Functions and Their Graphs

Transcription

1 Functions and Their Graphs. Rectangular Coordinates. Graphs of Equations. Linear Equations in Two Variables. Functions.5 Analzing Graphs of Functions. A Librar of Parent Functions.7 Transformations of Functions.9 Inverse Functions.8 Combinations of Functions:.0 Mathematical Modeling and Variation Composite Functions In Mathematics Functions show how one variable is related to another variable. In Real Life Functions are used to estimate values, simulate processes, and discover relationships. For instance, ou can model the enrollment rate of children in preschool and estimate the ear in which the rate will reach a certain number. Such an estimate can be used to plan measures for meeting future needs, such as hiring additional teachers and buing more books. (See Eercise, page.) Jose Luis Pelaez/Gett Images IN CAREERS There are man careers that use functions. Several are listed below. Financial analst Eercise 95, page 5 Biologist Eercise 7, page 9 Ta preparer Eample, page 0 Oceanographer Eercise 8, page

2 Chapter Functions and Their Graphs. RECTANGULAR COORDINATES What ou should learn Plot points in the Cartesian plane. Use the Distance Formula to find the distance between two points. Use the Midpoint Formula to find the midpoint of a line segment. Use a coordinate plane to model and solve real-life problems. Wh ou should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Eercise 70 on page, a graph represents the minimum wage in the United States from 950 to 009. The Cartesian Plane Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points in a plane called the rectangular coordinate sstem, or the Cartesian plane, named after the French mathematician René Descartes (59 50). The Cartesian plane is formed b using two real number lines intersecting at right angles, as shown in Figure.. The horizontal real number line is usuall called the -ais, and the vertical real number line is usuall called the -ais. The point of intersection of these two aes is the origin, and the two aes divide the plane into four parts called quadrants. Quadrant II Origin -ais Quadrant I (Vertical number line) (Horizontal number line) Quadrant III Quadrant IV -ais -ais Directed distance (, ) Directed distance -ais FIGURE. FIGURE. Ariel Skell/Corbis Each point in the plane corresponds to an ordered pair (, ) of real numbers and, called coordinates of the point. The -coordinate represents the directed distance from the -ais to the point, and the -coordinate represents the directed distance from the -ais to the point, as shown in Figure.. Directed distance from -ais, Directed distance from -ais (, ) (, ) FIGURE. (0, 0) (, ) (, 0) The notation, denotes both a point in the plane and an open interval on the real number line. The contet will tell ou which meaning is intended. Eample Plotting Points in the Cartesian Plane Plot the points,,,, 0, 0,, 0, and,. To plot the point,, imagine a vertical line through on the -ais and a horizontal line through on the -ais. The intersection of these two lines is the point,. The other four points can be plotted in a similar wa, as shown in Figure.. Now tr Eercise 7.

3 Section. Rectangular Coordinates The beaut of a rectangular coordinate sstem is that it allows ou to see relationships between two variables. It would be difficult to overestimate the importance of Descartes s introduction of coordinates in the plane. Toda, his ideas are in common use in virtuall ever scientific and business-related field. Eample Sketching a Scatter Plot, t Subscribers, N From 99 through 007, the numbers N (in millions) of subscribers to a cellular telecommunication service in the United States are shown in the table, where t represents the ear. Sketch a scatter plot of the data. (Source: CTIA-The Wireless Association) To sketch a scatter plot of the data shown in the table, ou simpl represent each pair of values b an ordered pair t, N and plot the resulting points, as shown in Figure.. For instance, the first pair of values is represented b the ordered pair 99,.. Note that the break in the t-ais indicates that the numbers between 0 and 99 have been omitted. Number of subscribers (in millions) FIGURE. N Now tr Eercise 5. Subscribers to a Cellular Telecommunication Service In Eample, ou could have let t represent the ear 99. In that case, the horizontal ais would not have been broken, and the tick marks would have been labeled through (instead of 99 through 007). t TECHNOLOGY The scatter plot in Eample is onl one wa to represent the data graphicall. You could also represent the data using a bar graph or a line graph. If ou have access to a graphing utilit, tr using it to represent graphicall the data given in Eample.

4 Chapter Functions and Their Graphs a + b = c The Pthagorean Theorem and the Distance Formula The following famous theorem is used etensivel throughout this course. a FIGURE.5 b c (, ) d Pthagorean Theorem For a right triangle with hpotenuse of length c and sides of lengths a and b, ou have a b c, as shown in Figure.5. (The converse is also true. That is, if a b c, then the triangle is a right triangle.) Suppose ou want to determine the distance d between two points, and, in the plane. With these two points, a right triangle can be formed, as shown in Figure.. The length of the vertical side of the triangle is, and the length of the horizontal side is. B the Pthagorean Theorem, ou can write d d. (, ) (, ) This result is the Distance Formula. The Distance Formula The distance d between the points, and, in the plane is FIGURE. d. Eample Finding a Distance Find the distance between the points, and,. Algebraic Let,, and,,. Then appl the Distance Formula. d Distance Formula Substitute for,,, and. Simplif. Simplif. Use a calculator. So, the distance between the points is about 5.8 units. You can use the Pthagorean Theorem to check that the distance is correct. Graphical Use centimeter graph paper to plot the points A, and B,. Carefull sketch the line segment from A to B. Then use a centimeter ruler to measure the length of the segment. 5 7 cm d? 5? 5 Now tr Eercise. Pthagorean Theorem Substitute for d. Distance checks. FIGURE.7 The line segment measures about 5.8 centimeters, as shown in Figure.7. So, the distance between the points is about 5.8 units.

5 Section. Rectangular Coordinates (5, 7) d = 5 (, ) FIGURE.8 d = 50 (, 0) d = You can review the techniques for evaluating a radical in Appendi A.. Eample Verifing a Right Triangle Show that the points,,, 0, and 5, 7 are vertices of a right triangle. The three points are plotted in Figure.8. Using the Distance Formula, ou can find the lengths of the three sides as follows. Because d d 0 5 d d d d ou can conclude b the Pthagorean Theorem that the triangle must be a right triangle. Now tr Eercise. The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, ou can simpl find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. The Midpoint Formula The midpoint of the line segment joining the points, and, is given b the Midpoint Formula Midpoint,. For a proof of the Midpoint Formula, see Proofs in Mathematics on page. Eample 5 Finding a Line Segment s Midpoint (9, ) (, 0) 9 ( 5, ) Midpoint FIGURE.9 Find the midpoint of the line segment joining the points 5, and 9,. Let, 5, and, 9,. M i dpoint, 5 9,, 0 Midpoint Formula Substitute for,,, and. Simplif. The midpoint of the line segment is, 0, as shown in Figure.9. Now tr Eercise 7(c).

6 Chapter Functions and Their Graphs Applications Eample Finding the Length of a Pass A football quarterback throws a pass from the 8-ard line, 0 ards from the sideline. The pass is caught b a wide receiver on the 5-ard line, 0 ards from the same sideline, as shown in Figure.0. How long is the pass? Distance (in ards) Football Pass 5 0 (0, 8) (0, 5) Distance (in ards) FIGURE.0 You can find the length of the pass b finding the distance between the points 0, 8 and 0, 5. d Distance Formula Substitute for,,, and Simplif. 99 Simplif. 0 Use a calculator. So, the pass is about 0 ards long. Now tr Eercise 57. In Eample, the scale along the goal line does not normall appear on a football field. However, when ou use coordinate geometr to solve real-life problems, ou are free to place the coordinate sstem in an wa that is convenient for the solution of the problem. Eample 7 Estimating Annual Revenue Sales (in billions of dollars) FIGURE. Barnes & Noble Sales (007, 5.) (00, 5.5) Midpoint (005, 5.) Barnes & Noble had annual sales of approimatel $5. billion in 005, and $5. billion in 007. Without knowing an additional information, what would ou estimate the 00 sales to have been? (Source: Barnes & Noble, Inc.) One solution to the problem is to assume that sales followed a linear pattern. With this assumption, ou can estimate the 00 sales b finding the midpoint of the line segment connecting the points 005, 5. and 007, 5.. M i dpoint, , 00, 5.5 Midpoint Formula Substitute for,, and. Simplif. So, ou would estimate the 00 sales to have been about $5.5 billion, as shown in Figure.. (The actual 00 sales were about $5. billion.) Now tr Eercise 59.

7 Section. Rectangular Coordinates 7 Eample 8 Translating Points in the Plane The triangle in Figure. has vertices at the points,,,, and,. Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure.. Paul Morrell Much of computer graphics, including this computer-generated goldfish tessellation, consists of transformations of points in a coordinate plane. One tpe of transformation, a translation, is illustrated in Eample 8. Other tpes include reflections, rotations, and stretches. FIGURE. FIGURE. 5 (, ) 5 7 To shift the vertices three units to the right, add to each of the -coordinates. To shift the vertices two units upward, add to each of the -coordinates. Original Point,,, (, ) (, ) Now tr Eercise. Translated Point,,,,, 5, The figures provided with Eample 8 were not reall essential to the solution. Nevertheless, it is strongl recommended that ou develop the habit of including sketches with our solutions even if the are not required. CLASSROOM DISCUSSION Etending the Eample Eample 8 shows how to translate points in a coordinate plane. Write a short paragraph describing how each of the following transformed points is related to the original point. Original Point,,, Transformed Point,,,

8 8 Chapter Functions and Their Graphs. VOCABULARY EXERCISES. Match each term with its definition. (a) -ais (i) point of intersection of vertical ais and horizontal ais (b) -ais (ii) directed distance from the -ais (c) origin (d) quadrants (iii) directed distance from the -ais (iv) four regions of the coordinate plane (e) -coordinate (v) horizontal real number line (f) -coordinate (vi) vertical real number line In Eercises, fill in the blanks. See for worked-out solutions to odd-numbered eercises.. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate sstem or the plane.. The is a result derived from the Pthagorean Theorem.. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the. SKILLS AND APPLICATIONS In Eercises 5 and, approimate the coordinates of the points. 5.. A In Eercises 7 0, plot the points in the Cartesian plane D B,,,, 0, 5,, 0, 0,,,,,,, 8, 0.5,, 5,,,.5,,,, C,,, B In Eercises, find the coordinates of the point.. The point is located three units to the left of the -ais and four units above the -ais.. The point is located eight units below the -ais and four units to the right of the -ais.. The point is located five units below the -ais and the coordinates of the point are equal.. The point is on the -ais and units to the left of the -ais. D C A In Eercises 5, determine the quadrant(s) in which, is located so that the condition(s) is (are) satisfied. 5. > 0 and < 0. < 0 and < 0 7. and > 0 8. > and 9. < 5 0. >. < 0 and > 0. > 0 and < 0. > 0. < 0 In Eercises 5 and, sketch a scatter plot of the data shown in the table. 5. NUMBER OF STORES The table shows the number of Wal-Mart stores for each ear from 000 through 007. (Source: Wal-Mart Stores, Inc.), Number of stores,

9 Section. Rectangular Coordinates 9. METEOROLOGY The table shows the lowest temperature on record (in degrees Fahrenheit) in Duluth, Minnesota for each month, where represents Januar. (Source: NOAA) In Eercises 7 8, find the distance between the points. 7.,,, 5 8.,, 8, 9.,,, 0.,,,.,,,. 8, 5, 0, 0.,, 5,.,,, 5.,,,.,,, 5 7..,.,.5, ,.,.9, 8. In Eercises 9, (a) find the length of each side of the right triangle, and (b) show that these lengths satisf the Pthagorean Theorem (, 5).. (0, ) 5 (9, ) (9, ) (, ) 8 Month, (, ) Temperature, (, 0) 8 (, 0) (, 5) (, ) (, 5) (5, ) In Eercises, show that the points form the vertices of the indicated polgon.. Right triangle:, 0,,,, 5. Right triangle:, ),, 5, 5, 5. Isosceles triangle:,,,,,. Isosceles triangle:,,, 9,, 7 In Eercises 7 5, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. 7.,, 9, 7 8.,,, 0 9., 0,, ,,, 8 5.,, 5, 5., 0, 0,,, 5, , 5.,.7, ,., 5., FLYING DISTANCE An airplane flies from Naples, Ital in a straight line to Rome, Ital, which is 0 kilometers north and 50 kilometers west of Naples. How far does the plane fl? 58. SPORTS A soccer plaer passes the ball from a point that is 8 ards from the endline and ards from the sideline. The pass is received b a teammate who is ards from the same endline and 50 ards from the same sideline, as shown in the figure. How long is the pass? SALES In Eercises 59 and 0, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc. in 005, given the sales in 00 and 007. Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.; Dollar Tree Stores, Inc.) 59. Big Lots Distance (in ards) (, 8) (50, ) Distance (in ards) Sales (in millions) $7 $5,,,

10 0 Chapter Functions and Their Graphs 0. Dollar Tree In Eercises, the polgon is shifted to a new position in the plane. Find the coordinates of the vertices of the polgon in its new position... (, ). Original coordinates of vertices: 7,,,,,, 7, Shift: eight units upward, four units to the right. Original coordinates of vertices: 5, 8,,, 7,, 5, Shift: units downward, 0 units to the left RETAIL PRICE In Eercises 5 and, use the graph, which shows the average retail prices of gallon of whole milk from 99 to 007. (Source: U.S. Bureau of Labor Statistics) Average price (in dollars per gallon) (, ) units (, ) units Sales (in millions) $800 $ units (, ) 7 (, ) 5 units (, 0) ( 5, ) Approimate the highest price of a gallon of whole milk shown in the graph. When did this occur?. Approimate the percent change in the price of milk from the price in 99 to the highest price shown in the graph. 7. ADVERTISING The graph shows the average costs of a 0-second television spot (in thousands of dollars) during the Super Bowl from 000 to 008. (Source: Nielson Media and TNS Media Intelligence) Cost of 0-second TV spot (in thousands of dollars) FIGURE FOR 7 (a) Estimate the percent increase in the average cost of a 0-second spot from Super Bowl XXXIV in 000 to Super Bowl XXXVIII in 00. (b) Estimate the percent increase in the average cost of a 0-second spot from Super Bowl XXXIV in 000 to Super Bowl XLII in ADVERTISING The graph shows the average costs of a 0-second television spot (in thousands of dollars) during the Academ Awards from 995 to 007. (Source: Nielson Monitor-Plus) Cost of 0-second TV spot (in thousands of dollars) (a) Estimate the percent increase in the average cost of a 0-second spot in 99 to the cost in 00. (b) Estimate the percent increase in the average cost of a 0-second spot in 99 to the cost in MUSIC The graph shows the numbers of performers who were elected to the Rock and Roll Hall of Fame from 99 through 008. Describe an trends in the data. From these trends, predict the number of performers elected in 00. (Source: rockhall.com) Number elected

11 Section. Rectangular Coordinates 70. LABOR FORCE Use the graph below, which shows the minimum wage in the United States (in dollars) from 950 to 009. (Source: U.S. Department of Labor) Minimum wage (in dollars) (a) Which decade shows the greatest increase in minimum wage? (b) Approimate the percent increases in the minimum wage from 990 to 995 and from 995 to 009. (c) Use the percent increase from 995 to 009 to predict the minimum wage in 0. (d) Do ou believe that our prediction in part (c) is reasonable? Eplain. 7. SALES The Coca-Cola Compan had sales of $9,805 million in 999 and $8,857 million in 007. Use the Midpoint Formula to estimate the sales in 00. Assume that the sales followed a linear pattern. (Source: The Coca-Cola Compan) 7. DATA ANALYSIS: EXAM SCORES The table shows the mathematics entrance test scores and the final eamination scores in an algebra course for a sample of 0 students (a) Sketch a scatter plot of the data. (b) Find the entrance test score of an student with a final eam score in the 80s. (c) Does a higher entrance test score impl a higher final eam score? Eplain. 7. DATA ANALYSIS: MAIL The table shows the number of pieces of mail handled (in billions) b the U.S. Postal Service for each ear from 99 through 008. (Source: U.S. Postal Service), TABLE FOR 7 Pieces of mail, (a) Sketch a scatter plot of the data. (b) Approimate the ear in which there was the greatest decrease in the number of pieces of mail handled. (c) Wh do ou think the number of pieces of mail handled decreased? 7. DATA ANALYSIS: ATHLETICS The table shows the numbers of men s M and women s W college basketball teams for each ear from 99 through 007. (Source: National Collegiate Athletic Association), Men s teams, M Women s teams, W (a) Sketch scatter plots of these two sets of data on the same set of coordinate aes.

12 Chapter Functions and Their Graphs (b) Find the ear in which the numbers of men s and women s teams were nearl equal. (c) Find the ear in which the difference between the numbers of men s and women s teams was the greatest. What was this difference? EXPLORATION 75. Aline segment has, as one endpoint and m, m as its midpoint. Find the other endpoint, of the line segment in terms of,, m, and m. 7. Use the result of Eercise 75 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectivel, (a),,, and (b) 5,,,. 77. Use the Midpoint Formula three times to find the three points that divide the line segment joining, and, into four parts. 78. Use the result of Eercise 77 to find the points that divide the line segment joining the given points into four equal parts. (a),,, (b),, 0, MAKE A CONJECTURE Plot the points,,, 5, and 7, on a rectangular coordinate sstem. Then change the sign of the -coordinate of each point and plot the three new points on the same rectangular coordinate sstem. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the -coordinate is changed. (b) The sign of the -coordinate is changed. (c) The signs of both the - and -coordinates are changed. 80. COLLINEAR POINTS Three or more points are collinear if the all lie on the same line. Use the steps below to determine if the set of points A,, B,, C, and the set of points A 8,, B 5,, C, are collinear. (a) For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship eists among these distances for each set of points? (b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line? (c) Compare our conclusions from part (a) with the conclusions ou made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearit. TRUE OR FALSE? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. 8. In order to divide a line segment into equal parts, ou would have to use the Midpoint Formula times. 8. The points 8,,,, and 5, represent the vertices of an isosceles triangle. 8. THINK ABOUT IT When plotting points on the rectangular coordinate sstem, is it true that the scales on the - and -aes must be the same? Eplain. 8. CAPSTONE Use the plot of the point 0, 0 in the figure. Match the transformation of the point with the correct plot. Eplain our reasoning. [The plots are labeled (i), (ii), (iii), and (iv).] (i) (iii) (a) 0, 0 (c) 0, 0 (, ) PROOF Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. ( b, c ) ( a + b, c) (0, 0) (ii) (iv) (b) 0, 0 (d) 0, 0 ( a, 0)