1. Graphing Reciprocal Functions
LESSON 8–3
Graphing Reciprocal Functions

2. Five-Minute Check (over Lesson 8–2) TEKS 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. TEKS

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

5. reciprocal function
hyperbola
Vocabulary

6. Concept

7. 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

8. Determine the values of x for which is not defined.
A. x = –7, x = 4
B. x = –14, x = 2
C. x = –4, x = 7
D. x = –2, x = 14
Example 1

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. Concept

16. 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

17. 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

18. 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

19. 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

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

21. 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

22. 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

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

24. 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

25. 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

26. 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

27. Graphing Reciprocal Functions
LESSON 8–3
Graphing Reciprocal Functions