You graphed polynomial functions Determine properties of reciprocal

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You graphed polynomial functions. • Determine properties of reciprocal functions. • Graph transformations of

You graphed polynomial functions. • Determine properties of reciprocal functions. • Graph transformations of reciprocal functions.

 • reciprocal function • hyperbola

• reciprocal function • hyperbola

Limitations on Domain Determine the values of x for which is not defined. Factor

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

Limitations on Domain Determine the values of x for which is not defined. Factor

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.

Determine Properties of Reciprocal Functions A. Identify the asymptotes, domain, and range of the

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.

Determine Properties of Reciprocal Functions From x = 2, as x-values decrease, f(x)-values approach

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:

Determine Properties of Reciprocal Functions From x = 2, as x-values decrease, f(x)-values approach

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 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.

Determine Properties of Reciprocal Functions B. Identify the asymptotes, domain, and range of the

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.

Determine Properties of Reciprocal Functions From x = – 2, as x-values decrease, f(x)-values

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:

Determine Properties of Reciprocal Functions From x = – 2, as x-values decrease, f(x)-values

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 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.

A. Identify the asymptotes of the function. A. x = 3 and f(x) =

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

A. Identify the asymptotes of the function. A. x = 3 and f(x) =

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

B. Identify the domain and range of the function. A. D = {x |

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

B. Identify the domain and range of the function. A. D = {x |

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

Graph Transformations A. Graph the function State the domain and range. This represents a

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

Graph Transformations Answer:

Graph Transformations Answer:

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

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

Graph Transformations B. Graph the function State the domain and range. This represents a

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.

Graph Transformations k = – 1: Answer: The graph is translated 1 unit down.

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

Graph Transformations k = – 1: The graph is translated 1 unit down. There

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}

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

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

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

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

B. State the domain and range of A. Domain: {x│x ≠ – 1}; Range:

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}

B. State the domain and range of A. Domain: {x│x ≠ – 1}; Range:

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}

Write Equations A. COMMUTING A commuter train has a nonstop service from one city

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 t = d for t. rt = d Original equation. Divide each side by r. d = 25

Write Equations Graph the equation Answer:

Write Equations Graph the equation Answer:

Write Equations Graph the equation Answer:

Write Equations Graph the equation Answer:

Write Equations B. COMMUTING A commuter train has a nonstop service from one city

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:

Write Equations B. COMMUTING A commuter train has a nonstop service from one city

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.

A. TRAVEL A commuter bus has a nonstop service from one city to another,

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.

A. TRAVEL A commuter bus has a nonstop service from one city to another,

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.

B. TRAVEL A commuter bus has a nonstop service from one city to another,

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.

B. TRAVEL A commuter bus has a nonstop service from one city to another,

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.