Graphing Reciprocal Functions LESSON 8–3 Graphing Reciprocal Functions

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

  TEKS

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

reciprocal function hyperbola Vocabulary

Concept

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

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

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

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

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

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

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

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

Concept

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

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

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

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

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

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

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

Write Equations Graph the equation Answer: Example 4A

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

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

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

Graphing Reciprocal Functions LESSON 8–3 Graphing Reciprocal Functions