1. BY: Mr. Nefalar HOW TO GRAPH Y=TAN(X)

2. Definition Asymptote – A straight line that is a limiting value of a function. A function approaches the asymptote, but never reaches it.

3. Asymptotes of A tan(Bx): + π = Bx 2 Note: there are actually 2 asymptotes in 1 period of tan. (a positive and a negative) π = Bx and - π = Bx 2 2 To solve for the asymptotes you just plug in B and solve for x.

4. Example 1 Graph the function. y = 3 tan(2x)- π /4 <x< 3 π /4 Step 1: Solve for the asymptotes: Asymptote formula: + π = Bx 2 Since B = 2 then we plug in 2 for B in the formula and solve for x. (next page)

5. Example 1 cont. …so the asymptotes are at + π and – π 4 4

6. Example 1 cont. Step 2, graph asymptotes: The asymptotes are + π and – π 4 4

7. Example 1 cont. Step 3 make a center point between the asymptotes:

8. Example 1 cont. Notice the distance from center point to asymptote is π /4 This says that the distance from each center point to adjacent asymptote is π /4.

9. Example 1 cont. Step 4: Mark the rest of the center points and asymptotes to fulfill the range: - π /4 <x< 3 π /4. Notice the pattern of asymptote  center pt  asymptote, etc.

10. Example 1 cont. Step 5: Mark a key pt between each center point and asymptote. The height of the key pt is the amplitude. For positive tan, right key pt is up and left key pt is down. For negative tan, left key pt is up and right key pt is down.

11. Example 1 cont. Step 6: Smoothly connect the dots.

12. Lets try another one…

13. Example 2 Graph the function. y = -tan(½ x)-2 π <x< π Step 1: Solve for the asymptotes: …so the asymptotes are at + π and – π

14. Example 2 cont. Step 2, graph asymptotes: The asymptotes are + π and – π

15. Example 2 cont. Step 3 make a center point between the asymptotes:

16. Example 2 cont. Notice the distance from center point to asymptote is π

17. Example 2 cont. Step 4: Mark the rest of the center points and asymptotes to fulfill the range: -2 π <x< π.

18. Example 2 cont. Step 5: Mark a key pt between each center point and asymptote. The height of the key pt is the amplitude. For positive tan, right key pt is up and left key pt is down. For negative tan, left key pt is up and right key pt is down.

19. Example 2 cont. Step 6: Smoothly connect the dots.

20. Example 2 cont. Why is there only a right half of the graph between -2 π and – π ?

21. Example 2 cont. Because the range said to only graph from -2 π <x< π so we don’t need to complete the left half of the graph to the left of -2 π

22. We will review this on Monday.