1-5 Study Guide and Intervention

Transcription

1 -5 Stud Guide and Intervention Parent Functions A parent function is the simplest of the functions in a famil. Parent Function Form Notes constant function f() = c graph is a horizontal line identit function f() = points on graph have coordinates (a, a) quadratic function f() = 2 graph is U-shaped cubic function f() = 3 graph is smmetric about the origin square root function f() = graph is in first quadrant reciprocal function f() = graph has two branches absolute value function f() = graph is V-shaped greatest integer function f() = defined as the greatest integer less than or equal to ; tpe of step function Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eample Describe the following characteristics of the graph of the parent function f() = 3 : domain, range, intercepts, smmetr, continuit, end behavior, and intervals on which the graph is increasing/decreasing. The graph confirms that D = { } and R = { }. The graph intersects the origin, so the -intercept is 0 and the -intercept is 0. It is smmetric about the origin and it is an odd function: f(-) = -f(). The graph is continuous because it can be traced without lifting the pencil off the paper. As decreases, approaches negative infinit, and as increases, approaches positive infinit. lim f() = - and lim f() = - The graph is alwas increasing, so it is increasing for (-, ). Eercise f() = 3 Describe the following characteristics of the graph of the parent function f() = 2 : domain, range, intercepts, smmetr, continuit, end behavior, and intervals on which the graph is increasing/decreasing. Lesson -5 Chapter 27 Glencoe Precalculus

2 -5 Stud Guide and Intervention (continued) Transformations of Parent Functions Parent functions can be transformed to create other members in a famil of graphs. Translations Reflections Dilations g() = f() + k is the graph of f() translated g() = f( - h) is the graph of f() translated g() = -f() is the graph of f() g() = f(-) is the graph of f() g() = a f() is the graph of f() g() = f(a) is the graph of f() k units up when k > 0. k units down when k < 0. h units right when h > 0. h units left when h < 0. reflected in the -ais. reflected in the -ais. epanded verticall if a >. compressed verticall if 0 < a <. compressed horizontall if a >. epanded horizontall if 0 < a <. Eample Identif the parent function f() of g() = - -, and describe how the graphs of g() and f() are related. Then graph f() and g() on the same aes. The graph of g() is the graph of the square root function f() = reflected in the -ais and then translated one unit down. Eercises Identif the parent function f() of g(), and describe how the graphs of g() and f() are related. Then graph f() and g() on the same aes.. g() = g() = g() = - - f() = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 2 Glencoe Precalculus

3 -5 Practice. Use the graph of f() = to graph g() = Use the graph of f() = to graph g() = Describe how the graph of f() = 2 and g() are related. Then write an equation for g(). Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. Identif the parent function f() of g() = Describe how the graphs of g() and f() are related. Then graph f() and g() on the same aes. 5. Graph f() = - if -3 + if -2 < 2. if 6 6. Use the graph of f() = 3 to graph g() = ( + ) Lesson -5 Chapter 29 Glencoe Precalculus

4 -5 Word Problem Practice. AREA The width w of a rectangular plot of land with fied area A is modeled b the function w(l) = A, where l is the l length. a. If the area is 000 square feet, describe the transformations of the parent function f() = used to graph w(). b. Describe a function of the length that could be used to find a minimum perimeter for a given area 3. TAXES Graph the ta rates for the different incomes b using a step function. Income Ta Rates for a Couple Filing Jointl Limits of Taable Income ($) Ta Rate (%) 0 to,200 5,20 to 99, ,60 to 5, ,75 to 27, ,05 and up 39.6 Source: Information Please Almanac c. Is the function ou found in part b a transformation of f()? Eplain. d. Find the minimum perimeter for an area of 000 square feet. 2. GOLF The path of the flight of a golf ball can be modeled b h() = , 0 where h() is the distance above the ground in ards and is the horizontal distance from the tee in ards. a. Describe the transformation of the parent function f() = 2 used to graph h(). b. Suppose the same shot was made from a tee located 0 ards behind the original tee. Rewrite h() to reflect this change. Ta Rate (%) Taable Income (thousands). HORIZON The function f() =.5 can be used to approimate the distance to the apparent horizon, or how far a person can see on a clear da, where f() is the distance in miles and is the person s elevation in feet. a. How does the graph of f() compare to the graph of its parent function? b. The function g() =.2 is also used to approimate the distance to the horizon. How does the graph of g() compare to the graph of its parent function? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 30 Glencoe Precalculus

5 -6 Stud Guide and Intervention Operations with Functions Two functions can be added, subtracted, multiplied, or divided to form a new function. For the new function, the domain consists of the intersection of the domains of the two functions, ecluding values that make a denominator equal to zero. Eample Given f() = and g() = + 2, find each function and its domain. a. (f + g)() b. ( g) f () (f + g) = f() + g() ( = g) f = f() g() = 2 - = ( - 3)( + 2) = = The domains of f and g are both The domains of f and g are both (-, ), so the domain of (f + g) is (-, ), but = -2 ields a zero in (-, ). the denominator of ( g) f. So, the domain is { -2, }. Eample 2 a. (f - g)() (f - g) = f() - g() = Given f() = 2-3 and g() =, find each function and its domain. b. (f g)() The domain of f is (-, ) and the domain of g is (, 0) (0, ), so the domain of (f - g) is (, 0) (0, ). Eercises g) (f g) = f() g() = ( 2-3) = - 3 The domain of f is (-, ) and the domain of g is (, 0) (0, ), so the domain of (f - g) is (, 0) (0, ). Find (f + g)(), (f - g)(), (f g)(), and ( f () for each f() and g(). State the domain of each new function.. f() = 2 -, g() = 2 2. f() = 2 + 7, g() = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 32 Glencoe Precalculus

6 -6 Stud Guide and Intervention (continued) Compositions of Functions In a function composition, the result of one function is used to evaluate a second function. Given functions f and g, the composite function f g can be described b the equation [f g]() = f[g()]. The domain of f g includes all -values in the domain of g for which g() is in the domain of f. Lesson -6 Eample and [g f](). Given f() = and g() = + 2, find [f g]() [f g]() = f[g()] Defi nition of composite functions = f( + 2) Replace g() with + 2. = 3( + 2) 2 + 2( + 2) - Substitute + 2 for in f(). = 3( ) Simplif. = [g f]() = g(f()) Defi nition of composite functions = g( ) Replace f() with = ( ) + 2 Substitute for in g(). = Simplif. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eercises For each pair of functions, find [f g](), [g f](), and [f g]().. f() = 2 +, g() = f() = 3 2, g() = 3. f() = 3, g() = 5. f() = 2, g() = f() = 3-5, g() = f() = 7. f() = 2-3, g() = - 2 -, g() = 2 -. f() = -, g() = + Chapter 33 Glencoe Precalculus

7 -6 Practice Find (f + g)(), (f - g)(), (f g)(), and ( g) f () for each f() and g(). State the domain of each new function.. f() = and g() = f() = 3 and g() = + For each pair of functions, find [f g](), [g f](), and [f g](3). 3. f() = + 5 and g() = - 3. f() = and g() = 3 5. f() = and g() = f() = and g() = 2 - Find f g. 7. f() = - 2. f() = - g() = 3 g() = Find two functions f and g such that h() = [f g](). Neither function ma be the identit function f() =. 9. h() = h() = RESTAURANT A group of three restaurant patrons order the same meal and drink and leave an % tip. Determine functions that represent the cost of all of the meals before tip, the actual tip, and the composition of the two functions that gives the cost for all of the meals including tip. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 3 Glencoe Precalculus

8 -6 Word Problem Practice. MARCHING BAND Band members form a circle of radius r when the music starts. The march outward as the pla. The function f(t) = 2.5t gives the radius of the circle in feet after t seconds. Using g(r) = πr 2 for the area of the circle, write a composite function that gives the area of the circle after t seconds. Then find the area, to the nearest tenth, after seconds.. TRAVEL Two travelers are budgeting mone for the same trip. The first traveler s budget (in dollars) can be represented b f() = The second traveler s budget (in dollars) can be represented b g() = , is the number of nights. a. Find (f + g)() and the relevant domain. b. What does the composite function in part a represent? Lesson CANDLES A hobbist makes and sells candles at a local market. The function c(h) = h gives the number of candles she has made after h hours. The function f(c) = c gives the cost of making c candles. a. Write the composite function that gives the cost of candle making after h hours. c. Find (f + g)(7) and eplain what the value represents. d. Repeat parts a c for (g - f)(). Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. b. A sale reduces the cost of making c candles b 0%. Write the sale function s() and the composite function that gives the cost of candle making after h hours if materials are purchased during the sale. 3. SCIENCE The function t() = gives the temperature in degrees Celsius of the liquid in a beaker after seconds. Decompose the function into two separate functions, s() and r(), so that s(r()) = t(). 5. POPULATION The function p() = predicts the population of elk in a forest for the ears 200 through 205 where is the number of ears since Decompose the function into two separate functions, a() and b(), so that [a b]() = p() and a() is a quadratic function and b() is a linear function. Chapter 35 Glencoe Precalculus

9 Answers (Lesson - and Lesson -5) Lesson -5 TI-Nspire Activit - Finding an Average Rate of Change Given a function, ou can draw two points on the function, connect the points with a line, and then find the slope of that line, giving ou the average rate of change for that interval. Eample For the function f () = , find the average rate of change for the interval [-2, 0]. Step : Add a GRAPHS & GEOMETRY page. Enter the function rule in the function entr line. Press / + G to hide the function entr line. -5 Stud Guide and Intervention Parent Functions A parent function is the simplest of the functions in a famil. Parent Function Form Notes constant function f() = c graph is a horizontal line identit function f() = points on graph have coordinates (a, a) quadratic function f() = 2 graph is U-shaped cubic function f() = 3 graph is smmetric about the origin square root function f() = graph is in first quadrant reciprocal function f() = graph has two branches absolute value function f() = graph is V-shaped greatest integer function f() = defined as the greatest integer less than or equal to ; tpe of step function Eample Describe the following characteristics of the graph of the parent function f() = 3 : domain, range, intercepts, smmetr, continuit, end behavior, and intervals on which the graph is increasing/decreasing. The graph confirms that D = { } and R = { }. The graph intersects the origin, so the -intercept is 0 and the -intercept is 0. It is smmetric about the origin and it is an odd function: f() = -f(). The graph is continuous because it can be traced without lifting the pencil off the paper. As decreases, approaches negative infinit, and as increases, approaches positive infinit. lim - f() = - and lim f() = The graph is alwas increasing, so it is increasing for (-, ). f() = 3 Eercise Describe the following characteristics of the graph of the parent function f() = 2 : domain, range, intercepts, smmetr, continuit, end behavior, and intervals on which the graph is increasing/decreasing. D = { }, R = { 0, }; -int: 0; -int: 0; smmetric with respect to -ais; even function; continuous; lim - decreasing for (-, 0) and increasing for (0, ) f() = and lim f() = ; Chapter 27 Glencoe Precalculus Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 2: Press b and choose POINTS & LINES > POINT ON. Place two points anwhere on the graph. Double-click on each -coordinate, changing one to -2 and the other to 0. The -coordinates will update. You ma need to adjust our viewing window to see the points. Step 3: Press b and choose POINTS & LINES > LINE. Connect the points on the graph. Step : Press b and select MEASUREMENT > SLOPE. Choose the line. The slope is -0, so the average rate of change for the interval [-2, 0] is -0. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eercises Use the method shown above to find the average rate of change of each function on the given interval.. f() = ; [-, -2] 2. f() = ; [-, -6] f() = ; [, 5]. f() = ; [-2, -] f() = ; [0, 2] 6. f() = ; [-2, -] f() = ; [, 2]. f() = 2 - ; [, 5] -9 6 Chapter 26 Glencoe Precalculus Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter A2 Glencoe Precalculus

10 Answers (Lesson -5) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Lesson -5 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Stud Guide and Intervention (continued) -5 Transformations of Parent Functions Parent functions can be transformed to create other members in a famil of graphs. k units up when k > 0. k units down when k < 0. g() = f() + k is the graph of f() translated Translations h units right when h > 0. h units left when h < 0. g() = f( - h) is the graph of f() translated reflected in the -ais. g() = -f() is the graph of f() Reflections reflected in the -ais. g() = f(-) is the graph of f() epanded verticall if a >. compressed verticall if 0 < a <. g() = a f() is the graph of f() Dilations -5 Practice. Use the graph of f() = to graph g() = Use the graph of f() = to graph g() = - 2. g() f() f() g() 3. Describe how the graph of f() = 2 and g() are related. Then write an equation for g(). g() is f() reflected in the -ais, translated unit right and unit up. g() = -( - ) 2 + g(). Identif the parent function f() of g() = Describe how the graphs of g() and f() are related. Then graph f() and g() on the same aes. The graph of g() is the graph of f() = stretched verticall and translated 2 units left and 3 units down. g() f() Graph f() = - if -3 + if -2 < 2. if 6 6. Use the graph of f() = 3 to graph g() = ( + ) 3. g() Chapter 29 Glencoe Precalculus compressed horizontall if a >. epanded horizontall if 0 < a <. g() = f(a) is the graph of f() Eample Identif the parent function f() of g() = - -, and describe how the graphs of g() and f() are related. Then graph f() and g() on the same aes. f() = The graph of g() is the graph of the square root function f() = reflected in the -ais and then translated one unit down. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. g() = - - Eercises Identif the parent function f() of g(), and describe how the graphs of g() and f() are related. Then graph f() and g() on the same aes.. g() = g() = The graph of g() is the graph of the square function f() = 2 epanded verticall and translated units down. The graph of g() is the graph of the absolute value function f() = compressed verticall and translated units left. Chapter 2 Glencoe Precalculus Answers Chapter A3 Glencoe Precalculus

11 Answers (Lesson -5) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Lesson -5 Word Problem Practice -5-5 Enrichment Rotations A rotation is a rigid transformation. A rotation turns a figure about a point a certain number of degrees. The rotation can be clockwise or counterclockwise. For this activit, assume all rotations are about the origin and in the counterclockwise direction. To rotate a point 90 about the origin, use the rule (, ) (-, ).. Rotate point A b 90 using the rule. Graph the point. Give the coordinates of A'. (-2, 3) 2. Rotate point A' b 90. Graph the point. Give the coordinates of A''. Then use the result to write a rule for rotating (, ) b 0. (-3, -2); (, ) (-, -) 3. Rotate point A'' b 90. Graph the point. Give the coordinates of A'''. Then use the result to write a rule for rotating (, ) b 270. (2, -3); (, ) (, -) ' To rotate a function, ou can plot several image points and then connect them. Graph each function. Then graph the function after it is rotated 90.. f() = f() = Graph each function. Then graph the function after it is rotated f() = f() = - -. The graph of the function f() = 2-3 is rotated 90. What function represents the rotated graph? f() = Chapter 3 Glencoe Precalculus 3. TAXES Graph the ta rates for the different incomes b using a step function.. AREA The width w of a rectangular plot of land with fied area A is modeled b the function w(l) = A, where l is the l length. Income Ta Rates for a Couple Filing Jointl Ta Rate (%) Limits of Taable Income ($) a. If the area is 000 square feet, describe the transformations of the parent function f() = used to graph w(). f() is epanded verticall. 0 to,200 5,20 to 99, ,60 to 5, ,75 to 27, ,05 and up 39.6 Source: Information Please Almanac Ta Rate (%) b. Describe a function of the length that could be used to find a minimum perimeter for a given area P(l) = 2l + 2 ( A l ) c. Is the function ou found in part b a transformation of f()? Eplain. No; sample answer: the are two different kinds of rational functions. d. Find the minimum perimeter for an area of 000 square feet ft Taable Income (thousands) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. HORIZON The function f() =.5 can be used to approimate the distance to the apparent horizon, or how far a person can see on a clear da, where f() is the distance in miles and is the person s elevation in feet. 2. GOLF The path of the flight of a golf ball can be modeled b h() = , where h() is the distance above the ground in ards and is the horizontal distance from the tee in ards. a. How does the graph of f() compare to the graph of its parent function? a. Describe the transformation of the parent function f() = 2 used to graph h(). It is the parent function compressed horizontall. b. The function g() =.2 is also used to approimate the distance to the horizon. How does the graph of g() compare to the graph of its parent function? h() is the graph of f() translated 0 units right, compressed verticall, refl ected in the -ais, and then translated 0 units up. It is the parent function epanded verticall. b. Suppose the same shot was made from a tee located 0 ards behind the original tee. Rewrite h() to reflect this change h() = - Chapter 30 Glencoe Precalculus Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter A Glencoe Precalculus

12 Answers (Lesson -6) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Lesson -6 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Stud Guide and Intervention -6 Operations with Functions Two functions can be added, subtracted, multiplied, or divided to form a new function. For the new function, the domain consists of the intersection of the domains of the two functions, ecluding values that make a denominator equal to zero. Eample Given f() = and g() = + 2, find each function and its domain. g) () ( g) f = f() g() = b. ( f a. (f + g)() (f + g) = f() + g() = = 2 - = - 3 ( - 3)( + 2) + 2 = The domains of f and g are both (-, ), but = -2 ields a zero in The domains of f and g are both (-, ), so the domain of (f + g) is (-, ). the denominator of ( g) f. So, the domain is { -2, }., find each function and its domain. -6 Stud Guide and Intervention (continued) Compositions of Functions In a function composition, the result of one function is used to evaluate a second function. Given functions f and g, the composite function f g can be described b the equation [f g]() = f[g()]. The domain of f g includes all -values in the domain of g for which g() is in the domain of f. Eample and [g f](). Given f() = and g() = + 2, find [f g]() [f g]() = f[g()] Defi nition of composite functions = f( + 2) Replace g() with + 2. = 3( + 2) 2 + 2( + 2) - Substitute + 2 for in f(). = 3( ) Simplif. = [g f]() = g(f()) Defi nition of composite functions = g( ) Replace f() with = ( ) + 2 Substitute for in g(). = Simplif. Eercises For each pair of functions, find [f g](), [g f](), and [f g]().. f() = 2 +, g() = f() = 3 2, g() = ; 2-5; ; ; f() = 3, g() = 5. f() = 2, g() = ; 5 3 ; ; + ; f() = 3-5, g() = f() = -, g() = ; ; 6 7. f() = 2-3, g() = ; ;. f() = -, g() = ; 2-5 ; -2 - ; - ; 0 Chapter 33 Glencoe Precalculus Given f() = 2-3 and g() = b. (f g)() Eample 2 a. (f - g)() (f - g) = f() - g() = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. (f g) = f() g() = ( 2-3) = - 3 The domain of f is (-, ) and the domain of g is (, 0) (0, ), so the domain of (f - g) is (, 0) (0, ). The domain of f is (-, ) and the domain of g is (, 0) (0, ), so the domain of (f - g) is (, 0) (0, ). Eercises () for each f() and g(). Find (f + g)(), (f - g)(), (f g)(), and ( g) f State the domain of each new function. 2. f() = 2 + 7, g() =. f() = 2 -, g() = 2 ; D = (-, 0) (0, ) ; D = [0, ) ; D = (-, 0) (0, ) ; D = [0, ) ; D = (-, 0) (0, ) ; D = [0, ) ; D = (0, ) ; D = (-, 0) (0, ) Chapter 32 Glencoe Precalculus Answers Chapter A5 Glencoe Precalculus

13 Answers (Lesson -6) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Lesson -6 Practice -6 Find (f + g)(), (f - g)(), (f g)(), and ( g) f () for each f() and g(). State the domain of each new function.. f() = and g() = f() = 3 and g() = , D = (-, ) 3 + +, D = [-, ) , D = (-, ) 3 - +, D = [-, ) , 3 +, D = [-, ) D = (-, ), D = (-, ) , D = { 6 5, } For each pair of functions, find [f g](), [g f](), and [f g](3). 3. f() = + 5 and g() = - 3. f() = and g() = 3-6 Word Problem Practice. MARCHING BAND Band members form a circle of radius r when the music starts. The march outward as the pla. The function f(t) = 2.5t gives the radius of the circle in feet after t seconds. Using g(r) = πr 2 for the area of the circle, write a composite function that gives the area of the circle after t seconds. Then find the area, to the nearest tenth, after seconds. g[f(t)] = 6.25πt 2 ; 3.2 ft 2 2. CANDLES A hobbist makes and sells candles at a local market. The function c(h) = h gives the number of candles she has made after h hours. The function f(c) = c gives the cost of making c candles. a. Write the composite function that gives the cost of candle making after h hours. f[c(h)] = 2 + h b. A sale reduces the cost of making c candles b 0%. Write the sale function s() and the composite function that gives the cost of candle making after h hours if materials are purchased during the sale. s() = 0.9; s{f[c(h)]} = h 3. SCIENCE The function t() = gives the temperature in degrees Celsius of the liquid in a beaker after seconds. Decompose the function into two separate functions, s() and r(), so that s(r()) = t(). Sample answer: s() = ; r() = 2. TRAVEL Two travelers are budgeting mone for the same trip. The first traveler s budget (in dollars) can be represented b f() = The second traveler s budget (in dollars) can be represented b g() = is the number of nights. a. Find (f + g)() and the relevant domain. (f + g)() = ; D = { 0, } b. What does the composite function in part a represent? the combined budget of both travelers c. Find (f + g)(7) and eplain what the value represents. $560; the combined amount that can be spent b the travelers on a 7-night trip d. Repeat parts a c for (g - f)(). a: (g - f)() = ; D = { 0, } b: how much more the second traveler can spend than the first c: $230; how much more the second traveler can spend on a 7-night trip 5. POPULATION The function p() = predicts the population of elk in a forest for the ears 200 through 205 where is the number of ears since Decompose the function into two separate functions, a() and b(), so that [a b]() = p() and a() is a quadratic function and b() is a linear function. Sample answer: a() = 2 2 ; b() = - 3 Chapter 35 Glencoe Precalculus + 2; + 2; ; ; f() = and g() = f() = and g() = ; ; ; ; 70 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find f g f() = - 2. f() = g() = 3 g() = { 2 3, } ; f g = 3-2 { ± 3, } ; f g = 2-3 Find two functions f and g such that h() = [f g](). Neither function ma be the identit function f() = h() = h() = Sample answer: f() = -, Sample answer: f() = 3, g() = 2-6 g() = +. RESTAURANT A group of three restaurant patrons order the same meal and drink and leave an % tip. Determine functions that represent the cost of all of the meals before tip, the actual tip, and the composition of the two functions that gives the cost for all of the meals including tip. f() = 3, where is the cost for one meal; g() =.; g(f()) = 3.5 Chapter 3 Glencoe Precalculus Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter A6 Glencoe Precalculus