1. Graphing Logarithmic functions
Transformations of the Parent Function Graph

2. 𝑦= 𝑙𝑜𝑔 2 𝑥 is the inverse of 𝑦= 2 𝑥

3. Logarithmic parent graph
Always passes through (1,0) and (b, 1)
Vertical asymptote x = 0
Domain: x > 0
Range: All real numbers

4. Transformations of the parent log
𝑦=𝑎• 𝑙𝑜𝑔 𝑏 𝑐 𝑥 −ℎ +𝑘
a<0 reflects over x-axis
multiply y-values by a
c<0 reflects over y-axis
divide x-values by c
𝑥 −ℎ translates right
+ k translates up
𝑥+ℎ translates left
- k translates down

5. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3

6. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3

7. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote:

8. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2

9. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2

10. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers

11. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
(1,0)
(___, 1)

12. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
(1,0)
(2, 1)

13. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a =
(1,0)
(2, 1)

14. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Add h =
to x
(1,0)
(2, 1)

15. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Add h =
to x
Add k =
to y
(1,0)
(2, 1)

16. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(2, 1)

17. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(2, 1)

18. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(2, 1)

19. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(2, 1)

20. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)

21. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)

22. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)

23. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)

24. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x ? - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)

25. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x > - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)

26. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x > - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)
Reflects over x-axis, dilation

27. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x > - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)
Reflects over x-axis, dilation,
translates left 2 units

28. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x > - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)
Reflects over x-axis, dilation,
translates left 2 units, translates down 3 units

29. Example 1 𝑦=−2 𝑙𝑜𝑔 2 (𝑥+2)−3 𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3 Asymptote: x = - 2
𝑎=−2 𝑏=2 𝑐=1 ℎ=−2 𝑘=−3
Asymptote: x = - 2
Domain: x > - 2
Range: all real numbers
Pts on parent graph
Mult. y
by a = -2
Add h =
to x
Add k = -3
to y
(1,0)
(1, 0)
(-1,0)
(-1, -3)
(2, 1)
(2, -2)
(0, -2)
(0, -5)
Reflects over x-axis, dilation,
translates left 2 units, translates down 3 units

30. Example 2 𝑦=3 𝑙𝑜𝑔 𝑥−3 +2

31. Example 2 𝑦=3 𝑙𝑜𝑔 𝑥−3 +2
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2

32. Example 2 𝑦=3 𝑙𝑜𝑔 𝑥−3 +2
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote:

33. Example 2 𝑦=3 𝑙𝑜𝑔 𝑥−3 +2
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3

34. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3

35. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers

36. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
(1,0)
(___ , 1)

37. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
(1,0)
(1/3 , 1)

38. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a =
(1,0)
(1/3 , 1)

39. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Add h =
to x
(1,0)
(1/3 , 1)

40. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Add h =
to x
Add k =
to y
(1,0)
(1/3 , 1)

41. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1/3 , 1)

42. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1/3 , 1)

43. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(1/3 , 1)

44. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(1/3 , 1)

45. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)

46. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)

47. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)

48. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)

49. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x ? 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)

50. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x > 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)

51. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x > 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)
Dilation

52. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x > 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)
Dilation, translates right 3 units

53. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x > 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)
Dilation, translates right 3 units, translates up 2 units

54. Example 2 𝑦=3 𝑙𝑜𝑔 1 3 𝑥−3 +2 𝑎=3 𝑏= 1 3 𝑐=1 ℎ=3 𝑘=2 Asymptote: x= 3
𝑎=3 𝑏= 𝑐=1 ℎ=3 𝑘=2
Asymptote: x= 3
Domain: x > 3
Range: all real numbers
Pts on parent graph
Mult. y
by a = 3
Add h =
to x
Add k = 2
to y
(1,0)
(1, 0)
(4,0)
(4, 2)
(1/3 , 1)
(1/3, 3)
(3 1/3, 3)
(3 1/3, 5)
Dilation, translates right 3 units, translates up 2 units

55. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3

56. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3
First, factor out the “c” = 2 (x – 1)

57. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3

58. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote:

59. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1

60. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1

61. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers

62. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
(1,0)
(___ , 1)

63. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
(1,0)
(3 , 1)

64. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c =
(1,0)
(3 , 1)

65. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c =
Add h =
to x
(1,0)
(3 , 1)

66. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c =
Add h =
to x
Add k =
to y
(1,0)
(3 , 1)

67. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(3 , 1)

68. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(3 , 1)

69. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(3 , 1)

70. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(3 , 1)

71. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)

72. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)

73. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)

74. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)

75. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x ? 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)

76. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x > 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)

77. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x > 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)
Dilation

78. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x > 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)
Dilation, translates right 1 unit

79. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x > 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)
Dilation, translates right 1 unit, translates up 3 units

80. Example 3 𝑦= 𝑙𝑜𝑔 3 2𝑥−2 +3 First, factor out the “c” = 2 (x – 1)
𝑎=1 𝑏=3 𝑐=2 ℎ=1 𝑘=3
Asymptote: x= 1
Domain: x > 1
Range: all real numbers
Pts on parent graph
Divide x
by c = 2
Add h = 1
to x
Add k = 3
to y
(1,0)
(1/2, 0)
(1 1/2,0)
(1 1/2, 3)
(3 , 1)
(1 1/2, 1)
(2 1/2, 1)
(2 1/2, 4)
Dilation, translates right 1 unit, translates up 3 units

81. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3

82. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3
First, factor out the “c” = -1 (x – 4)

83. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3

84. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote:

85. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4

86. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4

87. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers

88. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
(1,0)
(___ , 1)

89. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
(1,0)
(2 , 1)

90. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c =
(1,0)
(2 , 1)

91. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c =
Add h =
to x
(1,0)
(2 , 1)

92. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c =
Add h =
to x
Add k =
to y
(1,0)
(2 , 1)

93. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(2 , 1)

94. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(2 , 1)

95. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(2 , 1)

96. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(2 , 1)

97. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)

98. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)

99. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)

100. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)

101. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x ? 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)

102. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x < 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)

103. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x < 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)
Reflects over the y-axis

104. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x < 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)
Reflects over the y-axis, translates right 4 units

105. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x < 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)
Reflects over the y-axis, translates right 4 units, translates down 3 units

106. Example 4 𝑦= 𝑙𝑜𝑔 2 −𝑥+4 −3 First, factor out the “c” = -1 (x – 4)
𝑎=1 𝑏=2 𝑐=−1 ℎ=4 𝑘=−3
Asymptote: x= 4
Domain: x < 4
Range: all real numbers
Pts on parent graph
Divide x
by c = -1
Add h = 4
to x
Add k = -3
to y
(1,0)
(-1, 0)
(3,0)
(3, -3)
(2 , 1)
(-2, 1)
(2, -2)
Reflects over the y-axis, translates right 4 units, translates down 3 units

107. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1

108. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1
Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1

109. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1

110. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote:

111. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0

112. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0

113. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers

114. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
(1,0)
(___ , 1)

115. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
(1,0)
(4 , 1)

116. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a =
(1,0)
(4 , 1)

117. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Divide x
by c =
(1,0)
(4 , 1)

118. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Divide x
by c =
Add k =
to y
(1,0)
(4 , 1)

119. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(4 , 1)

120. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(4 , 1)

121. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(4 , 1)

122. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(4 , 1)

123. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)

124. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)

125. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)

126. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)

127. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x ? 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)

128. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x > 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)

129. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x > 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)
Reflects over the x-axis

130. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x > 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)
Reflects over the x-axis, dilation

131. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x > 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)
Reflects over the x-axis, dilation,
translates down 1 unit

132. Example 5 𝑦= −𝑙𝑜𝑔 4 2𝑥 −1 Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 4 2(𝑥+0) −1
𝑎=−1 𝑏=4 𝑐=2 ℎ=0 𝑘=−1
Asymptote: x= 0
Domain: x > 0
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Divide x
by c = 2
Add k =
to y
(1,0)
(1, 0)
(1/2, 0)
(1/2, -1)
(4 , 1)
(4, -1)
(2 , -1)
(2, -2)
Reflects over the x-axis, dilation,
translates down 1 unit

133. Example 6 𝑦= −𝑙𝑜𝑔 (𝑥+4)

134. Example 6 𝑦= −𝑙𝑜𝑔 (𝑥+4)
Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 𝑥+4 +0

135. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0

136. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote:

137. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4

138. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4

139. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers

140. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
(1,0)
( ___, 1)

141. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
(1,0)
(1/2 , 1)

142. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a =
(1,0)
(1/2 , 1)

143. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a =
Add h=
to x
(1,0)
(1/2 , 1)

144. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1/2 , 1)

145. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1/2 , 1)

146. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(1/2 , 1)

147. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)

148. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)

149. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)
(-3 1/2 , -1)

150. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x ? - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)
(-3 1/2 , -1)

151. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x > - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)
(-3 1/2 , -1)

152. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x > - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)
(-3 1/2 , -1)
Reflects over the x-axis

153. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x > - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)
(-3 1/2 , -1)
Reflects over the x-axis, translates left 4 units

154. Example 6 𝑦= −𝑙𝑜𝑔 1 2 (𝑥+4) Tip: Rewrite: 𝑦= −𝑙𝑜𝑔 1 2 𝑥+4 +0
𝑎=−1 𝑏= 𝑐=1 ℎ=−4 𝑘=0
Asymptote: x= -4
Domain: x > - 4
Range: all real numbers
Pts on parent graph
Mult. y
by a = -1
Add h= -4
to x
(1,0)
(1, 0)
(-3, 0)
(1/2 , 1)
(1/2, -1)
(-3 1/2 , -1)
Reflects over the x-axis, translates left 4 units