1. Splash Screen

2. Five-Minute Check (over Lesson 8–2) CCSS Then/Now New Vocabulary
Key Concept: Parent Function of Reciprocal Functions
Example 1: Limitations on Domain
Example 2: Determine Properties of Reciprocal Functions
Key Concept: Transformations of Reciprocal Functions
Example 3: Graph Transformations
Example 4: Real-World Example: Write Equations
Lesson Menu

3. Find the LCM of 13xy3 and 20x2y2z.
A. 260x2y3z
B. 130x2y3z
C. 58xy2z
D. xy2
5-Minute Check 1

4. Find the LCM of 13xy3 and 20x2y2z.
A. 260x2y3z
B. 130x2y3z
C. 58xy2z
D. xy2
5-Minute Check 1

5. A. B. C. D.
5-Minute Check 2

6. A. B. C. D.
5-Minute Check 2

7. A. B. C. D.
5-Minute Check 3

8. A. B. C. D.
5-Minute Check 3

9. Find the perimeter of the triangle. Express in simplest form.B. C. D.
5-Minute Check 4

10. Find the perimeter of the triangle. Express in simplest form.B. C. D.
5-Minute Check 4

11. ? A. x B. x(x + 2) C. (x + 2)(2x + 3) D. x(x + 2)(2x + 3)
5-Minute Check 5

12. ? A. x B. x(x + 2) C. (x + 2)(2x + 3) D. x(x + 2)(2x + 3)
5-Minute Check 5

13. Mathematical Practices 2 Reason abstractly and quantitatively.
Content Standards
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Mathematical Practices
2 Reason abstractly and quantitatively.
CCSS

14. You graphed polynomial functions.
Determine properties of reciprocal functions.
Graph transformations of reciprocal functions.
Then/Now

15. reciprocal function
hyperbola
Vocabulary

16. Concept

17. Determine the values of x for which is not defined.
Limitations on Domain
Determine the values of x for which is not defined.
Factor the denominator of the expression.
Answer:
Example 1

18. Determine the values of x for which is not defined.
Limitations on Domain
Determine the values of x for which is not defined.
Factor the denominator of the expression.
Answer: The function is undefined for x = –8 and x = 3.
Example 1

19. A. Identify the asymptotes, domain, and range of the function.
Determine Properties of Reciprocal Functions
A. Identify the asymptotes, domain, and range of the function.
Identify the x-values for which f(x) is undefined.
x – 2 = 0
x = 2
f(x) is not defined when x = 2. So, there is an asymptote at x = 2.
Example 2A

20. Determine Properties of Reciprocal Functions
From x = 2, as x-values decrease, f(x)-values approach 0, and as x-values increase, f(x)-values approach 0. So, there is an asymptote at f(x) = 0.
Answer:
Example 2A

21. Determine Properties of Reciprocal Functions
From x = 2, as x-values decrease, f(x)-values approach 0, and as x-values increase, f(x)-values approach 0. So, there is an asymptote at f(x) = 0.
Answer: There are asymptotes at x = 2 and f(x) = 0. The domain is all real numbers not equal to 2 and the range is all real numbers not equal to 0.
Example 2A

22. B. Identify the asymptotes, domain, and range of the function.
Determine Properties of Reciprocal Functions
B. Identify the asymptotes, domain, and range of the function.
Identify the x-values for which f(x) is undefined.
x + 2 = 0
x = –2
f(x) is not defined when x = –2. So, there is an asymptote at x = –2.
Example 2B

23. Determine Properties of Reciprocal Functions
From x = –2, as x-values decrease, f(x)-values approach 1, and as x-values increase, f(x)-values approach 1. So, there is an asymptote at f(x) = 1.
Answer:
Example 2B

24. Determine Properties of Reciprocal Functions
From x = –2, as x-values decrease, f(x)-values approach 1, and as x-values increase, f(x)-values approach 1. So, there is an asymptote at f(x) = 1.
Answer: There are asymptotes at x = –2 and f(x) = 1. The domain is all real numbers not equal to –2 and the range is all real numbers not equal to 1.
Example 2B

25. A. Identify the asymptotes of the function.
A. x = 3 and f(x) = 3
B. x = 0 and f(x) = –3
C. x = –3 and f(x) = –3
D. x = –3 and f(x) = 0
Example 2A

26. A. Identify the asymptotes of the function.
A. x = 3 and f(x) = 3
B. x = 0 and f(x) = –3
C. x = –3 and f(x) = –3
D. x = –3 and f(x) = 0
Example 2A

27. B. Identify the domain and range of the function.
D = {x | x ≠ –3}; R = {f(x) | f(x) ≠ –4}
B. D = {x | x ≠ 3}; R = {f(x) | f(x) ≠ 0}
C. D = {x | x ≠ 4}; R = {f(x) | f(x) ≠ –3}
D. D = {x | x ≠ 0}; R = {f(x) | f(x) ≠ 4}
Example 2B

28. B. Identify the domain and range of the function.
D = {x | x ≠ –3}; R = {f(x) | f(x) ≠ –4}
B. D = {x | x ≠ 3}; R = {f(x) | f(x) ≠ 0}
C. D = {x | x ≠ 4}; R = {f(x) | f(x) ≠ –3}
D. D = {x | x ≠ 0}; R = {f(x) | f(x) ≠ 4}
Example 2B

29. Concept

30. A. Graph the function State the domain and range.
Graph Transformations
A. Graph the function State the domain and range.
This represents a transformation of the graph of
a = –1: The graph is reflected across the x-axis.
h = –1: The graph is translated 1 unit left. There is an asymptote at x = –1.
k = 3: The graph is translated 3 units up. There is an asymptote at f(x) = 3.
Example 3A

31. Graph Transformations
Answer:
Example 3A

32. Answer: Domain: {x│x ≠ –1} Range: {f(x)│f(x) ≠ 3}
Graph Transformations
Answer: Domain: {x│x ≠ –1} Range: {f(x)│f(x) ≠ 3}
Example 3A

33. B. Graph the function State the domain and range.
Graph Transformations
B. Graph the function State the domain and range.
This represents a transformation of the graph of
a = –4: The graph is stretched vertically and reflected across the x-axis.
h = 2: The graph is translated 2 units right. There is an asymptote at x = 2.
Example 3B

34. Graph Transformations
k = –1: The graph is translated 1 unit down There is an asymptote at f(x) = –1.
Answer:
Example 3B

35. Answer: Domain: {x│x ≠ 2} Range: {f(x)│f(x) ≠ –1}
Graph Transformations
k = –1: The graph is translated 1 unit down There is an asymptote at f(x) = –1.
Answer: Domain: {x│x ≠ 2} Range: {f(x)│f(x) ≠ –1}
Example 3B

36. A. Graph the function B.
C. D.
Example 3A

37. A. Graph the function B.
C. D.
Example 3A

38. B. State the domain and range of
A. Domain: {x│x ≠ –1}; Range: {f(x)│f(x) ≠ –2}
B. Domain: {x│x ≠ 4}; Range: {f(x)│f(x) ≠ 2}
C. Domain: {x│x ≠ 1}; Range: {f(x)│f(x) ≠ –2}
D. Domain: {x│x ≠ –1}; Range: {f(x)│f(x) ≠ 2}
Example 3B

39. B. State the domain and range of
A. Domain: {x│x ≠ –1}; Range: {f(x)│f(x) ≠ –2}
B. Domain: {x│x ≠ 4}; Range: {f(x)│f(x) ≠ 2}
C. Domain: {x│x ≠ 1}; Range: {f(x)│f(x) ≠ –2}
D. Domain: {x│x ≠ –1}; Range: {f(x)│f(x) ≠ 2}
Example 3B

40. Solve the formula r = d for t. t
Write Equations
A. COMMUTING A commuter train has a nonstop service from one city to another, a distance of about 25 miles. Write an equation to represent the travel time between these two cities as a function of rail speed. Then graph the equation.
Solve the formula r = d for t. t
r = d Original equation. t
Divide each side by r.
d = 25
Example 4A

41. Write Equations
Graph the equation
Answer:
Example 4A

42. Write Equations
Graph the equation
Answer:
Example 4A

43. Write Equations
B. COMMUTING A commuter train has a nonstop service from one city to another, a distance of about 25 miles. Explain any limitations to the range and domain in this situation.
Answer:
Example 4B

44. Write Equations
B. COMMUTING A commuter train has a nonstop service from one city to another, a distance of about 25 miles. Explain any limitations to the range and domain in this situation.
Answer: The range and domain are limited to all real numbers greater than 0 because negative values do not make sense. There will be further restrictions to the domain because the train has minimum and maximum speeds at which it can travel.
Example 4B

45. A. TRAVEL A commuter bus has a nonstop service from one city to another, a distance of about 76 miles. Write an equation to represent the travel time between these two cities as a function of rail speed.
A. B.
C. D.
Example 4A

46. A. TRAVEL A commuter bus has a nonstop service from one city to another, a distance of about 76 miles. Write an equation to represent the travel time between these two cities as a function of rail speed.
A. B.
C. D.
Example 4A

47. B. TRAVEL A commuter bus has a nonstop service from one city to another, a distance of about 76 miles. Graph the equation to represent the travel time between these two cities as a function of rail speed.
A. B.
C. D.
Example 4

48. B. TRAVEL A commuter bus has a nonstop service from one city to another, a distance of about 76 miles. Graph the equation to represent the travel time between these two cities as a function of rail speed.
A. B.
C. D.
Example 4

49. End of the Lesson