Attributes and Transformations of Reciprocal Functions VOCABULARY


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1 TEKS FOCUS  Attributes and Transformations of Reciprocal Functions VOCABULARY TEKS (6)(G) Analze the effect on the graphs of f () = when f () is replaced b af (), f (b), f (  c), and f () + d for specific positive and negative real values of a, b, c, and d. TEKS ()(C) Select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS ()(A), ()(E), ()(G), ()(A), (6)(H), (6)(K), (6)(L), (7)(I) Branch Each piece of a discontinuous graph is called a branch. Rational function A rational function f () can be written as f () = P() Q(), where P() and Q() are polnomial functions. Reciprocal function A reciprocal function belongs to the famil whose parent function is f () = where 0. Number sense the understanding of what numbers mean and how the are related ESSENTIAL UNDERSTANDING Transformations of the parent reciprocal function include stretches, compressions (or shrinks), reflections, and horizontal and vertical translations. Concept Summar Reciprocal Function Famil Parent Function f () = This function and its transformations are part of the rational functions. Translation = + d =  c d 7 0 shifts up d units c 7 0 shifts to the right c units d 6 0 shifts down d units c 6 0 shifts to the left c units Stretch, Compression, and Reflection = a = b a 7 vertical stretch b 7 horizontal compression (shrink) 0 6 a 6 vertical compression (shrink) 0 6 b 6 horizontal stretch a 6 0 reflection across ais b 6 0 reflection across ais 57
2 Problem TEKS Process Standard ()(E) Graphing an Inverse Variation Function What values should ou choose for? Choose values of that divide nicel into. Make a table of points that are eas to graph. What is the graph of =, 0? Identif the  and intercepts and the asmptotes of the graph. Also, state the domain and range of the function in interval notation. Step Make a table of values that includes positive and negative values of Step Graph the points. 6 6 O Notice how the values get closer to zero as the absolute values of get larger The absolute values of get ver large as approaches zero. Step 3 Connect the points with a smooth curve. cannot be zero, so there is no intercept. The numerator is never zero, so is never 0. There is no intercept. The ais is a horizontal asmptote. The ais is a vertical asmptote. Knowing the asmptotes provides ou with the basic shape of the graph. The domain in interval notation consists of the intervals (, 0) and (0, ). The range in interval notation consists of the intervals (, 0) and (0, ) O 6 5 Lesson  Attributes and Transformations of Reciprocal Functions
3 Problem Analzing the Graph of af () When f () = For each given value of a, how do the graphs of = and = a compare? What is the effect of a on the graph? A a = 6 The graph (in red) of = 6 is a stretch of the graph of = (in black) b the factor 6. B a = 0.5 The graph (in blue) of = 0.5 is a shrink of the graph of = (in black) b the factor. 6 O 0.5 C a = 6 How does the negative sign affect the graph? The values have signs that are opposite those in part A. The graph in A reflects across the ais. 6 The graph of =  6 is the stretch b the factor 6 in part A followed b a reflection across the ais. Problem 3 TEKS Process Standard ()(G) Analzing the Graph of f () + d When f () = How does f() + d transform other parent functions? The number d translates the graph of a parent function verticall. What are the graphs of each rational function? How are these graphs different from the parent function f () =? Give the domain and range in inequalities. A = + O To obtain the graph of = +, ou can shift the graph of = up units. Notice that this will also shift the horizontal asmptote up units to =, and change the range of the function to the set of values such that 7 or 6. The domain will not change, so the domain is the set of values such that 7 0 or 6 0. continued on net page 59
4 Problem 3 continued B = 3 To obtain the graph of =  3, ou can shift the graph of = down 3 units. Notice that this will also shift the horizontal asmptote down 3 units to =3, and the range of the function is the set of values such that 73 or 63. The domain is the set of values such that 7 0 or 6 0. O Problem Analzing the Graph of f ( c) When f () = How does f( c) transform other parent functions? The number c translates the graph of a parent function horizontall. What are the graphs of each rational function? How are these graphs different from the parent function f () =? Give the domain and range in set notation. A = To obtain the graph of = , ou can shift the graph of = right unit. Notice that this will also shift the vertical asmptote right unit to =, and change the domain of the function to { 0 }. The range of the function will not change, and is { 0 0}. O B = + To obtain the graph of = +, ou can shift the graph of = left units. Notice that this will also shift the vertical asmptote left units to =. The domain of the function is { 0 }. The range is { 0 0}. O 60 Lesson  Attributes and Transformations of Reciprocal Functions
5 Problem 5 Analzing the Graph of f (b) When f () = How does f(b) transform other parent functions? Depending on what number b represents, it stretches or compresses the graph of a parent function horizontall. Analze the effects on the graph of f () = for each transformed function. How does the graph of the parent function change? A = 3 Complete a table of values to help ou sketch the graph f() = = 3 O Multipling b a constant will stretch or compress the graph of the parent function horizontall. In this eample, the graph is compressed horizontall b a factor of 3. B = Complete a table of values to help ou sketch the graph. O f() = You can rewrite the function as f () = . Multipling b a negative constant will reflect the graph across the ais. This graph is also compressed horizontall b a factor of. 6
6 Problem 6 Writing the Equation of a Transformation How can ou get started? Identif the asmptotes of the graph. Multiple Choice This graph of a function is a translation of the graph of =. What is an equation for the function? = = = = The asmptotes are =3 and =. Thus c =3 and d =. = a  c + d Write a general form containing all the tpes of transformations ou see in the graph. = +  (3) Substitute for a, c, and d. = Simplif. The correct choice is A. O Problem 7 TEKS Process Standard ()(C) Using a Reciprocal Function Clubs The rowing club is renting a 57passenger bus for a da trip. The cost of the bus is $750. Five passengers will be chaperones. If the students who attend share the bus cost equall, what function models the cost per student C with respect to the number of students n who attend? What is the domain of the function? How man students must ride the bus to make the cost per student no more than $0? The bus holds 57 passengers. The bus costs $750. Five riders are chaperones who pa nothing for the bus. A function for the cost per student The number of students needed so that the cost does not eceed $0 per student Write a reciprocal function for the situation. Graph the function and solve an inequalit using the $0 limit. Is the domain " 5? No; the domain is the set of possible numbers of students, so onl positive integers make sense. To share the cost equall, divide 750 b the number of students, n, who attend. The function that models the cost per student is C = 750 n. The bus has a capacit of 57 passengers and there will be 5 chaperones. The maimum number of students is 575 = 5. The domain is the integers from to 5. Use a graphing calculator to solve the inequalit 750 n 0. Let Y = 750 and Y = 0. continued on net page 6 Lesson  Attributes and Transformations of Reciprocal Functions
7 Problem 7 continued Change the window dimensions to get a closer look at the graph. Use the intersect feature. For all values greater than or equal to 3, the cost is less than $0. The number of people must be a whole number. Intersection X= 37.5 Y = 0 If 37, the cost will be more than $0. At least 3 students must ride the bus. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to. Graph each function. Identif the  and intercepts and the asmptotes of the graph. Also, state the domain and the range of the function in interval notation.. = 5. = 3. =  3. =0 Select Tools to Solve Problems ()(C) Graph the equations = and = a using the given value of a. Then identif the effect of a on the graph. 5. a = 6. a = 7. a = 0.5. a = 0.75 Sketch the asmptotes and the graph of each function. Identif the domain and range in set notation. 9. = = = = = = =  6. = Write an equation for the transformation of f () = that has the given asmptotes. State the domain and range in inequalities. 7. = 0 and =. = and = 3 9. = and = 0. Appl Mathematics ()(A) The weight P in pounds that a beam can safel carr is inversel proportional to the distance D in feet between the supports of the beam. For a certain tpe of wooden beam, P = 900 D. What distance between supports is needed to carr 00 lb?. Appl Mathematics ()(A) A high school decided to spend $750 on student academic achievement awards. At least 5 awards will be given, the should be equal in value, and each award should not be less than $50. Write and sketch a function that models the relationship between the number a of awards and the cost c of each award. What are the domain and range of the function? 63
8 . Create Representations to Communicate Mathematical Ideas ()(E) Write an equation for a horizontal translation of =. Then write an equation for a vertical translation of =. Identif the horizontal and vertical asmptotes of the graph of each function. Use a graph of f () = to determine the maimum and minimum value of the function over the interval. 3. [, ]. [3, 5] 5. [, 0.5] 6. Over the interval [a, b], the minimum value of f () = is f (b) and the maimum value is f (a). Describe the characteristics of the real numbers a and b. Eplain our reasoning. Analze the effect on the graph of f () = b the function g(). 7. g() = 5. g() = You sketch the graphs of the functions =, =, and = at the same time using a graphing calculator. How can ou identif each function? 30. Describe the graph of = b where 0 6 b 6. How is this graph different from the parent function =? Sketch the graph of each function. 3. = = = 3. 0 = 35. Eplain Mathematical Ideas ()(G) Eplain how knowing the asmptotes of a translation of = can help ou graph the function. Include an eample. 36. Appl Mathematics ()(A) The formula p = a models the relationship between atmospheric pressure p in inches of mercur and altitude a in miles. Use the data shown with the photo. At which location does the model predict the pressure to be about 3.93 in. of mercur? (Hint: mi = 50 ft.) A. Sahara Desert B. Kalahari Desert C. Mt. Kilimanjaro D. Vinson Massif Vinson Massif alt. 6,60 ft Select Tools to Solve Problems ()(C) Graph each pair of functions. Find the approimate point(s) of intersection. 37. = 6 , = 6 3. = 3 +, = 39. = , = 6. Sahara Desert average alt. 500 ft Kalahari Desert average alt. 300 ft Mt. Kilimanjaro alt. 9,30 ft 6 Lesson  Attributes and Transformations of Reciprocal Functions
9 0. Connect Mathematical Ideas ()(F) How will the domain and the range of the parent function f () = change after the translation of its graph b 3 units up and b 5 units to the left? State the domain and range in inequalities. Use Multiple Representations to Communicate Mathematical Ideas ()(D) Compare each pair of graphs and find an points of intersection.. = and = 0 0. = and = 3. = 0 0 and =. Find two reciprocal functions such that the minimum distance from the origin to the graph of each function is. 5. Write each equation in the form = k + c, and sketch the graph. a. = 36 b. =   b c. = d.  = TEXAS Test Practice 6. What is an equation for the translation of = that has asmptotes at = 3 and =5? A. = B. = C. = D. = The graph at the right shows which inequalit? F G..5 + Ú 5 H J O. If p and q var inversel, and p = 0 when q =, what is q when p =? A. 0 B. 5 C.  5 D Which equation represents the inverse of the graph at the right? F. = log 3 H. = log 3 G. = log 3 J. = log What is b if the graph of = 7b includes the point (, )? O 65
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