BIG IDEAS MATH. Oklahoma Edition. Ron Larson Laurie Boswell. Erie, Pennsylvania BigIdeasLearning.com

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1 BIG IDEAS MATH Oklahoma Edition Ron Larson Laurie Boswell Erie, Pennslvania BigIdeasLearning.com

2 Linear Functions, Linear Sstems, and Matrices Interval Notation and Set Notation Parent Functions and Transformations Transformations of Linear and Absolute Value Functions Modeling with Linear Functions Solving Linear Sstems Solving Linear Sstems Using Technolog Basic Matri Operations Pizza Shop (p. 8) Prom (p. 7) SEE the Big Idea Café Epenses (p. ) Dirt Bike (p. ) Swimming (p. ) Book.indb 7//7 : PM

3 Maintaining Mathematical Proficienc Evaluating Epressions Eample Evaluate the epression ( ). ( ) = (9 ) Evaluate the power within parentheses. Evaluate. = 8 Multipl within parentheses. = Divide. = Subtract ( + ) ( + ). ( + 8) Transformations of Figures Eample Reflect the black rectangle in the -ais. Then translate the new rectangle units to the left and unit down. A B Move each verte units left and unit down. D C D C D C Take the opposite of each -coordinate. A B A B Graph the transformation of the figure. 7. Translate the rectangle unit right and units up. 8. Reflect the triangle in the -ais. Then translate units left. 9. Translate the trapezoid units down. Then reflect in the -ais.. ABSTRACT REASONING Give an eample to show wh the order of operations is important when evaluating a numerical epression. Is the order of transformations of figures important? Justif our answer. Dnamic Solutions available at BigIdeasMath.com

4 Mathematical Actions and Processes Mathematicall profi cient students use technological tools to eplore concepts. Using a Graphing Calculator Core Concept Standard and Square Viewing Windows A tpical screen on a graphing calculator has a height-to-width ratio of to. This means that when ou view a graph using the standard viewing window of to (on each ais), the graph will not be shown in its true perspective. WINDOW Xmin=- Xma= Xscl= Ymin=- Yma= Yscl= This is the standard viewing window. To view a graph in its true perspective, ou need to change to a square viewing window, where the tick marks on the -ais are spaced the same as the tick marks on the -ais. WINDOW Xmin=-9 Xma=9 Xscl= Ymin=- Yma= Yscl= This is a square viewing window. Using a Graphing Calculator Use a graphing calculator to graph =. SOLUTION In the standard viewing window, notice that the tick marks on the -ais are closer together than those on the -ais. This implies that the graph is not shown in its true perspective. In a square viewing window, notice that the tick marks on both aes have the same spacing. This implies that the graph is shown in its true perspective. Monitoring Progress Use a graphing calculator to graph the equation using the standard viewing window and a square viewing window. Describe an differences in the graphs.. =. = +. = +. =. =. =. This is the graph in the standard viewing window. This is the graph in a square viewing window. Determine whether the viewing window is square. Eplain , , , 8.,.,., Chapter Linear Functions, Linear Sstems, and Matrices

5 . Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers? A collection of objects is called a set. You can use braces { } to represent a set b listing its members or b using set-builder notation to define the set in terms of the properties of its members. For instance, the set of the numbers,, and can be denoted as {,, } List the members of the set in braces. and the set of all odd whole numbers can be denoted as { is a whole number and is odd} Set-builder notation which is read The set of all real numbers such that is a whole number and is odd. If all of the members of a set A are also members of a set B, then set A is a subset of set B. For instance, if set A = {a, b} and set B = {a, b, c, d}, then set A is a subset of set B. Writing Subsets in Set Notation ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, ou need to connect and communicate mathematical ideas. Work with a partner. Write all the nonempt subsets of each set. a. {, } b. {c, d} c. {,, } d. {e, f, g, h} Writing Subsets in Set Notation Work with a partner. Write each given subset of the real numbers in set-builder notation. Describe each set-subset relationship among these sets. a. the integers b. the whole numbers c. the natural numbers d. the rational numbers e. the irrational numbers f. the positive integers Writing Subsets in Set Notation Work with a partner. Write each indicated set of numbers using either braces to list its members or set-builder notation. Eplain our choice of notation. a. the whole numbers through b. the real numbers through c. the prime whole numbers d. the integers through Communicate Your Answer. When is it convenient to use set-builder notation to represent a set of numbers?. What are some relationships between subsets of the real numbers? Section. Interval Notation and Set Notation

6 . Lesson What You Will Learn Core Vocabular set, p. subset, p. endpoints, p. bounded interval, p. unbounded interval, p. set-builder notation, p. Represent intervals using interval notation. Represent intervals using set-builder notation. Using Interval Notation In mathematics, a collection of objects is called a set. You can use braces { } to represent a set b listing its members or elements. For instance, the set {,, } A set with three members contains the three numbers,, and. A set with no elements, the empt set (or null set), can be represented b empt braces, or with the smbol. Man other sets are also described in words, such as the set of real numbers. If all the members of a set A are also members of a set B, then set A is a subset of set B. The set of natural numbers {,,,,...} is a subset of the set of real numbers. The diagram shows several important subsets of the real numbers. UNDERSTANDING MATHEMATICAL TERMS The smbols represent subsets of the real numbers. R: Real numbers Q: Rational numbers Z: Integers W: Whole numbers N: Natural numbers Real Numbers (R) Rational Numbers (Q) Integers (Z) Whole Numbers (W) Natural Numbers (N) Irrational Numbers Man subsets of the real numbers can be represented as intervals on the real number line. Core Concept Bounded Intervals on the Real Number Line Let a and b be two real numbers such that a < b. Then a and b are the endpoints of four different bounded intervals on the real number line, as shown below. A bracket or closed circle indicates that the endpoint is included in the interval and a parenthesis or open circle indicates that the endpoint is not included in the interval. Inequalit Interval Notation Graph a b a, b a < < b (a, b) a < b a, b) a < b (a, b a a a a b b b b The length of an bounded interval, a, b, (a, b), a, b), or (a, b, is the distance between its endpoints: b a. An bounded interval has a fi nite length. An interval that does not have a finite length is called unbounded or infi nite. Chapter Linear Functions, Linear Sstems, and Matrices

7 Core Concept Unbounded Intervals on the Real Number Line Let a and b be real numbers. Each interval on the real number line shown below is called an unbounded interval. Inequalit Interval Notation Graph a a, ) > a (a, ) a a b (, b < b (, b) (, ) b b The smbols (infi nit) and (negative infi nit) are used to represent the unboundedness of intervals such as 7, ) and (, 7. Because these smbols do not represent real numbers, the are alwas enclosed b a parenthesis. Writing Interval Notation Write each interval in interval notation. a. b. > c. d. SOLUTION a. The graph of is the bounded interval,. b. The graph of > is the unbounded interval (, ). c. The graph represents all the real numbers between and, including the endpoint. This is the bounded interval, ). d. The graph represents all the real numbers less than or equal to. This is the unbounded interval (,. Monitoring Progress Write the interval in interval notation.. 7 < <.. Section. Interval Notation and Set Notation

8 Using Set-Builder Notation Another wa to represent intervals is to write them in set-builder notation. Core Concept Set-Builder Notation Set-builder notation uses smbols to define a set in terms of the properties of the members of the set. Set-builder notation { < b} Words the set of all real numbers such that is less than b Set-builder Notation Graph { a or > b} a b { a} a Using Set-Builder Notation Sketch the graph of each set of numbers. a. { < } b. { or > } SOLUTION a. The real numbers in the set satisf both > and. b. The real numbers in the set satisf either or >. UNDERSTANDING MATHEMATICAL TERMS The smbol denotes membership in a set. The epression Z means that is a member (or element) of the set of integers. Writing Set-Builder Notation Write the set of numbers in set-builder notation. a. the set of all integers greater than b. (, ) or (, ) SOLUTION a. is greater than and is an integer. { > and Z} b. can be an real number ecept. { } Monitoring Progress Sketch the graph of the set of numbers.. { < }. { or } Write the set of numbers in set-builder notation.. (, or (, ) 7. the set of all integers ecept Chapter Linear Functions, Linear Sstems, and Matrices

9 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE Two real numbers a and b are the of four different intervals on the real number line.. WHICH ONE DOESN T BELONG? The graph of which set of numbers does not belong with the other three? Eplain. (, > and { < } the set of all integers greater than and less than or equal to Monitoring Progress and Modeling with Mathematics In Eercises, use braces to list the elements of the set.. the set of whole numbers less than. the set of odd whole numbers less than. the set of integers greater than. the set of integers less than 8 In Eercises 7, write the interval in interval notation. (See Eample.) 7. < < 9 8. < < the real numbers from through. the real numbers between and In Eercises 7, sketch the graph of the set of numbers. (See Eample.) 7. { < < } 8. { } 9. { < or > }. { } In Eercises 8, write the set of numbers in set-builder notation. (See Eample.)., ). (, 98. (, or, ). (, or, ). the set of all integers less than. the set of all real numbers greater than 9 and less than 7. the set of all real numbers ecept 8. the set of all whole numbers ecept 9. ERROR ANALYSIS Describe and correct the error in rewriting the interval (, 8 in set-builder notation. { < 8}. ERROR ANALYSIS Describe and correct the error in rewriting the interval 7, ) in set-builder notation. { 7 or < } Section. Interval Notation and Set Notation 7

10 . MODELING WITH MATHEMATICS The elevation relative to sea level in the United States ranges from 8 feet in Death Valle, California to, feet on Mount McKinle, Alaska. Write the range of elevations in interval notation and in set-builder notation.. MODELING WITH MATHEMATICS The main floor of an auditorium ranges from feet below the stage to 8 feet above the stage. The floor of the balcon ranges from to 7 feet above the stage. Write the range of the floor levels relative to the stage in interval notation and in set-builder notation.. MAKING AN ARGUMENT Your friend sas that it is impossible to write the set { and, and W} in interval notation. Is our friend correct? Eplain.. NUMBER SENSE Write each set using braces to list the elements in interval notation and in set-builder notation. If not possible, eplain wh. a. the set of even whole numbers b. the set of real numbers less than c. the set of real numbers or more units from. THOUGHT PROVOKING Eplain how ou can add the same number to each member of the set of whole numbers to produce another important subset of the real numbers. 7. MATHEMATICAL CONNECTIONS You are marking a rectangular paintball zone that must be meters wide and have a perimeter of at least meters but not more than meters. Find the interval for the length of the rectangular paintball zone.. HOW DO YOU SEE IT? The graphs show the legal driving speeds (in miles per hour) on two different roadwas. A Legal speeds (miles per hour) B 7 8 Legal speeds (miles per hour) a. Write the legal driving speeds shown in Graph A in interval notation, in set-builder notation, and in words. b. Write the legal driving speeds shown in Graph B in interval notation, in set-builder notation, and in words. c. One of the roadwas is a state highwa and the other is a residential street. Which graph represents each roadwa? 8. MATHEMATICAL CONNECTIONS You have gallons of roof coating to appl to the roof of a mobile home that is feet wide. Twent gallons covers 7 to square feet. Find the interval for the length that ou will cover before ou need to bu more roof coating. ft Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Complete the table of values for the function f. Then graph the function. 9. f() =. f() = + f() f(). f() =. f() = 8 f() f() 8 Chapter Linear Functions, Linear Sstems, and Matrices

11 . Parent Functions and Transformations Essential Question What are the characteristics of some of the basic parent functions? Identifing Basic Parent Functions JUSTIFYING CONCLUSIONS To be proficient in math, ou need to justif our conclusions and communicate them clearl to others. Work with a partner. Graphs of eight basic parent functions are shown below. Classif each function as constant, linear, absolute value, quadratic, square root, cubic, reciprocal, or eponential. Justif our reasoning. a. b. c. d. e. f. g. h. Communicate Your Answer. What are the characteristics of some of the basic parent functions?. Write an equation for each function whose graph is shown in Eploration. Then use a graphing calculator to verif that our equations are correct. Section. Parent Functions and Transformations 9

12 . Lesson What You Will Learn Core Vocabular parent function, p. transformation, p. translation, p. reflection, p. vertical stretch, p. vertical shrink, p. Previous function domain range slope scatter plot Identif families of functions. Describe transformations of parent functions. Describe combinations of transformations. Identifing Function Families Functions that belong to the same famil share ke characteristics. The parent function is the most basic function in a famil. Functions in the same famil are transformations of their parent function. Core Concept Parent Functions Famil Constant Linear Absolute Value Quadratic Rule f() = f() = f() = f() = Graph Domain All real numbers All real numbers All real numbers All real numbers Range = All real numbers LOOKING FOR STRUCTURE You can also use function rules to identif functions. The onl variable term in f is an -term, so it is an absolute value function. Identifing a Function Famil Identif the function famil to which f belongs. Compare the graph of f to the graph of its parent function. SOLUTION The graph of f is V-shaped, so f is an absolute value function. The graph is shifted up and is narrower than the graph of the parent absolute value function. The domain of each function is all real numbers, but the range of f is and the range of the parent absolute value function is. f() = + Monitoring Progress. Identif the function famil to which g belongs. Compare the graph of g to the graph of its parent function. g() = ( ) Chapter Linear Functions, Linear Sstems, and Matrices

13 Describing Transformations A transformation changes the size, shape, position, or orientation of a graph. A translation is a transformation that shifts a graph horizontall and/or verticall but does not change its size, shape, or orientation. REMEMBER The slope-intercept form of a linear equation is = m + b, where m is the slope and b is the -intercept. Graphing and Describing Translations Graph g() = and its parent function. Then describe the transformation. SOLUTION The function g is a linear function with a slope of and a -intercept of. So, draw a line through the point (, ) with a slope of. The graph of g is units below the graph of the parent linear function f. f() = So, the graph of g() = is a vertical translation units down of the graph of the parent linear function. (, ) g() = A reflection is a transformation that flips a graph over a line called the line of refl ection. A reflected point is the same distance from the line of reflection as the original point but on the opposite side of the line. REMEMBER The function p() = is written in function notation, where p() is another name for. Graphing and Describing Reflections Graph p() = and its parent function. Then describe the transformation. SOLUTION The function p is a quadratic function. Use a table of values to graph each function. = = f() = p() = The graph of p is the graph of the parent function flipped over the -ais. So, p() = is a reflection in the -ais of the parent quadratic function. Monitoring Progress Graph the function and its parent function. Then describe the transformation.. g() = +. h() = ( ). n() = Section. Parent Functions and Transformations

14 Another wa to transform the graph of a function is to multipl all of the -coordinates b the same positive factor (other than ). When the factor is greater than, the transformation is a vertical stretch. When the factor is greater than and less than, it is a vertical shrink. Graphing and Describing Stretches and Shrinks Graph each function and its parent function. Then describe the transformation. a. g() = b. h() = REASONING ABSTRACTLY To visualize a vertical stretch, imagine pulling the points awa from the -ais. To visualize a vertical shrink, imagine pushing the points toward the -ais. SOLUTION a. The function g is an absolute value function. Use a table of values to graph the functions. g() = = = The -coordinate of each point on g is two times the -coordinate of the corresponding point on the parent function. f() = So, the graph of g() = is a vertical stretch of the graph of the parent absolute value function. b. The function h is a quadratic function. Use a table of values to graph the functions. = = f() = h() = The -coordinate of each point on h is one-half of the -coordinate of the corresponding point on the parent function. Chapter Linear Functions, Linear Sstems, and Matrices So, the graph of h() = is a vertical shrink of the graph of the parent quadratic function. Monitoring Progress Graph the function and its parent function. Then describe the transformation.. g() =. h() = 7. c() =.

15 Combinations of Transformations You can use more than one transformation to change the graph of a function. Describing Combinations of Transformations Use a graphing calculator to graph g() = + and its parent function. Then describe the transformations. SOLUTION 8 The function g is an absolute value function. f The graph shows that g() = + is a reflection in the -ais followed b a translation units left and units down of the graph of the parent absolute value function. g Modeling with Mathematics Time (seconds), Height (feet), The table shows the height of a dirt bike seconds after jumping off a ramp. What tpe of function can ou use to model the data? Estimate the height after.7 seconds. SOLUTION. Understand the Problem You are asked to identif the tpe of function that can model the table of values and then to find the height at a specific time.. Make a Plan Create a scatter plot of the data. Then use the relationship shown in the scatter plot to estimate the height after.7 seconds.. Solve the Problem Create a scatter plot. The data appear to lie on a curve that resembles a quadratic function. Sketch the curve. So, ou can model the data with a quadratic function. The graph shows that the height is about feet after.7 seconds.. Look Back To check that our solution is reasonable, analze the values in the table. Notice that the heights decrease after second. Because.7 is between. and, the height must be between feet and 8 feet. 8 < < Monitoring Progress Use a graphing calculator to graph the function and its parent function. Then describe the transformations. 8. h() = + 9. d() = ( ). The table shows the amount of fuel in a chainsaw over time. What tpe of function can ou use to model the data? When will the tank be empt? Time (minutes), Fuel remaining (fluid ounces), 9 Section. Parent Functions and Transformations

16 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The function f() = is the of f() =.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. What are the vertices of the figure after a reflection in the -ais, followed b a translation units right? What are the vertices of the figure after a translation units up and units right? What are the vertices of the figure after a translation units right, followed b a reflection in the -ais? What are the vertices of the figure after a translation units up, followed b a reflection in the -ais? Monitoring Progress and Modeling with Mathematics In Eercises, identif the function famil to which f belongs. Compare the graph of f to the graph of its parent function. (See Eample.).. f() = f() = f() = + f() = 7. MODELING WITH MATHEMATICS At 8: a.m., the temperature is F. The temperature increases F each hour for the net 7 hours. Graph the temperatures over time t (t = represents 8: a.m.). What tpe of function can ou use to model the data? Eplain. 8. MODELING WITH MATHEMATICS You purchase a car from a dealership for $,. The trade-in value of the car each ear after the purchase is given b the function f() =,. What tpe of function models the trade-in value? In Eercises 9 8, graph the function and its parent function. Then describe the transformation. (See Eamples and.) 9. g() = +. f() =. f() =. h() = ( + ). g() =. f() = +. h() =. g() = Chapter Linear Functions, Linear Sstems, and Matrices 7. f() = 8. f() =

17 In Eercises 9, graph the function and its parent function. Then describe the transformation. (See Eample.) 9. f() =. g() =. f() =. h() =. h() =. g() =. h() =. f() = In Eercises 7, use a graphing calculator to graph the function and its parent function. Then describe the transformations. (See Eample.) 7. f() = + 8. h() = + 9. h() =. f() = +. g() =. f() =. f() = ( + ) +. g() = ERROR ANALYSIS In Eercises and, identif and correct the error in describing the transformation of the parent function... 8 The graph is a reflection in the -ais and a vertical shrink of the parent quadratic function. The graph is a translation units right of the parent absolute value function, so the function is f() = +. MATHEMATICAL CONNECTIONS In Eercises 7 and 8, find the coordinates of the figure after the transformation. 7. Translate units 8. Reflect in the -ais. down. B A C D USING TOOLS In Eercises 9, identif the function famil and describe the domain and range. Use a graphing calculator to verif our answer. 9. g() = +. h() = + A C. g() = +. f() = +. f() =. f() = + B. MODELING WITH MATHEMATICS The table shows the speeds of a car as it travels through an intersection with a stop sign. What tpe of function can ou use to model the data? Estimate the speed of the car when it is ards past the intersection. (See Eample.) Displacement from sign (ards), Speed (miles per hour),. THOUGHT PROVOKING In the same coordinate plane, sketch the graph of the parent quadratic function and the graph of a quadratic function that has no -intercepts. Describe the transformation(s) of the parent function. 7. USING STRUCTURE Graph the functions f() = and g() =. Are the equivalent? Eplain. Section. Parent Functions and Transformations

18 8. HOW DO YOU SEE IT? Consider the graphs of f, g, and h. h f g a. Does the graph of g represent a vertical stretch or a vertical shrink of the graph of f? Eplain our reasoning. b. Describe how to transform the graph of f to obtain the graph of h. 9. MAKING AN ARGUMENT Your friend sas two different translations of the graph of the parent linear function can result in the graph of f() =. Is our friend correct? Eplain.. DRAWING CONCLUSIONS A person swims at a constant speed of meter per second. What tpe of function can be used to model the distance the swimmer travels? If the person has a -meter head start, what tpe of transformation does this represent? Eplain.. PROBLEM SOLVING You are plaing basketball with our friends. The height (in feet) of the ball above the ground t seconds after a shot is released from our hand is modeled b the function f(t) = t + t +.. a. Without graphing, identif the tpe of function that models the height of the basketball. b. What is the value of t when the ball is released from our hand? Eplain our reasoning. c. How man feet above the ground is the ball when it is released from our hand? Eplain.. MODELING WITH MATHEMATICS The table shows the batter lives of a computer over time. What tpe of function can ou use to model the data? Interpret the meaning of the -intercept in this situation. Time (hours), Batter life remaining, 8% % % % 8 %. REASONING Compare each function with its parent function. State whether it contains a horizontal translation, vertical translation, both, or neither. Eplain our reasoning. a. f() = b. f() = ( 8) c. f() = + + d. f() =. CRITICAL THINKING Use the values,,, and in the correct bo so the graph of each function intersects the -ais. Eplain our reasoning. a. f() = + b. f() = c. f() = + d. f() = Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Determine whether the ordered pair is a solution of the equation.. f() = + ; (, ). f() = ; (, ) 7. f() = ; (, ) 8. f() = ; (, 8) Find the -intercept and the -intercept of the graph of the equation. 9. =. = +. + =. = 8 Chapter Linear Functions, Linear Sstems, and Matrices

19 USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to visualize results and eplore consequences.. Transformations of Linear and Absolute Value Functions Essential Question How do the graphs of = f() + k, = f( h), and = f() compare to the graph of the parent function f? Work with a partner. Compare the graph of the function = + k to the graph of the parent function f() =. Transformations of the Parent Absolute Value Function Transformation Parent function = = + = Transformations of the Parent Absolute Value Function Work with a partner. Compare = the graph of the function = h to the graph of the parent function f() =. Transformation Parent function = + = Transformation of the Parent Absolute Value Function Work with a partner. Compare the graph of the function = = to the graph of the parent function f() =. Transformation Parent function = Communicate Your Answer. How do the graphs of = f () + k, = f( h), and = f() compare to the graph of the parent function f?. Compare the graph of each function to the graph of its parent function f. Use a graphing calculator to verif our answers are correct. a. = b. = + c. = d. = + e. = ( ) f. = Section. Transformations of Linear and Absolute Value Functions 7

20 . Lesson What You Will Learn Write functions representing translations and reflections. Write functions representing stretches and shrinks. Write functions representing combinations of transformations. Translations and Reflections You can use function notation to represent transformations of graphs of functions. Core Concept Horizontal Translations The graph of = f( h) is a horizontal translation of the graph of = f(), where h. Vertical Translations The graph of = f() + k is a vertical translation of the graph of = f(), where k. = f( h), h < = f() = f() + k, k > = f() = f( h), h > Subtracting h from the inputs before evaluating the function shifts the graph left when h < and right when h >. = f() + k, k < Adding k to the outputs shifts the graph down when k < and up when k >. Writing Translations of Functions Let f() = +. a. Write a function g whose graph is a translation units down of the graph of f. b. Write a function h whose graph is a translation units to the left of the graph of f. SOLUTION a. A translation units down is a vertical translation that adds to each output value. g() = f() + ( ) Add to the output. = + + ( ) Substitute + for f(). = Simplif. The translated function is g() =. Check h f g b. A translation units to the left is a horizontal translation that subtracts from each input value. h() = f( ( )) Subtract from the input. = f( + ) Add the opposite. = ( + ) + Replace with + in f(). = + Simplif. The translated function is h() = +. 8 Chapter Linear Functions, Linear Sstems, and Matrices

21 STUDY TIP When ou reflect a function in a line, the graphs are smmetric about that line. Core Concept Reflections in the -ais The graph of = f() is a reflection in the -ais of the graph of = f(). = f() = f() Reflections in the -ais The graph of = f( ) is a reflection in the -ais of the graph of = f(). = f( ) = f() Multipling the outputs b changes their signs. Multipling the inputs b changes their signs. Writing Reflections of Functions Let f() = + +. a. Write a function g whose graph is a reflection in the -ais of the graph of f. b. Write a function h whose graph is a reflection in the -ais of the graph of f. SOLUTION a. A reflection in the -ais changes the sign of each output value. g() = f() Multipl the output b. = ( + + ) Substitute + + for f(). = + Distributive Propert The reflected function is g() = +. Check b. A reflection in the -ais changes the sign of each input value. h() = f( ) Multipl the input b. f h = + + Replace with in f(). = ( ) + Factor out. g = + Product Propert of Absolute Value = + Simplif. The reflected function is h() = +. Monitoring Progress Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check our answer.. f() = ; translation units up. f() = ; translation units to the right. f() = + ; reflection in the -ais. f() = + ; reflection in the -ais Section. Transformations of Linear and Absolute Value Functions 9

22 STUDY TIP The graphs of = f( a) and = a f() represent a stretch or shrink and a reflection in the - or -ais of the graph of = f(). Stretches and Shrinks In the previous section, ou learned that vertical stretches and shrinks transform graphs. You can also use horizontal stretches and shrinks to transform graphs. Core Concept Horizontal Stretches and Shrinks The graph of = f(a) is a horizontal stretch or shrink b a factor of of the graph of a = f(), where a > and a. Multipling the inputs b a before evaluating the function stretches the graph horizontall (awa from the -ais) when < a <, and shrinks the graph horizontall (toward the -ais) when a >. Vertical Stretches and Shrinks The graph of = a f() is a vertical stretch or shrink b a factor of a of the graph of = f(), where a > and a. Multipling the outputs b a stretches the graph verticall (awa from the -ais) when a >, and shrinks the graph verticall (toward the -ais) when < a <. = f(a), a > = f() = f(a), < a < The -intercept stas the same. = a f(), a > = f() = a f(), < a < The -intercept stas the same. Writing Stretches and Shrinks of Functions Let f() =. Write (a) a function g whose graph is a horizontal shrink of the graph of f b a factor of, and (b) a function h whose graph is a vertical stretch of the graph of f b a factor of. SOLUTION a. A horizontal shrink b a factor of multiplies each input value b. g() = f() Multipl the input b. Check = Replace with in f(). g h f The transformed function is g() =. b. A vertical stretch b a factor of multiplies each output value b. h() = f() Multipl the output b. = ( ) Substitute for f(). = Distributive Propert The transformed function is h() =. Monitoring Progress Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check our answer.. f() = + ; horizontal stretch b a factor of. f() = ; vertical shrink b a factor of Chapter Linear Functions, Linear Sstems, and Matrices

23 Combinations of Transformations You can write a function that represents a series of transformations on the graph of another function b appling the transformations one at a time in the stated order. Combining Transformations Let the graph of g be a vertical shrink b a factor of. followed b a translation units up of the graph of f() =. Write a rule for g. SOLUTION Check Step First write a function h that represents the vertical shrink of f. h() =. f() Multipl the output b.. f =. Substitute for f(). g Step Then write a function g that represents the translation of h. 8 g() = h() + Add to the output. 8 =. + Substitute. for h(). The transformed function is g() =. +. Modeling with Mathematics You design a computer game. Your revenue for downloads is given b f() =. Your profit is $ less than 9% of the revenue for downloads. Describe how to transform the graph of f to model the profit. What is our profit for downloads? SOLUTION. Understand the Problem You are given a function that represents our revenue and a verbal statement that represents our profit. You are asked to find the profit for downloads.. Make a Plan Write a function p that represents our profit. Then use this function to find the profit for downloads.. Solve the Problem profit = 9% revenue Vertical shrink b a factor of.9 p() =.9 f() Translation units down =.9 Substitute for f(). =.8 Simplif. f p To find the profit for downloads, evaluate p when =. p() =.8() = =.8 Your profit is $ for downloads. X= Y=. Look Back The vertical shrink decreases the slope, and the translation shifts the graph units down. So, the graph of p is below and not as steep as the graph of f. Monitoring Progress 7. Let the graph of g be a translation units down followed b a reflection in the -ais of the graph of f () =. Write a rule for g. Use a graphing calculator to check our answer. 8. WHAT IF? In Eample, our revenue function is f() =. How does this affect our profit for downloads? Section. Transformations of Linear and Absolute Value Functions

24 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The function g() = is a horizontal of the function f() =.. WHICH ONE DOESN'T BELONG? Which transformation does not belong with the other three? Eplain our reasoning. Translate the graph of f() = + up units. Stretch the graph of f() = + verticall b a factor of. Shrink the graph of f() = + horizontall b a factor of. Translate the graph of f() = + left unit. Monitoring Progress and Modeling with Mathematics In Eercises 8, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check our answer. (See Eample.). f() = ; translation units to the left. f() = + ; translation units to the right. PROBLEM SOLVING You open a café. The function f() = represents our epected net income (in dollars) after being open weeks. Before ou open, ou incur an etra epense of $,. What transformation of f is necessar to model this situation? How man weeks will it take to pa off the etra epense?. f() = + + ; translation units down. f() = 9; translation units up 7. f() = + 8. f() = + f g 9. WRITING Describe two different translations of the graph of f that result in the graph of g. g f In Eercises, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check our answer. (See Eample.). f() = + ; reflection in the -ais. f() = ; reflection in the -ais f() = g() =. f() = ; reflection in the -ais. f() = + ; reflection in the -ais. f() = + ; reflection in the -ais. f() = + ; reflection in the -ais Chapter Linear Functions, Linear Sstems, and Matrices

25 In Eercises 7, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check our answer. (See Eample.) 7. f() = + ; vertical stretch b a factor of 8. f() = + ; vertical shrink b a factor of 9. f() = + ; horizontal shrink b a factor of. f() = + ; horizontal stretch b a factor of. f() = + (, ) g. f() = f (, ) (, ) In Eercises 7, write a function g whose graph represents the indicated transformations of the graph of f. (See Eample.) 7. f() = ; vertical stretch b a factor of followed b a translation unit up 8. f() = ; translation units down followed b a vertical shrink b a factor of 9. f() = ; translation units to the right followed b a horizontal stretch b a factor of. f() = ; reflection in the -ais followed b a translation units to the right. f() =. f() = 8 f g 8 f g g f 8 ERROR ANALYSIS In Eercises and, identif and correct the error in writing the function g whose graph represents the indicated transformations of the graph of f. ANALYZING RELATIONSHIPS In Eercises, match the graph of the transformation of f with the correct equation shown. Eplain our reasoning. f. f () = ; translation units to the right followed b a translation units up g() = f () = ; translation units down followed b a vertical stretch b a factor of g() =... MAKING AN ARGUMENT Your friend claims that when writing a function whose graph represents a combination of transformations, the order is not important. Is our friend correct? Justif our answer. A. = f() B. = f() C. = f( + ) D. = f() + Section. Transformations of Linear and Absolute Value Functions

26 . MODELING WITH MATHEMATICS During a recent period of time, bookstore sales have been declining. The sales (in billions of dollars) can be modeled b the function f(t) = 7 t + 7., where t is the number of ears since. Suppose sales decreased at twice the rate. How can ou transform the graph of f to model the sales? Eplain how the sales in are affected b this change. (See Eample.) MATHEMATICAL CONNECTIONS For Eercises 7, describe the transformation of the graph of f to the graph of g. Then find the area of the shaded triangle. 7. f() = 8. f() = f g g. HOW DO YOU SEE IT? Consider the graph of f() = m + b. Describe the effect each transformation has on the slope of the line and the intercepts of the graph. f a. Reflect the graph of f in the -ais. b. Shrink the graph of f verticall b a factor of. c. Stretch the graph of f horizontall b a factor of. 9. f() = +. f() = f. REASONING The graph of g() = + is a reflection in the -ais, vertical stretch b a factor of, and a translation units down of the graph of its parent function. Choose the correct order for the transformations of the graph of the parent function to obtain the graph of g. Eplain our reasoning. Maintaining Mathematical Proficienc Evaluate the function for the given value of.. f() = + ; = 7. f() = ; = 8. f() = + ; = 9. f() = ; = Create a scatter plot of the data. g f. ABSTRACT REASONING The functions f() = m + b and g() = m + c represent two parallel lines. a. Write an epression for the vertical translation of the graph of f to the graph of g. b. Use the definition of slope to write an epression for the horizontal translation of the graph of f to the graph of g. f g. THOUGHT PROVOKING You are planning a cross-countr biccle trip of miles. Your distance d (in miles) from the halfwa point can be modeled b d = 7, where is the time (in das) and = represents June. Your plans are altered so that the model is now a right shift of the original model. Give an eample of how this can happen. Sketch both the original model and the shifted model.. CRITICAL THINKING Use the correct value,, or with a, b, and c so the graph of g() = a b + c is a reflection in the -ais followed b a translation one unit to the left and one unit up of the graph of f() = +. Eplain our reasoning. Reviewing what ou learned in previous grades and lessons. 8. f() 9 f() Chapter Linear Functions, Linear Sstems, and Matrices

27 . Modeling with Linear Functions Essential Question How can ou use a linear function to model and analze a real-life situation? Modeling with a Linear Function MODELING WITH MATHEMATICS To be proficient in math, ou need to routinel interpret our results in the contet of the situation. Work with a partner. A compan purchases a copier for $,. The spreadsheet shows how the copier depreciates over an 8-ear period. a. Write a linear function to represent the value V of the copier as a function of the number t of ears. b. Sketch a graph of the function. Eplain wh this tpe of depreciation is called straight line depreciation. c. Interpret the slope of the graph in the contet of the problem. Modeling with Linear Functions Work with a partner. Match each description of the situation with its corresponding graph. Eplain our reasoning A Year, t 7 8 B Value, V $, $,7 $9, $8, $7, $,7 $, $, $, a. A person gives $ per week to a friend to repa a $ loan. b. An emploee receives $. per hour plus $ for each unit produced per hour. c. A sales representative receives $ per da for food plus $. for each mile driven. d. A computer that was purchased for $7 depreciates $ per ear. A. B. 8 8 C. D Communicate Your Answer. How can ou use a linear function to model and analze a real-life situation?. Use the Internet or some other reference to find a real-life eample of straight line depreciation. a. Use a spreadsheet to show the depreciation. b. Write a function that models the depreciation. c. Sketch a graph of the function. Section. Modeling with Linear Functions

28 . Lesson Core Vocabular line of fit, p. 8 line of best fit, p. 9 correlation coefficient, p. 9 Previous slope slope-intercept form point-slope form scatter plot What You Will Learn Write equations of linear functions using points and slopes. Find lines of fit and lines of best fit. Writing Linear Equations Core Concept Writing an Equation of a Line Given slope m and -intercept b Given slope m and a point (, ) Use slope-intercept form: = m + b Use point-slope form: = m( ) Given points (, ) and (, ) First use the slope formula to find m. Then use point-slope form with either given point. Distance (miles) Asteroid DA 8 (, ) Time (seconds) REMEMBER An equation of the form = m indicates that and are in a proportional relationship. Writing a Linear Equation from a Graph The graph shows the distance Asteroid DA travels in seconds. Write an equation of the line and interpret the slope. The asteroid came within 7, miles of Earth in Februar,. About how long does it take the asteroid to travel that distance? SOLUTION From the graph, ou can see the slope is m = =.8 and the -intercept is b =. Use slope-intercept form to write an equation of the line. = m + b Slope-intercept form =.8 + Substitute.8 for m and for b. The equation is =.8. The slope indicates that the asteroid travels.8 miles per second. Use the equation to find how long it takes the asteroid to travel 7, miles. 7, =.8 Substitute 7, for. 8 Divide each side b.8. Because there are seconds in hour, it takes the asteroid about hour to travel 7, miles. Monitoring Progress. The graph shows the remaining balance on a car loan after making monthl paments. Write an equation of the line and interpret the slope and -intercept. What is the remaining balance after paments? Balance (thousands of dollars) 8 (, ) (, 8) Car Loan Number of paments Chapter Linear Functions, Linear Sstems, and Matrices

29 Modeling with Mathematics Lakeside Inn Number of students, Total cost, $ $8 $ 7 $ $7 Two prom venues charge a rental fee plus a fee per student. The table shows the total costs for different numbers of students at Lakeside Inn. The total cost (in dollars) for students at Sunview Resort is represented b the equation = +. Which venue charges less per student? How man students must attend for the total costs to be the same? SOLUTION. Understand the Problem You are given an equation that represents the total cost at one venue and a table of values showing total costs at another venue. You need to compare the costs.. Make a Plan Write an equation that models the total cost at Lakeside Inn. Then compare the slopes to determine which venue charges less per student. Finall, equate the cost epressions and solve to determine the number of students for which the total costs are equal.. Solve the Problem First find the slope using an two points from the table. Use (, ) = (, ) and (, ) = (, 8). m = 8 = = = Write an equation that represents the total cost at Lakeside Inn using the slope of and a point from the table. Use (, ) = (, ). = m( ) Point-slope form = ( ) Substitute for m,, and. = Distributive Propert = + Add to each side. Equate the cost epressions and solve. + = + Set cost epressions equal. = Combine like terms. = Divide each side b. Comparing the slopes of the equations, Sunview Resort charges $ per student, which is less than the $ per student that Lakeside Inn charges. The total costs are the same for students.. Look Back Notice that the table shows the total cost for students at Lakeside Inn is $. To check that our solution is correct, verif that the total cost at Sunview Resort is also $ for students. = () + Substitute for. = Simplif. Monitoring Progress. WHAT IF? Maple Ridge charges a rental fee plus a $ fee per student. The total cost is $9 for students. Describe the number of students that must attend for the total cost at Maple Ridge to be less than the total costs at the other two venues. Use a graph to justif our answer. Section. Modeling with Linear Functions 7

30 Finding Lines of Fit and Lines of Best Fit Data do not alwas show an eact linear relationship. When the data in a scatter plot show an approimatel linear relationship, ou can model the data with a line of fit. Core Concept Finding a Line of Fit Step Create a scatter plot of the data. Step Sketch the line that most closel appears to follow the trend given b the data points. There should be about as man points above the line as below it. Step Choose two points on the line and estimate the coordinates of each point. These points do not have to be original data points. Step Write an equation of the line that passes through the two points from Step. This equation is a model for the data. Finding a Line of Fit Femur length, Height, The table shows the femur lengths (in centimeters) and heights (in centimeters) of several people. Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate the height of a person whose femur is centimeters long. SOLUTION Step Create a scatter plot of the data. The data show a linear relationship. Step Sketch the line that most closel appears to fit the data. One possibilit is shown. Step Choose two points on the line. For the line shown, ou might choose (, 7) and (, 9). Step Write an equation of the line. First, find the slope. m = 9 7 = = =. Use point-slope form to write an equation. Use (, ) = (, 7). = m( ) Point-slope form 7 =.( ) Substitute for m,, and. 7 =. =. + 7 Use the equation to estimate the height of the person. Height (centimeters) Distributive Propert Add 7 to each side. Human Skeleton 8 (, 7) (, 9) Femur length (centimeters) =.() + 7 Substitute for. = 7. Simplif. The approimate height of a person with a -centimeter femur is 7. centimeters. 8 Chapter Linear Functions, Linear Sstems, and Matrices

31 The line of best fit is the line that lies as close as possible to all of the data points. Man technolog tools have a linear regression feature that ou can use to find the line of best fit for a set of data. The correlation coefficient, denoted b r, is a number from to that measures how well a line fits a set of data pairs (, ). When r is near, the points lie close to a line with a positive slope. When r is near, the points lie close to a line with a negative slope. When r is near, the points do not lie close to an line. Using a Graphing Calculator humerus femur Use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data in Eample. Estimate the height of a person whose femur is centimeters long. Compare this height to our estimate in Eample. SOLUTION Step Enter the data into two lists. L L L()= L Step Use the linear regression feature. The line of best fit is =. +. The value of r is close to. LinReg =a+b a=.7 b=.9987 r = r= Step Graph the regression equation with the scatter plot. Step Use the trace feature to find the value of when =. =. + ATTENDING TO PRECISION Be sure to analze the data values to help ou select an appropriate viewing window for our graph. X= Y= The approimate height of a person with a -centimeter femur is centimeters. This is less than the estimate found in Eample. Monitoring Progress. The table shows the humerus lengths (in centimeters) and heights (in centimeters) of several females. Humerus length, 8 7 Height, 9 a. Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate the height of a female whose humerus is centimeters long. b. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Estimate the height of a female whose humerus is centimeters long. Compare this height to our estimate in part (a). Section. Modeling with Linear Functions 9

32 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The linear equation = + is written in form.. VOCABULARY A line of best fit has a correlation coefficient of.98. What can ou conclude about the slope of the line? Monitoring Progress and Modeling with Mathematics In Eercises 8, use the graph to write an equation of the line and interpret the slope. (See Eample.). Tipping. Gasoline Tank Tip (dollars) (, ) 8 Cost of meal (dollars) Fuel (gallons) 8 9 (9, 9) Distance (miles) 9. MODELING WITH MATHEMATICS Two newspapers charge a fee for placing an advertisement in their paper plus a fee based on the number of lines in the advertisement. The table shows the total costs for different length advertisements at the Dail Times. The total cost (in dollars) for an advertisement that is lines long at the Greenville Journal is represented b the equation = +. Which newspaper charges less per line? How man lines must be in an advertisement for the total costs to be the same? (See Eample.) Dail Times. Savings Account Balance (dollars) (, ) Time (weeks) 7. Tping Speed Words tped (, ) (, ) Time (minutes). Tree Growth 8. Tree height (feet) Volume (cubic feet) Age (ears) Swimming Pool (, ) (, 8) Time (hours) Number of lines, Total cost, PROBLEM SOLVING While on vacation in Canada, ou notice that temperatures are reported in degrees Celsius. You know there is a linear relationship between Fahrenheit and Celsius, but ou forget the formula. From science class, ou remember the freezing point of water is C or F, and its boiling point is C or F. a. Write an equation that represents degrees Fahrenheit in terms of degrees Celsius. b. The temperature outside is C. What is this temperature in degrees Fahrenheit? c. Rewrite our equation in part (a) to represent degrees Celsius in terms of degrees Fahrenheit. d. The temperature of the hotel pool water is 8 F. What is this temperature in degrees Celsius? Chapter Linear Functions, Linear Sstems, and Matrices

33 ERROR ANALYSIS In Eercises and, describe and correct the error in interpreting the slope in the contet of the situation.. Balance (dollars) Savings Account (, ) (, ) Year The slope of the line is, so after 7 ears, the balance is $7. 7. MODELING WITH MATHEMATICS The data pairs (, ) represent the average annual tuition (in dollars) for public colleges in the United States ears after. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in. Interpret the slope and -intercept in this situation. (See Eample.) (,,8), (,,7), (,,88) (,,7), (,,8), (,,97) 8. MODELING WITH MATHEMATICS The table shows the numbers of tickets sold for a concert when different prices are charged. Write an equation of a line of fit for the data. Does it seem reasonable to use our model to predict the number of tickets sold when the ticket price is $8? Eplain.. Income (dollars) 8 Earnings (, ) (, ) Hours The slope is, so the income is $ per hour. Ticket price (dollars), 7 Tickets sold, 9 USING TOOLS In Eercises 9, use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Find and interpret the correlation coefficient. 9.. In Eercises, determine whether the data show a linear relationship. If so, write an equation of a line of fit. Estimate when = and eplain its meaning in the contet of the situation. (See Eample.).... Minutes walking, Calories burned, 7 7 Months, 9 8 Hair length (in.), 7 Hours, Batter life (%), 8 Shoe size, 8 8. Heart rate (bpm), OPEN-ENDED Give two real-life quantities that have (a) a positive correlation, (b) a negative correlation, and (c) approimatel no correlation. Eplain. Section. Modeling with Linear Functions

34 . HOW DO YOU SEE IT? You secure an interest-free loan to purchase a boat. You agree to make equal monthl paments for the net two ears. The graph shows the amount of mone ou still owe. Loan balance (hundreds of dollars) Boat Loan 8 Time (months) a. What is the slope of the line? What does the slope represent? b. What is the domain and range of the function? What does each represent? c. How much do ou still owe after making paments for months?. MATHEMATICAL CONNECTIONS Which equation has a graph that is a line passing through the point (8, ) and is perpendicular to the graph of = +? A = B = + 7 C = 7 D = 7. PROBLEM SOLVING You are participating in an orienteering competition. The diagram shows the position of a river that cuts through the woods. You are currentl miles east and mile north of our starting point, the origin. What is the shortest distance ou must travel to reach the river? 8 North = + East 7. MAKING AN ARGUMENT A set of data pairs has a correlation coefficient r =.. Your friend sas that because the correlation coefficient is positive, it is logical to use the line of best fit to make predictions. Is our friend correct? Eplain our reasoning. 8. THOUGHT PROVOKING Points A and B lie on the line = +. Choose coordinates for points A, B, and C where point C is the same distance from point A as it is from point B. Write equations for the lines connecting points A and C and points B and C. 9. ABSTRACT REASONING If and have a positive correlation, and and z have a negative correlation, then what can ou conclude about the correlation between and z? Eplain.. ANALYZING RELATIONSHIPS Data from North American countries show a positive correlation between the number of personal computers per capita and the average life epectanc in the countr. a. Does a positive correlation make sense in this situation? Eplain. b. Is it reasonable to conclude that giving residents of a countr personal computers will lengthen their lives? Eplain. Maintaining Mathematical Proficienc Solve the sstem of linear equations in two variables b elimination or substitution.. + = 7. + =. + = = 9 = = Reviewing what ou learned in previous grades and lessons. = = 8. = + = = + = Chapter Linear Functions, Linear Sstems, and Matrices

35 . Solving Linear Sstems Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent when it has at least one solution. A linear sstem is inconsistent when it has no solution. Recognizing Graphs of Linear Sstems Work with a partner. Match each linear sstem with its corresponding graph. Eplain our reasoning. Then classif the sstem as consistent or inconsistent. a. = b. = c. = + = + = + = A. B. C. Solving Sstems of Linear Equations Work with a partner. Solve each linear sstem b substitution or elimination. Then use the graph of the sstem below to check our solution. a. + = b. + = c. + = = + = + = FINDING AN ENTRY POINT To be proficient in math, ou need to look for entr points to the solution of a problem. Communicate Your Answer. How can ou determine the number of solutions of a linear sstem?. Suppose ou were given a sstem of three linear equations in three variables. Eplain how ou would approach solving such a sstem.. Appl our strateg in Question to solve the linear sstem. + + z = Equation z = Equation + z = Equation Section. Solving Linear Sstems

36 . Lesson Core Vocabular linear equation in three variables, p. sstem of three linear equations, p. solution of a sstem of three linear equations, p. ordered triple, p. Previous sstem of two linear equations What You Will Learn Visualize solutions of sstems of linear equations in three variables. Solve sstems of linear equations in three variables algebraicall. Solve real-life problems. Visualizing Solutions of Sstems A linear equation in three variables,, and z is an equation of the form a + b + cz = d, where a, b, and c are not all zero. The following is an eample of a sstem of three linear equations in three variables. + 8z = Equation + + z = Equation + z = Equation A solution of such a sstem is an ordered triple (,, z) whose coordinates make each equation true. The graph of a linear equation in three variables is a plane in three-dimensional space. The graphs of three such equations that form a sstem are three planes whose intersection determines the number of solutions of the sstem, as shown in the diagrams below. Eactl One Solution The planes intersect in a single point, which is the solution of the sstem. Infinitel Man Solutions The planes intersect in a line. Ever point on the line is a solution of the sstem. The planes could also be the same plane. Ever point in the plane is a solution of the sstem. No Solution There are no points in common with all three planes. Chapter Linear Functions, Linear Sstems, and Matrices

37 Solving Sstems of Equations Algebraicall The algebraic methods ou used to solve sstems of linear equations in two variables can be etended to solve a sstem of linear equations in three variables. LOOKING FOR STRUCTURE The coefficient of in Equation makes a convenient variable to eliminate. ANOTHER WAY In Step, ou could also eliminate to get two equations in and z, or ou could eliminate z to get two equations in and. Core Concept Solving a Three-Variable Sstem Step Rewrite the linear sstem in three variables as a linear sstem in two variables b using the substitution or elimination method. Step Solve the new linear sstem for both of its variables. Step Substitute the values found in Step into one of the original equations and solve for the remaining variable. When ou obtain a false equation, such as =, in an of the steps, the sstem has no solution. When ou do not obtain a false equation, but obtain an identit such as =, the sstem has infinitel man solutions. Solving a Three-Variable Sstem (One Solution) Solve the sstem. + + z = Equation SOLUTION + z = 7 Equation + z = Equation Step Rewrite the sstem as a linear sstem in two variables. + + z = Add times Equation to + 8z = Equation (to eliminate ). + z = New Equation + z = z = 9 Add times Equation to Equation (to eliminate ). 7z = New Equation Step Solve the new linear sstem for both of its variables. + z = Add new Equation 7z = and new Equation. z = 8 z = Solve for z. = Substitute into new Equation or to find. Step Substitute = and z = into an original equation and solve for. + z = Write original Equation. ( ) + () = Substitute for and for z. = Solve for. The solution is =, =, and z =, or the ordered triple (,, ). Check this solution in each of the original equations. Section. Solving Linear Sstems

38 Solving a Three-Variable Sstem (No Solution) Solve the sstem. + + z = Equation + + z = Equation + z = Equation SOLUTION Step Rewrite the sstem as a linear sstem in two variables. z = Add times Equation + + z = to Equation. = 7 Because ou obtain a false equation, the original sstem has no solution. ANOTHER WAY Subtracting Equation from Equation gives z =. After substituting for z in each equation, ou can see that each is equivalent to = +. Solving a Three-Variable Sstem (Man Solutions) Solve the sstem. + z = Equation SOLUTION z = Equation + z = Equation Step Rewrite the sstem as a linear sstem in two variables. + z = z = Add Equation to Equation (to eliminate z). = New Equation z = + z = Add Equation to Equation (to eliminate z). = 8 New Equation Step Solve the new linear sstem for both of its variables. + = 8 Add times new Equation = 8 to new Equation. = Because ou obtain the identit =, the sstem has infinitel man solutions. Step Describe the solutions of the sstem using an ordered triple. One wa to do this is to solve new Equation for to obtain = +. Then substitute + for in original Equation to obtain z =. So, an ordered triple of the form (, +, ) is a solution of the sstem. Monitoring Progress Solve the sstem. Check our solution, if possible.. + z =. + z =. + + z = 8 + z = 7 + z = + z = z = z = + + z =. In Eample, describe the solutions of the sstem using an ordered triple in terms of. Chapter Linear Functions, Linear Sstems, and Matrices

39 Solving Real-Life Problems Solving a Multi-Step Problem B LAWN B B B A A A B An amphitheater charges $7 for each seat in Section A, $ for each seat in Section B, and $ for each lawn seat. There are three times as man seats in Section B as in Section A. The revenue from selling all, seats is $87,. How man seats are in each section of the amphitheater? STAGE SOLUTION Step Write a verbal model for the situation. Number of seats in B, = Number of seats in A, Number of seats in A, Number of + seats in B, Number of + lawn seats, z = Total number of seats 7 Number of seats in A, + Number of seats in B, + Number of lawn seats, z = Total revenue Step Write a sstem of equations. = Equation + + z =, Equation z = 87, Equation Step Rewrite the sstem in Step as a linear sstem in two variables b substituting for in Equations and. + + z =, Write Equation. + + z =, Substitute for. + z =, New Equation z = 87, Write Equation. 7 + () + z = 87, Substitute for. + z = 87, New Equation STUDY TIP When substituting to find values of other variables, choose original or new equations that are easiest to use. Step Solve the new linear sstem for both of its variables. z = 9, Add times new Equation + z = 87, to new Equation. = 8, = Solve for. = Substitute into Equation to find. z = 7, Substitute into Equation to find z. The solution is =, =, and z = 7,, or (,, 7,). So, there are seats in Section A, seats in Section B, and 7, lawn seats. Monitoring Progress. WHAT IF? On the first da,, tickets sold, generating $, in revenue. The number of seats sold in Sections A and B are the same. How man lawn seats are still available? Section. Solving Linear Sstems 7

40 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY The solution of a sstem of three linear equations is epressed as a(n).. WRITING Eplain how ou know when a linear sstem in three variables has infinitel man solutions. Monitoring Progress and Modeling with Mathematics In Eercises 8, solve the sstem using the elimination method. (See Eample.). + z =. + z = + + z = + z = 7 + z = 9 + z =. + z = 9. + z = z = z = z = + z = z = 8. + z = + z = 7 + z = + z = + + z =. + z =. + z = + z = + z = + z = z = z =. + z = + z = + z = z = z = 7. MODELING WITH MATHEMATICS Three orders are placed at a pizza shop. Two small pizzas, a liter of soda, and a salad cost $; one small pizza, a liter of soda, and three salads cost $; and three small pizzas, a liter of soda, and two salads cost $. How much does each item cost? ERROR ANALYSIS In Eercises 9 and, describe and correct the error in the first step of solving the sstem of linear equations. + z = z = + z = z = z = + z = 7 + z = 8 + z = + z = 8. MODELING WITH MATHEMATICS Sam s Furniture Store places the following advertisement in the local newspaper. Write a sstem of equations for the three combinations of furniture. What is the price of each piece of furniture? Eplain. SAM S Furniture Store Sofa and love seat In Eercises, solve the sstem using the elimination method. (See Eamples and.). + z =. + z = + z = z = + z = + = Sofa and two chairs Sofa, love seat, and one chair 8 Chapter Linear Functions, Linear Sstems, and Matrices

41 In Eercises 9 8, solve the sstem of linear equations using the substitution method. (See Eample.) z =. z = z = + + z = 7 + z = z = z =. + = + + z = + + z = + z = 9 + z =. + z =. = + z = + = z = + z =. + z =. z = + + z = z = + + z = 8 + z = z = 8. + z = z = + z = z = + z = 9. PROBLEM SOLVING The number of left-handed people in the world is one-tenth the number of righthanded people. The percent of right-handed people is nine times the percent of left-handed people and ambidetrous people combined. What percent of people are ambidetrous?. WRITING Eplain when it might be more convenient to use the elimination method than the substitution method to solve a linear sstem. Give an eample to support our claim.. REPEATED REASONING Using what ou know about solving linear sstems in two and three variables, plan a strateg for how ou would solve a sstem that has four linear equations in four variables. MATHEMATICAL CONNECTIONS In Eercises and, write and use a linear sstem to answer the question.. The triangle has a perimeter of feet. What are the lengths of sides, m, and n? = m m n = + m. What are the measures of angles A, B, and C? A (A C) A B (A + B) C. MODELING WITH MATHEMATICS Use a sstem of linear equations to model the data in the following newspaper article. Solve the sstem to find how man athletes finished in each place. Lawrence High prevailed in Saturda s track meet with the help of individual-event placers earning a combined 8 points. A first-place finish earns points, a secondplace finish earns points, and a third-place finish earns point. Lawrence had a strong second-place showing, with as man second place finishers as first- and third-place finishers combined.. OPEN-ENDED Consider the sstem of linear equations below. Choose nonzero values for a, b, and c so the sstem satisfies the given condition. Eplain our reasoning. + + z = a + b + cz = + z = a. The sstem has no solution. b. The sstem has eactl one solution. c. The sstem has infinitel man solutions.. MAKING AN ARGUMENT A linear sstem in three variables has no solution. Your friend concludes that it is not possible for two of the three equations to have an points in common. Is our friend correct? Eplain our reasoning. Section. Solving Linear Sstems 9

42 7. PROBLEM SOLVING A contractor is hired to build an. HOW DO YOU SEE IT? Determine whether the apartment comple. Each 8-square-foot unit has a bedroom, kitchen, and bathroom. The bedroom will be the same size as the kitchen. The owner orders 98 square feet of tile to completel cover the floors of two kitchens and two bathrooms. Determine how man square feet of carpet is needed for each bedroom. BATHROOM sstem of equations that represents the circles has no solution, one solution, or infinitel man solutions. Eplain our reasoning. a. b. KITCHEN. CRITICAL THINKING Find the values of a, b, and c so that the linear sstem shown has (,, ) as its onl solution. Eplain our reasoning. BEDROOM Total Area: 8 ft + z = a + z = b + z = c 8. THOUGHT PROVOKING Does the sstem of linear equations have more than one solution? Justif our answer.. ANALYZING RELATIONSHIPS Determine which arrangement(s) of the integers,, and produce a solution of the linear sstem that consist of onl integers. Justif our answer. + + z = + z = z = + z = + + z = + z = 9. PROBLEM SOLVING A florist must make identical bridesmaid bouquets for a wedding. The budget is $, and each bouquet must have flowers. Roses cost $. each, lilies cost $ each, and irises cost $ each. The florist wants twice as man roses as the other two tpes of flowers combined.. ABSTRACT REASONING Write a linear sstem to represent the first three pictures below. Use the sstem to determine how man tangerines are required to balance the apple in the fourth picture. Note: The first picture shows that one tangerine and one apple balance one grapefruit. a. Write a sstem of equations to represent this situation, assuming the florist plans to use the maimum budget b. Solve the sstem to find how man of each tpe of flower should be in each bouquet. c. Suppose there is no limitation on the total cost of the bouquets. Does the problem still have eactl one solution? If so, find the solution. If not, give three possible solutions Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Simplif.. ( ). (m + ). (z ) 7. ( ) Write a function g described b the given transformation of f() =. 8. translation units to the left 9. reflection in the -ais. translation units up. vertical stretch b a factor of Book.indb Chapter Linear Functions, Linear Sstems, and Matrices 7//7 : PM

43 . Solving Linear Sstems Using Technolog Essential Question How can ou represent algebraic epressions using a coefficient matri? A matri is a rectangular arrangement of numbers. The dimensions of a matri with m rows and n columns are m n. So, the dimensions of matri A are. A = rows columns Writing Coefficient Matrices Work with a partner. Match each set of algebraic epressions with its coefficient matri. Eplain our reasoning. Sample Algebraic + + z Coefficient epressions: + z matri: a. + b. + z + + c. z d. + +z z z A. C. B. D. Writing Coefficient Matrices ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, ou need to analze mathematical relationships to connect mathematical ideas. Work with a partner. Write and enter the coefficient matri for each set of epressions into a graphing calculator. a. z b. z c. + + z + z + + 9z Communicate Your Answer. How can ou represent algebraic epressions using a coefficient matri?. Write the algebraic epressions that are represented b the coefficient matri. MATRIXA Section. Solving Linear Sstems Using Technolog

44 . Lesson What You Will Learn Core Vocabular matri, p. dimensions of a matri, p. elements of a matri, p. augmented matri, p. Previous sstem of three linear equations Write augmented matrices for sstems of linear equations. Use technolog to solve sstems of linear equations in three variables. Writing Augmented Matrices for Sstems A matri is a rectangular arrangement of numbers. The dimensions of a matri with m rows and n columns are m n (read m b n ). So, the dimensions of matri A are. The numbers in the matri are its elements. A = columns rows The element in the first row and third column is. A matri derived from a sstem of linear equations (each written in standard form with the constant term on the right) is the augmented matri of the sstem. For eample, the sstem below can be represented b the given augmented matri. Sstem: + z = Augmented + + z = matri: + z = coefficients constants Before ou write an augmented matri, make sure each equation in the sstem is written in standard form. Include zeros for the coefficients of an missing variables. This determines the order of the constants and coefficients in the augmented matri. Writing an Augmented Matri Write an augmented matri for the sstem. Then state the dimensions. SOLUTION = + = Begin b rewriting each equation in the sstem in standard form. + = + = Net, use the coefficients and constants as elements of the augmented matri. The augmented matri has two rows and three columns, so the dimensions are Chapter Linear Functions, Linear Sstems, and Matrices

45 Writing an Augmented Matri COMMON ERROR Because the second equation does not have a z-term, the coefficient of z is. Write an augmented matri for the sstem. Then state the dimensions. = 7 z 9 + = + z = SOLUTION Begin b rewriting each equation in the sstem in standard form. + z = z = + z = Net, use the coefficients and constants as elements of the augmented matri The augmented matri has three rows and four columns, so the dimensions are. Monitoring Progress Write an augmented matri for the sstem. Then state the dimensions.. 8 =. + = z z = + = z = + z = + + 8z = + = Solving Sstems of Equations Using Technolog Man technolog tools have matri features that ou can use to solve a sstem of linear equations. The augmented matri in Eample, rewritten in reduced row-echelon form, is shown below. Observe that the solution to this sstem is (,, z) = (.,., ). You can verif this solution in the original sstem Core Concept Solving a Linear Sstem Using Technolog Step Write an augmented matri for the linear sstem. Step Enter the augmented matri into our graphing calculator. Step Use the reduced row-echelon form feature to rewrite the sstem. Step Interpret the result from Step to solve the linear sstem. Section. Solving Linear Sstems Using Technolog

46 Solving a Sstem Using Technolog Use a graphing calculator to solve the sstem. + + z = + + z = z = REMEMBER An m n matri has m rows and n columns. SOLUTION Step Write an augmented matri for the linear sstem. Step Enter the dimensions and elements of the augmented matri into our graphing calculator NAMES :A :B :C :D :E :F 7:G MATH EDIT MATRIXA Step Use the reduced row-echelon form feature to rewrite the sstem. NAMES MATH :randm( 7:augment 8:Matr list( 9:List matr( :cumsum A:ref( B:rref( EDIT rref(a - Step Converting the matri back to a sstem of linear equations, ou have: = = z = The solution is =, =, and z =, or the ordered triple (,, ). Check this solution in each of the original equations. Check + + z = + + z = z = + + =? ( ) + () + =? () =? = = = Monitoring Progress Use a graphing calculator to solve the sstem.. + z =. + z =. + z = + z = + z = z = 8 + z = + z = + z = Chapter Linear Functions, Linear Sstems, and Matrices

47 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETING THE SENTENCE A matri derived from a sstem of linear equations is called the matri of the sstem.. WRITING Describe how to find the solution of a sstem of linear equations in three variables using technolog. Monitoring Progress and Modeling with Mathematics In Eercises, write an augmented matri for the sstem. Then state the dimensions. (See Eamples and.). = 7. = = 7 = z =. + z = z = + + 9z = 8 z 8 = z = z = 8. + z = 9 z = + 8z = z = + + 9z = 9. + = z = z z = 8 + z = + 9 z = + z = ERROR ANALYSIS In Eercises and, describe and correct the error in writing an augmented matri for the sstem below z = 8 + = 9z + = The augmented matri is: The augmented matri is: In Eercises, use a graphing calculator to solve the sstem. (See Eample.). + + z = 7. + z = z = 7 + z = + z = 9 + z=. + + z = 9. + z = z = + + z = z = + 9z = 7. + z = z = + + z = + + z = + z = + z = z = z = z = + z = + + z = z =. + + z =. + + z = + + z = z = + + z = z = 8. MODELING WITH MATHEMATICS A compan sells three tpes of gift baskets. The basic basket has two movie passes and one package of microwave popcorn, and costs $.. The medium basket has two movie passes, two packages of microwave popcorn, and one DVD, and costs $7. The super basket has four movie passes, three packages of microwave popcorn, and two DVDs, and costs $7.. a. Write an augmented matri to represent the situation. b. Use a graphing calculator to find the cost of each basket item. Section. Solving Linear Sstems Using Technolog

48 . MODELING WITH MATHEMATICS You go shopping at a local department store with our friend and cousin. You bu one pair of jeans, four pairs of shorts, and two shirts for $8. Your friend bus two pairs of jeans, one pair of shorts, and three shirts for $7. Your cousin bus one pair of jeans, two pairs of shorts, and one shirt for $. a. Write an augmented matri to represent the situation. b. Use a graphing calculator to find the cost of each piece of clothing.. MODELING WITH MATHEMATICS You have 8 coins in nickels, dimes, and quarters with a combined value of $.. There are twice as man quarters as dimes. a. Write an augmented matri to represent the situation. b. Use a graphing calculator to find the number of each tpe of coin.. HOW DO YOU SEE IT? Write a sstem of equations for the augmented matri below MAKING AN ARGUMENT Your friend states that the number of rows in an augmented matri of the sstem will alwas be the same as the number of variables in the sstem. Is our friend correct? Eplain our reasoning. 8. USING TOOLS Use a graphing calculator to solve the sstem of four linear equations in four variables. w + + z = + 7z = w z = w + + z = 9. REASONING Is it possible to write more than one augmented matri for a sstem of linear equations? Eplain our reasoning.. MATHEMATICAL CONNECTIONS The sum of the measures of the angles in ABC is 8. The sum of the measures of angle B and angle C is twice the measure of angle A. The measure of angle B is less than the measure of angle C. a. Write an augmented matri to represent the situation. b. Use a graphing calculator to find the measures of the three angles.. ABSTRACT REASONING Let a, b, and c be real numbers. Classif the linear sstem represented b each matri as consistent or inconsistent. Eplain our reasoning. a. b. c a b c a b a b. THOUGHT PROVOKING Write an augmented matri, not in reduced row-echelon form, for a sstem that has eactl one solution, (,, z) = (,, ). Justif our answer. Maintaining Mathematical Proficienc Solve the inequalit. Graph the solution.. >. z w + 7 w +. r + < r + 7 Reviewing what ou learned in previous grades and lessons Graph the inequalit in a coordinate plane. 7. < 8. > Chapter Linear Functions, Linear Sstems, and Matrices

49 .7 Basic Matri Operations Essential Question How can ou add or subtract two matrices with the same dimensions? Adding and Subtracting Matrices Work with a partner. Enter matri A and matri B into a graphing calculator. A = 8 9 B = Use a graphing calculator to perform the indicated operation. a. A + B b. A B 7 c. Compare each element in the matri in part (a) with the corresponding elements in matri A and matri B. What do ou notice? d. Compare each element in the matri in part (b) with the corresponding elements in matri A and matri B. What do ou notice? Adding and Subtracting Matrices Work with a partner. Use the results of Eploration to perform the indicated operation without a calculator. Then, use a graphing calculator to check our answer. a. b LOOKING FOR A PATTERN To be proficient in math, ou need to investigate relationships, observe patterns, and use our observations to write general rules. Communicate Your Answer. How can ou add or subtract two matrices with the same dimensions?. Use matri A in Eploration and C = 7 to find A + C and A C. Can ou add or subtract two matrices with different dimensions?. In addition to adding and subtracting matrices, ou can also multipl a matri b a real number. Use a graphing calculator to find each product. Then write a rule describing how to multipl a matri b a real number. a. 7 b. 9 8 Section.7 Basic Matri Operations 7

50 .7 Lesson Core Vocabular scalar, p. 9 scalar multiplication, p. 9 Previous matri dimensions of a matri elements of a matri What You Will Learn Add and subtract matrices. Multipl a matri b a scalar. Use matrices to solve real-life problems. Adding and Subtracting Matrices Core Concept Adding and Subtracting Matrices To add or subtract two matrices with the same dimensions, add or subtract their corresponding elements. Adding Matrices a b c d + e f g h = a + e b + f c + g d + h Subtracting Matrices a c b d e g f h = a e c g b f d h The sum or difference of two matrices with different dimensions is undefined. Adding and Subtracting Matrices Perform the indicated operation, if possible. a. + b c. 8 SOLUTION a. b = = 7 = 8 = + + ( ) ( ) 7 ( ) 8 8 c. The dimensions of the first matri are, and the dimensions of the second matri are. Because the matrices have different dimensions, the sum is undefined. 8 Chapter Linear Functions, Linear Sstems, and Matrices

51 Core Concept Scalar Multiplication When working with matrices, a real number is often called a scalar. To multipl a matri b a scalar, multipl each element in the matri b the scalar. This process is called scalar multiplication. Multipling a Matri b a Scalar k k a c b d = ka kc kb kd Perform the indicated operation(s). a. b. + 7 Multipling a Matri b a Scalar STUDY TIP The order of operations for matri epressions is similar to that for real numbers. In Eample (b), perform scalar multiplication before matri addition. SOLUTION a. = b. = () () () + ( ) () () Monitoring Progress 7 = () ( ) () () + + = + ( ) = + = 7 9 Perform the indicated operation(s), if possible Section.7 Basic Matri Operations 9

52 Solving Real-Life Problems Matrices are useful for organizing data and for performing the same operations on large numbers of data values. Modeling with Mathematics A high school district track and field organization used the same standard qualifing times for its -meter run district championships in two consecutive ears. The table below shows the numbers of bos and girls in the three class levels who qualified for the district championships in the two ears. Chapter Linear Functions, Linear Sstems, and Matrices Qualifiers: -Meter Run Class Bos 8 Girls 8 8 Organize the data using two matrices, one for and one for. Then find and interpret a matri giving the average number of qualifiers in each class and gender. SOLUTION Step Organize the data using two matrices. Step Class Class Class (A) (B) Bos Girls Bos Girls To find the average number of qualifiers in each class, add matri A and matri B, and then multipl the result b. ( (A + B) = 8 ) = = 8 Step Interpret the matri from Step. Over the two ears, the average number of qualifiers for the district championships in the -meter run were 7 bos and 8 girls in Class, bos and girls in Class, and bos and girls in Class. Monitoring Progress Monitoring Progress 7. Repeat Eample b organizing the data into two matrices. Compare the results.

53 .7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE You can add or subtract two matrices when the have the same.. WRITING Eplain how to multipl a matri b a scalar. Monitoring Progress and Modeling with Mathematics In Eercises, perform the indicated operation, if possible. (See Eample.) ERROR ANALYSIS In Eercises 7 and 8, describe and correct the error in adding the matrices = = 7. In Eercises 9, perform the indicated operation(s). (See Eample.) In Eercises 7, use the given matrices to evaluate the epression. A = 9 7, B = 7, C =.. 7. A + B 8. B A 9. B C. C + A. A + B + C. B C + A... MODELING WITH MATHEMATICS The table below shows the numbers (in thousands) of vehicles sold b tpe and compan in and. (See Eample.) Sales Tpe Compan A Car 8 Truck 78 Utilit Compan B Car 9 Truck 7 Utilit 79 7 a. Organize the information using two matrices that represent the numbers of vehicles sold b tpe and compan in and. b. Find a matri giving the average sales for each tpe and compan. c. Interpret the matri from part (b). Section.7 Basic Matri Operations

54 . MODELING WITH MATHEMATICS The table below shows the numbers (in thousands) of vehicles sold b tpe and compan in and. Sales Tpe Compan A Car 9 Motorccle Truck 9 Utilit 9 8 Compan B Car 9 Motorccle 9 Truck 8 Utilit 89 9 a. Organize the information using two matrices that represent the numbers of vehicles sold b tpe and compan in and. b. Find a matri giving the average sales for each tpe and compan. c. Interpret the matri from part (b).. MAKING AN ARGUMENT Your friend sas that the sum of 7 and is 7. Is our friend correct? Eplain.. HOW DO YOU SEE IT? A corporation has three factories that manufacture acoustic guitars and electric guitars. The production levels are represented b matri A. A = 7 Factor A B C Acoustic 7 Electric Guitar tpe a. Interpret the element in the second row and second column. b. How could ou find the production levels when production increases b %? 7. MODELING WITH MATHEMATICS The table below shows the populations (in thousands) of a cit b age group and gender for and. Populations Age Group Male Female a. Organize the information using two matrices that represent the populations b age group and gender for and. b. Find a matri giving the change in the population for each age group and gender from to. c. Interpret the matri from part (b). REASONING In Eercises 8, find the matri X that makes the equation true. 8. X + = 9. X = 9. X + =. X = 9. THOUGHT PROVOKING Write two matrices A and B such that A + B = 7 8 and 8 A B =. Maintaining Mathematical Proficienc Factor the polnomial. Reviewing what ou learned in previous grades and lessons Chapter Linear Functions, Linear Sstems, and Matrices

55 Chapter Review Dnamic Solutions available at BigIdeasMath.com. Interval Notation and Set Notation (pp. 8) a. Write in interval notation. The graph of is the unbounded interval, ). b. Write the set of integers from through 8 in set-builder notation. is in the interval, 8 and is an integer: { 8, Z} Write the interval in interval notation.. 8. >. 8 <. Write the set of integers less than or greater than in set-builder notation.. Parent Functions and Transformations (pp. 9 ) Graph g() = ( ) + and its parent function. Then describe the transformation. The function g is a quadratic function. f g The graph of g is a translation units right and unit up of the graph of the parent quadratic function. Graph the function and its parent function. Then describe the transformation.. f() = +. g() = 7. h() = 8. h() = 9. f() =. g() = ( + ). Transformations of Linear and Absolute Value Functions (pp. 7 ) Let the graph of g be a translation units to the right followed b a reflection in the -ais of the graph of f() =. Write a rule for g. Step First write a function h that represents the translation of f. h() = f( ) Subtract from the input. = Replace with in f(). Step Then write a function g that represents the reflection of h. g() = h( ) Multipl the input b. = Replace with in h(). = ( + ) Factor out. = + Product Propert of Absolute Value = + Simplif. The transformed function is g() = +. Chapter Chapter Review

56 Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check our answer.. f() = ; reflection in the -ais followed b a translation units to the left. f() = ; vertical shrink b a factor of followed b a translation units up. f() = ; translation units down followed b a reflection in the -ais. Modeling with Linear Functions (pp. ) The table shows the numbers of ice cream cones sold for different outside temperatures (in degrees Fahrenheit). Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate how man ice cream cones are sold when the temperature is F. Temperature, Number of cones, Step Create a scatter plot of the data. The data show a linear relationship. Step Sketch the line that appears to most closel fit the data. One possibilit is shown. Step Choose two points on the line. For the line shown, ou might choose (7, 7) and (9, 7). Step Write an equation of the line. First, find the slope. Number of cones Ice Cream Cones Sold (9, 7) (7, 7) 8 m = 7 7 = 9 7 = =. Use point-slope form to write an equation. Use (, ) = (7, 7). 8 Temperature ( F) = m( ) Point-slope form 7 =.( 7) Substitute for m,, and. 7 =. Distributive Propert =. + Add 7 to each side. Use the equation to estimate the number of ice cream cones sold. =.() + Substitute for. = Simplif. Approimatel ice cream cones are sold when the temperature is F. Chapter Linear Functions, Linear Sstems, and Matrices

57 Write an equation of the line.. The table shows the total number (in billions) of U.S. movie admissions each ear for ears. Use a graphing calculator to find an equation of the line of best fit for the data. Year, 8 Admissions, You ride our bike and measure how far ou travel. After minutes, ou travel. miles. After minutes, ou travel. miles. Write an equation to model our distance. How far can ou ride our bike in minutes?. Solving Linear Sstems (pp. ) Solve the sstem. + z = Equation + z = Equation + z = Equation Step Rewrite the sstem as a linear sstem in two variables. + z = Add Equation to + z = Equation (to eliminate z). + = New Equation + z = + z = Add times Equation to Equation (to eliminate z). + = 9 New Equation Step Solve the new linear sstem for both of its variables. = Add times new Equation + = 9 to new Equation. = = Solve for. = Substitute into new Equation or to find. Step Substitute = and = into an original equation and solve for z. + z = Write original Equation. ( ) + z = Substitute for and for. z = Solve for z. The solution is =, =, and z =, or the ordered triple (,, ). Chapter Chapter Review

58 Solve the sstem. Check our solution, if possible z = 7. z = z = + + z = z = 9 + z = = z + + z = + z = 9. + z =. + z =. + z = + + 7z = = + + z = + 9 z = + z = z = 8. A school band performs a spring concert for a crowd of people. The revenue for the concert is $. There are more adults at the concert than students. How man of each tpe of ticket are sold? BAND CONCERT STUDENTS - $ ADULTS - $7 CHILDREN UNDER - $. Solving Linear Sstems Using Technolog (pp. ) Use a graphing calculator to solve the sstem. 8 + z = 7 Equation + + z = 7 Equation + z = Equation Step Step Step Write an augmented matri for the linear sstem Enter the dimensions and elements of the augmented matri into our graphing calculator. Use the reduced row-echelon form feature to rewrite the sstem. MATRIXA rref(a 7 - Step Converting back to a sstem of linear equations, ou have =, = 7, and z =. The solution to the sstem is (, 7, ). Use a graphing calculator to solve the sstem.. + z =. + z =. + z = + z = + z = + z = + z = 8 + z = + z = Chapter Linear Functions, Linear Sstems, and Matrices