Graphs and Functions

CHAPTER Graphs and Functions. Graphing Equations. Introduction to Functions. Graphing Linear Functions. The Slope of a Line. Equations of Lines Integrated Review Linear Equations in Two Variables.6 Graphing Piecewise- Defined Functions and Shifting and Reflecting Graphs of Functions.7 Graphing Linear Inequalities The linear equations and inequalities we eplored in Chapter are statements about a single variable. This chapter ea mines statements about two variables: linear equations and inequalities in two variables. We focus particularl on graphs of those equations and inequalities that lead to the notion of relation and to the notion of function, perhaps the single most important and useful concept in all of mathematics. We define online courses as courses in which at least 80% of the content is delivered online. Although there are man tpes of course deliver used b instructors, the bar graph below shows the increase in percent of students taking at least one online course. Notice that the two functions, f() and g(), both approimate the percent of students taking at least one online course. Also, for both functions, is the number of ears since 000. In Section., Eercises 8 86, we use these functions to predict the growth of online courses. Percent 0 0 0 Percent of Students Taking at Least One Online Course f().7. g() 0.07.9.9 0 0 00 00 00 00 006 007 008 009 Year Source: The College Board: Trends in Higher Education Series 6

Section. Graphing Equations 7. Graphing Equations S Plot Ordered Pairs. Determine Whether an Ordered Pair of Numbers Is a Solution to an Equation in Two Variables. Graph Linear Equations. Graph Nonlinear Equations. Graphs are widel used toda in newspapers, magazines, and all forms of newsletters. A few eamples of graphs are shown here. Percent 0 0 Percent of People Who Go to the Movies 0 7 8 9 0 9 0 9 60 & up Source: Motion Picture Association of America Ages Number of W-iFi-Enabled Cell Phones in U.S. (in millions) 60 0 0 00 80 60 0 0 Projected Growth of Wi-Fi-Enabled Cell Phones in U.S. 009 00 *0 *0 *0 *0 *0 Year Note: Data is from Chapter opener, shown here as a broken-line graph * (projected) To help us understand how to read these graphs, we will review their basis the rectangular coordinate sstem. Plotting Ordered Pairs on a Rectangular Coordinate Sstem One wa to locate points on a plane is b using a rectangular coordinate sstem, which is also called a Cartesian coordinate sstem after its inventor, René Descartes (96 60). A rectangular coordinate sstem consists of two number lines that intersect at right angles at their 0 coordinates. We position these aes on paper such that one number line is horizontal and the other number line is then vertical. The horizontal number line is called the -ais (or the ais of the abscissa), and the vertical number line is called the -ais (or the ais of the ordinate). The point of intersection of these aes is named the origin. Notice in the left figure on the net page that the aes divide the plane into four regions. These regions are called quadrants. The top-right region is quadrant I. Quadrants II, III, and IV are numbered counterclockwise from the first quadrant as shown. The -ais and the -ais are not in an quadrant. Each point in the plane can be located, or plotted, or graphed b describing its position in terms of distances along each ais from the origin. An ordered pair, represented b the notation (, ), records these distances.

8 CHAPTER Graphs and Functions -ais Quadrant II Quadrant I Origin A(, ) Origin Quadrant III Quadrant IV -ais B(, ) For eample, the location of point A in the above figure on the right is described as units to the left of the origin along the -ais and units upward parallel to the -ais. Thus, we identif point A with the ordered pair -,. Notice that the order of these numbers is critical. The -value - is called the -coordinate and is associated with the -ais. The -value is called the -coordinate and is associated with the -ais. Compare the location of point A with the location of point B, which corresponds to the ordered pair, -. Can ou see that the order of the coordinates of an ordered pair matters? Also, two ordered pairs are considered equal and correspond to the same point if and onl if their -coordinates are equal and their -coordinates are equal. Keep in mind that each ordered pair corresponds to eactl one point in the real plane and that each point in the plane corresponds to eactl one ordered pair. Thus, we ma refer to the ordered pair (, ) as the point (, ). EXAMPLE Plot each ordered pair on a Cartesian coordinate sstem and name the quadrant or ais in which the point is located. a., - b. 0, c. -, d. -, 0 e. a -, -b f..,. Solution The si points are graphed as shown. a., - lies in quadrant IV. b. 0, is on the -ais. c. -, lies in quadrant II. d. -, 0 is on the -ais. e. a -, -b is in quadrant III. f..,. is in quadrant I. (, ) (, 0) ( q, ) (0, ) (.,.) (, ) Plot each ordered pair on a Cartesian coordinate sstem and name the quadrant or ais in which the point is located. a., - b. 0, - c. -, d., 0 e. a -, -b f..,. Notice that the -coordinate of an point on the -ais is 0. For eample, the point with coordinates -, 0 lies on the -ais. Also, the -coordinate of an point on the -ais is 0. For eample, the point with coordinates 0, lies on the -ais. These points that lie on the aes do not lie in an quadrants.

Section. Graphing Equations 9 CONCEPT CHECK Which of the following correctl describes the location of the point, - 6 in a rectangular coordinate sstem? a. units to the left of the -ais and 6 units above the -ais b. units above the -ais and 6 units to the left of the -ais c. units to the right of the -ais and 6 units below the -ais d. units below the -ais and 6 units to the right of the -ais Man tpes of real-world data occur in pairs. Stud the graph below and notice the paired data 0, 7 and the corresponding plotted point, both in blue. Percent of Sales Completed Using Cards* 60 0 (0, 7) Percent 0 0 0 0 98 99 00 0 (projected) Year Source: The Nilson Report *These include credit or debit cards, prepaid cards, and EBT (electronic benefits transfer) cards. This paired data point, 0, 7, means that in the ear 0, it is predicted that 7% of sales will be completed using some tpe of card (credit, debit, etc.). Determining Whether an Ordered Pair Is a Solution Solutions of equations in two variables consist of two numbers that form a true statement when substituted into the equation. A convenient notation for writing these numbers is as ordered pairs. A solution of an equation containing the variables and is written as a pair of numbers in the order,. If the equation contains other variables, we will write ordered pair solutions in alphabetical order. EXAMPLE Determine whether 0, -,, 9, and, -6 are solutions of the equation - =. Solution To check each ordered pair, replace with the -coordinate and with the -coordinate and see whether a true statement results. Let = 0 and = -. - = 0 - - 0 + = True Let = and = 9. - = - 9-9 -6 = False Let = and = -6. - = - -6 6 + 6 = True Thus,, 9 is not a solution of - =, but both 0, - and, -6 are solutions. Answer to Concept Check: c Determine whether,, 0, 6, and, - are solutions of the equation + = 8.

0 CHAPTER Graphs and Functions Graphing Linear Equations The equation - =, from Eample, actuall has an infinite number of ordered pair solutions. Since it is impossible to list all solutions, we visualize them b graphing. A few more ordered pairs that satisf - = are, 0,, -,,, and, -9. These ordered pair solutions along with the ordered pair solutions from Eample are plotted on the following graph. The graph of - = is the single line containing these points. Ever ordered pair solution of the equation corresponds to a point on this line, and ever point on this line corresponds to an ordered pair solution. (, ) (, 0) 6 7 - = (, ) # - = 0 # - 0 = 6 (, 6) 7 - # - - = 8-6 # 9 - -6 = (, 9) 0-9 # - -9 = 0 - # 0 - - = (0, ) The equation - = is called a linear equation in two variables, and the graph of ever linear equation in two variables is a line. Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B = C where A and B are not both 0. This form is called standard form. Some eamples of equations in standard form: - = -. +.6 = 0 Helpful Hint Remember: A linear equation is written in standard form when all of the variable terms are on one side of the equation and the constant is on the other side. Man real-life applications are modeled b linear equations. Suppose ou have a part-time job at a store that sells office products. Your pa is $000 plus 0% or of the price of the products ou sell. If we let represent products sold and represent monthl salar, the linear equation that models our salar is = 000 + (Although this equation is not written in standard form, it is a linear equation. To see this, subtract from both sides.)

Section. Graphing Equations Some ordered pair solutions of this equation are below. Products Sold 0 000 000 000 000 0,000 Monthl Salar 000 00 00 600 800 000 For eample, we sa that the ordered pair (000, 00) is a solution of the equation = 000 + because when is replaced with 000 and is replaced with 00, a true statement results. = 000 + 00 000 + 000 Let = 000 and = 00. 00 000 + 00 00 = 00 True A portion of the graph of = 000 + is shown in the net eample. Since we assume that the smallest amount of product sold is none, or 0, then must be greater than or equal to 0. Therefore, onl the part of the graph that lies in quadrant I is shown. Notice that the graph gives a visual picture of the correspondence between products sold and salar. Helpful Hint A line contains an infinite number of points and each point corresponds to an ordered pair that is a solution of its corresponding equation. EXAMPLE Use the graph of = 000 + to answer the following questions. a. If the salesperson sells $8000 of products in a particular month, what is the salar for that month? b. If the salesperson wants to make more than $000 per month, what must be the total amount of products sold? Solution a. Since is products sold, find 8000 along the -ais and move verticall up until ou reach a point on the line. From this point on the line, move horizontall to the left until ou reach the -ais. Its value on the -ais is 600, which means if $8000 worth of products is sold, the salar for the month is $600. 00 Monthl Salar (in dollars) 000 600 00 800 00 000 (000, 600) (000, 800) (000, 00) (000, 00) (0, 000) (0,000, 000) 0 000 000 6000 8000 0,000,000 Products Sold (in dollars) b. Since is monthl salar, find 000 along the -ais and move horizontall to the right until ou reach a point on the line. Either read the corresponding -value from

CHAPTER Graphs and Functions the labeled ordered pair or move verticall downward until ou reach the -ais. The corresponding -value is 0,000. This means that $0,000 worth of products sold gives a salar of $000 for the month. For the salar to be greater than $000, products sold must be greater that $0,000. Use the graph in Eample to answer the following questions. a. If the salesperson sells $6000 of products in a particular month, what is the salar for that month? b. If the salesperson wants to make more than $800 per month, what must be the total amount of products sold? Recall from geometr that a line is determined b two points. This means that to graph a linear equation in two variables, just two solutions are needed. We will find a third solution, just to check our work. To find ordered pair solutions of linear equations in two variables, we can choose an -value and find its corresponding -value, or we can choose a -value and find its corresponding -value. The number 0 is often a convenient value to choose for and for. EXAMPLE Graph the equation = - +. Solution This is a linear equation. (In standard form it is + =.) Find three ordered pair solutions, and plot the ordered pairs. The line through the plotted points is the graph. Since the equation is solved for, let s choose three -values. We ll choose 0,, and then - for to find our three ordered pair solutions. Let = 0 = - + = - # 0 + = Simplif. Let = = - + = - # + = - Simplif. The three ordered pairs (0, ),, -, and -, are listed in the table, and the graph is shown. 0 - - Let = - = - + = - - + = Simplif. (, ) 7 6 (0, ) (, ) Graph the equation = - -. Notice that the graph crosses the -ais at the point (0, ). This point is called the -intercept. (You ma sometimes see just the number called the -intercept.) This graph also crosses the -ais at the point a, 0b. This point is called the -intercept. (You ma also see just the number called the -intercept.) Since ever point on the -ais has an -value of 0, we can find the -intercept of a graph b letting = 0 and solving for. Also, ever point on the -ais has a -value of 0. To find the -intercept, we let = 0 and solve for.

Section. Graphing Equations Finding - and -Intercepts To find an -intercept, let = 0 and solve for. To find a -intercept, let = 0 and solve for. We will stud intercepts further in Section.. EXAMPLE Graph the linear equation =. Helpful Hint Notice that b using multiples of for, we avoid fractions. Helpful Hint Since the equation = is solved for, we choose -values for finding points. This wa, we simpl need to evaluate an epression to find the -value, as shown. Solution To graph, we find ordered pair solutions, plot the ordered pairs, and draw a line through the plotted points. We will choose -values and substitute in the equation. To avoid fractions, we choose -values that are multiples of. To find the -intercept, we let = 0. If = 0, then = 0, or 0. If = 6, then = 6, or. If = -, then = -, or -. This graph crosses the -ais at (0, 0) and the -ais at (0, 0). This means that the -intercept is (0, 0) and that the -intercept is (0, 0). = 0 0 6 - - Graph the linear equation = -. (0, 0) (, ) a (6, ) 6 7 Graphing Nonlinear Equations Not all equations in two variables are linear equations, and not all graphs of equations in two variables are lines. EXAMPLE 6 Graph =. Solution This equation is not linear because the term does not allow us to write it in the form A + B = C. Its graph is not a line. We begin b finding ordered pair solutions. Because this graph is solved for, we choose -values and find corresponding -values. If = -, then = -, or 9. If = -, then = -, or. If = -, then = -, or. If = 0, then = 0, or 0. If =, then =, or. If =, then =, or. If =, then =, or 9. - 9 - - 0 0 9 (, 9) (, ) (, ) 9 8 7 6 (, ) (, ) (, 9) Verte (0, 0) Stud the table a moment and look for patterns. Notice that the ordered pair solution (0, 0) contains the smallest -value because an other -value squared will give a positive result. This means that the point (0, 0) will be the lowest point on the graph. Also notice that all other -values correspond to two different -values. For eample, = 9, and also - = 9. This means that the graph will be a mirror image of itself across the -ais. Connect the plotted points with a smooth curve to sketch the graph.

CHAPTER Graphs and Functions This curve is given a special name, a parabola. We will stud more about parabolas in later chapters. 6 Graph =. EXAMPLE 7 Graph the equation = 0 0. Solution This is not a linear equation since it cannot be written in the form A + B = C. Its graph is not a line. Because we do not know the shape of this graph, we find man ordered pair solutions. We will choose -values and substitute to find corresponding -values. If = -, then = 0-0, or. If = -, then = 0-0, or. If = -, then = 0-0, or. If = 0, then = 0 0 0, or 0. If =, then = 0 0, or. If =, then = 0 0, or. If =, then = 0 0, or. - - - 0 0 (, ) (, ) (, ) (, ) (, ) (, ) (0, 0) Again, stud the table of values for a moment and notice an patterns. From the plotted ordered pairs, we see that the graph of this absolute value equation is V-shaped. 7 Graph = - 0 0. Graphing Calculator Eplorations In this section, we begin a stud of graphing calculators and graphing software packages for computers. These graphers use the same point plotting technique that we introduced in this section. The advantage of this graphing technolog is, of course, that graphing calculators and computers can find and plot ordered pair solutions much faster than we can. Note, however, that the features described in these boes ma not be available on all graphing calculators. The rectangular screen where a portion of the rectangular coordinate sstem is displaed is called a window. We call it a standard window for graphing when both the - and -aes displa coordinates between -0 and 0. This information is often displaed in the window menu on a graphing calculator as Xmin = -0 Xma = 0 Xscl = The scale on the @ais is one unit per tick mark. Ymin = -0 Yma = 0 Yscl = The scale on the @ais is one unit per tick mark. To use a graphing calculator to graph the equation = - +, press the Y = ke and enter the kestrokes

Section. Graphing Equations - +. c (Check our owner s manual to make sure the negative ke is pressed here and not the subtraction ke.) The top row should now read Y = - +. Net, press the GRAPH ke, and the displa should look like this: 0 0 0 0 Use a standard window and graph the following equations. (Unless otherwise stated, we will use a standard window when graphing.). = -. + 7.9. = - +.8. = -. = -. = 0-0 + 6. = 0 + 0-7. = + 8. = + Vocabular, Readiness & Video Check 0 0 Use the choices below to fill in each blank. Some choices ma not be used. line parabola origin V-shaped. The intersection of the -ais and -ais is a point called the.. The rectangular coordinate sstem has quadrants and aes.. The graph of a single ordered pair of numbers is how man points?. The graph of A + B = C, where A and B are not both 0, is a(n).. The graph of = looks. 6. The graph of = is a. Martin-Ga Interactive Videos See Video. Watch the section lecture video and answer the following questions. 7. Several points are plotted in Eamples. Where do ou start when using this method to plot a point? How does the st coordinate tell ou to move? How does the nd coordinate tell ou to move? Include the role of signs in our answer. 8. Based on Eamples and, complete the following statement. An ordered pair is a solution of an equation in variables if, when the variables are replaced with their ordered pair values, a statement results. 9. From Eample 6 and the lecture before, what is the graph of an equation? If ou need onl two points to determine a line, wh are three ordered pair solutions or points found for a linear equation? 0. Based on Eamples 7 and 8, complete the following statements. When graphing a nonlinear equation, first recognize it as a nonlinear equation and know that the graph is a line. If ou don t know the of the graph, plot enough points until ou see a pattern.

6 CHAPTER Graphs and Functions. Eercise Set Plot each point and name the quadrant or ais in which the point lies. See Eample.. (, )., -. -,. -, -. a, -b 6. a -, 6 b 7. (0,.) 8. -., 0 9. -, - 0. -., 0 Determine the coordinates of each point on the graph. See Eample.. Point C D. Point D. Point E. Point F E B A. Point B 6. Point A C F Determine whether each ordered pair is a solution of the given equation. See Eample. 7. = - ; 0,, -, -8 8. = - + 7;,, -, 9. -6 + = -6;, 0, a, 6 b 0. - = 9; 0,, a, -b. = ;,,, 8. = 0 0 ; -,, 0,. = ;, 8,, 9. = ; -,,, 6. = + ;,,, 6. = - ;, -, 8, 6 MIXED 0 0 0 0 Determine whether each equation is linear or not. Then graph the equation b finding and plotting ordered pair solutions. See Eamples through 7. 7. + = 8. - = 8 9. = 0. = 6. = -. = 6 -. = +. = +. - = 6. - = 7 7. = 8. = 9. = - 0. = +. = -. = -. = - +. = - +. = 0 + 0 6. = 0-0 7. = (Hint: Let = -, -, -, 0,,.) 8. = - (Hint: Let = -, -, -, 0,,.) 9. = - 0 0 0. = -. = -. = -. = - +. = - + REVIEW AND PREVIEW Solve the following equations. See Section... - + = 6-6 6. + 7 + = + 0 7. + = 0 8. 6 + = Solve the following inequalities. See Section.. 9. - 60. - 7 8 6. - 7 + 6. 9 + 8 6 - CONCEPT EXTENSIONS Without graphing, visualize the location of each point. Then give its location b quadrant or - or -ais. 6., - 6. -, 7 6. 0, -00 66. -87, 0 67. -0, -0 68. (0, 0) Given that is a positive number and that is a positive number, determine the quadrant or ais in which each point lies. 69., - 70. -, 7. (, 0) 7. 0, - 7. -, - 7. (0, 0) Solve. See the Concept Check in this section. 7. Which correctl describes the location of the point -,. in a rectangular coordinate sstem? a. unit to the right of the -ais and. units above the -ais b. unit to the left of the -ais and. units above the -ais c. unit to the left of the -ais and. units below the -ais d. unit to the right of the -ais and. units below the -ais 76. Which correctl describes the location of the point a0, - b in a rectangular coordinate sstem? a. on the -ais and unit to the left of the -ais b. on the -ais and unit to the right of the -ais c. on the -ais and unit above the -ais d. on the -ais and unit below the -ais

Section. Graphing Equations 7 For Eercises 77 through 80, match each description with the graph that best illustrates it. 77. Moe worked 0 hours per week until the fall semester started. He quit and didn t work again until he worked 60 hours a week during the holida season starting mid-december. 78. Kawana worked 0 hours a week for her father during the summer. She slowl cut back her hours to not working at all during the fall semester. During the holida season in December, she started working again and increased her hours to 60 hours per week. 79. Wend worked from Jul through Februar, never quitting. She worked between 0 and 0 hours per week. 80. Bartholomew worked from Jul through Februar. During the holida season between mid-november and the beginning of Januar, he worked 0 hours per week. The rest of the time, he worked between 0 and 0 hours per week. a. b. c. 60 0 0 0 0 0 60 0 0 0 0 0 A A S O N D J F S O N D J F d. 60 0 0 0 0 0 60 0 0 0 0 0 A A S O N D J F S O N D J F This broken-line graph shows the hourl minimum wage and the ears it increased. Use this graph for Eercises 8 through 8. Hourl Minimum Wage (in dollars) 8.0 7. 7.0 6. 6.0..0..0..0 $.0 The Federal Hourl Minimum Wage $.80 $. $.7 $. $6. $.8 $7. $. 980 8 8 86 88 90 9 9 96 98 00 0 0 06 08 0 Year Source: U.S. Department of Labor 8. What was the first ear that the minimum hourl wage rose above $.00? 8. What was the first ear that the minimum hourl wage rose above $6.00? 8. Wh do ou think that this graph is shaped the wa it is? 8. The federal hourl minimum wage started in 98 at $0.. How much will it have increased b 0? 8. Graph = - + 7. Let = 0,,,, to generate ordered pair solutions. 86. Graph = + +. Let = -, -, -, 0, to generate ordered pair solutions. 87. The perimeter of a rectangle whose width is a constant inches and whose length is inches is given b the equation = + 6 a. Draw a graph of this equation. b. Read from the graph the perimeter of a rectangle whose length is inches. inches inches 88. The distance traveled in a train moving at a constant speed of 0 miles per hour is given b the equation = 0 where is the time in hours traveled. a. Draw a graph of this equation. b. Read from the graph the distance traveled after 6 hours. For income ta purposes, the owner of Cop Services uses a method called straight-line depreciation to show the loss in value of a cop machine he recentl purchased. He assumes that he can use the machine for 7 ears. The following graph shows the value of the machine over the ears. Use this graph to answer Eercises 89 through 9. Value (in thousands of dollars) 7 6 6 7 Years 89. What was the purchase price of the cop machine? 90. What is the depreciated value of the machine in 7 ears? 9. What loss in value occurred during the first ear? 9. What loss in value occurred during the second ear? 9. Wh do ou think that this method of depreciating is called straight-line depreciation? 9. Wh is the line tilted downward? 9. On the same set of aes, graph =, = -, and = +. What patterns do ou see in these graphs? 96. On the same set of aes, graph =, =, and = -. Describe the differences and similarities in these graphs.

8 CHAPTER Graphs and Functions Write each statement as an equation in two variables. Then graph each equation. 97. The -value is more than three times the -value. 98. The -value is - decreased b twice the -value. 99. The -value is more than the square of the -value. 00. The -value is decreased b the square of the -value. Use a graphing calculator to verif the graphs of the following eercises. 0. Eercise 9 0. Eercise 0 0. Eercise 7 0. Eercise 8. Introduction to Functions S Define Relation, Domain, and Range. Identif Functions. Use the Vertical Line Test for Functions. Find the Domain and Range of a Function. Use Function Notation. Defining Relation, Domain, and Range Recall our eample from the last section about products sold and monthl salar. We modeled the data given b the equation = 000 +. This equation describes a relationship between -values and -values. For eample, if = 000, then this equation describes how to find the -value related to = 000. In words, the equation = 000 + sas that 000 plus of the -value gives the corresponding -value. The -value of 000 corresponds to the -value of 000 + # 000 = 00 for this equation, and we have the ordered pair (000, 00). There are other was of describing relations or correspondences between two numbers or, in general, a first set (sometimes called the set of inputs) and a second set (sometimes called the set of outputs). For eample, First Set: Input Correspondence Second Set: Output People in a certain cit Each person s age, to The set of nonnegative the nearest ear integers A few eamples of ordered pairs from this relation might be (Ana, ), (Bob, 6), (Tre, ), and so on. Below are just a few other was of describing relations between two sets and the ordered pairs that the generate. Correspondence First Set: Input a c e Second Set: Output Ordered Pairs a,, c,, e, Ordered Pairs -, -, (, ), (, ),, - Some Ordered Pairs (, ), (0, ), and so on Relation, Domain, and Range A relation is a set of ordered pairs. The domain of the relation is the set of all first components of the ordered pairs. The range of the relation is the set of all second components of the ordered pairs. For eample, the domain for our relation on the left above is a, c, e6 and the range is, 6. Notice that the range does not include the element of the second set.