1. Splash Screen

2. Concept

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

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

5. Example 2A 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.

6. Example 2B Determine Properties of Reciprocal Functions 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. B. Identify the asymptotes, domain, and range of the function.

7. Example 2B 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.

8. Example 2A 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 A. Identify the asymptotes of the function.

9. Example 2B A.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} B. Identify the domain and range of the function.

10. Concept

11. Example 3A Graph Transformations 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. a=–1:The graph is reflected across the x-axis. This represents a transformation of the graph of A. Graph the function State the domain and range.

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

13. Example 3B Graph Transformations 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. This represents a transformation of the graph of B. Graph the function State the domain and range.

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

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

16. Example 3B 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} B. State the domain and range of

17. Example 4A 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. Divide each side by r. d = 25 r= dOriginal equation. t Solve the formula r = d for t. t

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

19. Example 4B 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.

20. Example 4A 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.

21. Example 4 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.

22. Homework P. 549 # 2 – 34 even

23. End of the Lesson