a cotangent graph and a sine graph. This is the graph for y = sin x.

7. Next  Back To see how the sine and cotangent graphs are related, look at what happens when the graph for y = cot x is superimposed over y = sin x.

8. Next  Back In the diagram below, y = sin x is drawn in gray while y = cot x is drawn in black. Notice that the cotangent graph has VERTICAL asymptotes (indicated by broken lines) everywhere the sine graph touches the x-axis.

9. Next  Back y = tan x. y = cot x. For tangent and cotangent graphs, the distance between any two consecutive vertical asymptotes represents one complete period.

10. Next  Back y = tan x. y = cot x. One complete period is highlighted on each of these graphs. For both y = tan x and y = cot x, the period is π. (From the beginning of a cycle to the end of that cycle, the distance along the x-axis is π.)

11. Next  Back For y = tan x, there is no phase shift. The y-intercept is located at the point (0,0). We will call that point, the key point.

12. Next  Back A tangent graph has a phase shift if the key point is shifted to the left or to the right.

13. Next  Back For y = cot x, there is no phase shift. Y = cot x has a vertical asymptote located along the y-axis. We will call that asymptote, the key asymptote.

14. Next  Back A cotangent graph has a phase shift if the key asymptote is shifted to the left or to the right.

15. Next  Back y = a tan b (x - c). For a tangent graph which has no vertical shift, the equation for the graph can be written as For a cotangent graph which has no vertical shift, the equation for the graph can be written as y = a cot b (x - c). c indicates the phase shift, also known as the horizontal shift. a indicates whether the graph reflects about the x-axis. b affects the period.

16. Next  Back y = a tan b (x - c) y = a cot b (x - c) Unlike sine or cosine graphs, tangent and cotangent graphs have no maximum or minimum values. Their range is (-∞, ∞), so amplitude is not defined. However, it is important to determine whether a is positive or negative. When a is negative, the tangent or cotangent graph will “flip” or reflect about the x-axis.

17. Next  Back Notice the behavior of y = tan x. Notice what happens to each section of the graph as it nears its asymptotes. As each section nears the asymptote on its left, the y-values approach - ∞. As each section nears the asymptote on its right, the y-values approach + ∞.

18. Next  Back Notice what happens to each section of the graph as it nears its asymptotes. As each section nears the asymptote on its left, the y-values approach + ∞. As each section nears the asymptote on its right, the y-values approach - ∞. Notice the behavior of y = cot x.

19. Next  Back This is the graph for y = tan x. y = - tan x Consider the graph for y = - tan x In this equation a, the numerical coefficient for the tangent, is equal to -1. The fact that a is negative causes the graph to “flip” or reflect about the x-axis.

20. Next  Back This is the graph for y = cot x. y = - 2cot x Consider the graph for y = - 2 cot x In this equation a, the numerical coefficient for the cotangent, is equal to -2. The fact that a is negative causes the graph to “flip” or reflect about the x-axis.

21. Next  Back y = a tan b (x - c) y = a cot b (x - c) b affects the period of the tangent or cotangent graph. For tangent and cotangent graphs, the period can be determined by Conversely, when you already know the period of a tangent or cotangent graph, b can be determined by

22. Next  Back A complete period (including two consecutive vertical asymptotes) has been highlighted on the tangent graph below. The distance between the asymptotes in this graph is. Therefore, the period of this graph is also. For all tangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

23. Next  Back We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis. Use, the period of this tangent graph, to calculate b. An equation for this graph can be written as or.

24. Next  Back A complete period (including two consecutive vertical asymptotes) has been highlighted on the cotangent graph below. The distance between the asymptotes is. Therefore, the period of this graph is also. For all cotangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

25. Next  Back We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis. Use, the period of this cotangent graph, to calculate b. An equation for this graph can be written as or.