Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 8 Chapter Graphs, Functions, and Models.. Introduction Polnomial to Functions Graphing and Modeling Plot points. Determine whether an ordered pair is a solution of an equation. Graph equations. Find the distance between two points in the plane and find the midpoint of a segment. Find an equation of a circle with a given center and radius, and given an equation of a circle, find the center and the radius. Graph equations of circles. Graphs Graphs provide a means of displaing, interpreting, and analzing data in a visual format. It is not uncommon to open a newspaper or magazine and encounter graphs. Eamples of bar, circle, and line graphs are shown below. Americans Without Health Insurance More than 7 million Americans do not have health insurance. Percentage of uninsured b age 0 7 8 4 44 4 4 64 7% % 8% 7% 6% Source: U.S. Bureau of the Census 00 00 Land Mines in Afghanistan Where land mines are found in Afghanistan: Agricultural land Roads 6% Residential areas 7% 4% % % Irrigation sstems 6% Other Grazing land Sources: United Nations; Landmine Monitor; howstuffworks.com Indianapolis Life 00 Festival Mini-Marathon Number of participants 000 0000 000 000 000 0 77 80 8 90 9 00 0 Year II I (, +) (+, +) (, ) (0, 0) III IV (, ) (+, ) Man real-world situations can be modeled, or described mathematicall, using equations in which two variables appear. We use a plane to graph a pair of numbers. To locate points on a plane, we use two perpendicular number lines, called aes, which intersect at 0, 0. We call this point the origin. The horizontal ais is called the -ais, and the vertical ais is called the -ais.(other variables, such as a and b, can also be used.) The aes divide the plane into four regions, called quadrants, denoted b Roman numerals and numbered counterclockwise from the upper right. Arrows show the positive direction of each ais. Each point, in the plane is described b an ordered pair. The first number,, indicates the point s horizontal location with respect to the -ais, and the second number,, indicates the point s vertical location with
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 9 respect to the -ais. We call the first coordinate, -coordinate, or abscissa. We call the second coordinate, -coordinate, or ordinate. Such a representation is called the Cartesian coordinate sstem in honor of the French mathematician and philosopher René Descartes (96 60). In the first quadrant, both coordinates of a point are positive. In the second quadrant, the first coordinate is negative and the second is positive. In the third quadrant, both coordinates are negative, and in the fourth quadrant, the first coordinate is positive and the second is negative. (, ) (0, 4) 4 (, 4) (4, ) (, 0) (0, 0) 4 4 ( 4, ) (, 4) 4 EXAMPLE Graph and label the points,, 4,,, 4, 4,,, 4, 0, 4,, 0,and 0, 0. Solution To graph or plot,,we note that the -coordinate,, tells us to move from the origin units to the left of the -ais. Then we move units up from the -ais.* To graph the other points, we proceed in a similar manner. (See the graph at left.) Note that the point 4, is different from the point, 4. Solutions of Equations Equations in two variables, like 8, have solutions, that are ordered pairs such that when the first coordinate is substituted for and the second coordinate is substituted for, the result is a true equation. The first coordinate in an ordered pair generall represents the variable that occurs first alphabeticall. EXAMPLE Determine whether each ordered pair is a solution of 8. a), 7 b), 4 Solution We substitute the ordered pair into the equation and determine whether the resulting equation is true. a) 8 7? 8 8 We substitute for and 7 for (alphabetical order). FALSE The equation 8 is false, so, 7 is not a solution. b) 8 4? 8 6 8 8 We substitute for and 4 for. TRUE The equation 8 8 is true, so, 4 is a solution. *We first saw notation such as, in Section R.. There the notation represented an open interval. Here the notation represents an ordered pair. The contet in which the notation appears usuall makes the meaning clear.
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 60 Chapter Graphs, Functions, and Models We can also perform these substitutions on a graphing calculator. When we substitute for and 7 for,we get. Since 8,, 7 is not a solution of the equation. When we substitute for and 4 for, we get 8, so, 4 is a solution. ( ) 7 4 8 Graphs of Equations The equation considered in Eample actuall has an infinite number of solutions. Since we cannot list all the solutions, we will make a drawing, called a graph, that represents them. Shown at left are some suggestions for drawing graphs. To Graph an Equation To graph an equation is to make a drawing that represents the solutions of that equation. Graphs of equations of the tpe A B C are straight lines. Man such equations can be graphed convenientl using intercepts. The -intercept of the graph of an equation is the point at which the graph crosses the -ais. The -intercept is the point at which the graph crosses the -ais. We know from geometr that onl one line can be drawn through two given points. Thus, if we know the intercepts, we can graph the line. To ensure that a computational error has not been made, it is a good idea to calculate and plot a third point as a check. - and -Intercepts An -intercept is a point a,0. To find a,let 0 and solve for. A -intercept is a point 0, b. To find b,let 0 and solve for. EXAMPLE Graph: 8. Solution The graph is a line. To find ordered pairs that are solutions of this equation, we can replace either or with an number and then solve for the other variable. In this case, it is convenient to find the intercepts of the graph. For instance, if is replaced with 0, then 0 8 8 6. Dividing b
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 6 Suggestions for Drawing Graphs. Calculate solutions and list the ordered pairs in a table.. Use graph paper.. Draw aes and label them with the variables. 4. Use arrows on the aes to indicate positive directions.. Scale the aes; that is, label the tick marks on the aes. Consider the ordered pairs found in step () when choosing the scale. 6. Plot the ordered pairs, look for patterns, and complete the graph. Label the graph with the equation being graphed. Thus, 0, 6 is a solution. It is the -intercept of the graph. If is replaced with 0, then 0 8 8 9. Dividing b Thus, 9, 0 is a solution. It is the -intercept of the graph. We find a third solution as a check. If is replaced with, then 8 8 8 Subtracting 8. Dividing b Thus,, 8 is a solution. We list the solutions in a table and then plot the points. Note that the points appear to lie on a straight line., 0 6 0, 6 9 0 9, 0 8, 8 -intercept 9 8 7 (0, 6) 4 4 6 7 8 9 -intercept 4 Were we to graph additional solutions of 8, the would be on the same straight line. Thus, to complete the graph, we use a straightedge to draw a line, as shown in the figure. This line represents all solutions of the equation. Ever point on the line represents a solution; ever solution is represented b a point on the line. When graphing some equations, it is easier to first solve for and then find ordered pairs. We can use the addition and multiplication principles to solve for. 4 8 (, h) (9, 0) GCM EXAMPLE 4 Graph:. Solution We first solve for : Subtracting on both sides. Multipling b on both sides
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 6 Chapter Graphs, Functions, and Models B choosing multiples of for, we can avoid fraction values when calculating.for eample, if we choose for,we get. The following table lists a few more points. We plot the points and draw the graph.,, 0 0,, 4 (0, ) (, ) 4 4 (, ) 4, or Plot Plot Plot \Y (/)X \Y \Y \Y4 \Y \Y6 \Y7 In the equation, the value of depends on the value chosen for, so is said to be the independent variable and the dependent variable. We can graph an equation on a graphing calculator. Man calculators require an equation to be entered in the form. In such a case, if the equation is not initiall given in this form, it must be solved for before it is entered. For the equation in Eample 4, we enter on the equation-editor, or, screen in the form, as shown in the window at left. Net, we determine the portion of the -plane that will appear on the calculator s screen. That portion of the plane is called the viewing window. The notation used in this tet to denote a window setting consists of four numbers L,R,B,T, which represent the Left and Right endpoints of the -ais and the Bottom and Top endpoints of the -ais, respectivel. The window with the settings,,, is the standard viewing window. On some graphing calculators, the standard window can be selected quickl using the ZSTANDARD feature from the ZOOM menu. WINDOW Xmin Xma Xscl Ymin Yma Yscl Xres Xmin and Xma are used to set the left and right endpoints of the -ais, respectivel; Ymin and Yma are used to set the bottom and top endpoints of the -ais. The settings Xscl and Yscl give the scales for the aes. For eample, Xscl and Yscl means that there is unit between tick marks on each of the aes.
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 6 After entering the equation and choosing a viewing window, we can then draw the graph shown at left. GCM EXAMPLE Graph: 9. Solution Note that this is not of the form A B C, thus it is not linear. We make a table of values, plot enough points to obtain an idea of the shape of the curve, and connect them with a smooth curve. It is important to scale the aes to include most of the ordered pairs listed in the table. Here it is appropriate to use a larger scale on the -ais than on the -ais. 9, 4, 4, 0 0, 6, 6 4 4,,, 4, 4 0 8 6 4 4 6 8 0 0 Select values for. Compute values for. Plot Plot Plot \Y X 9X \Y \Y \Y4 \Y \Y6 \Y7 Figure TABLE SETUP TblStart Tbl Indpnt: Auto Ask Depend: Auto Ask Figure A graphing calculator can be used to create a table of ordered pairs that are solutions of an equation. For the equation in Eample, 9,we first enter the equation on the equation-editor screen (see Fig. ). Then we set up a table in AUTO mode b designating a value for TBLSTART and a value for TBL. The calculator will produce a table starting with the value of TBLSTART and continuing b adding TBL to suppl succeeding -values. For the equation 9, we let TBLSTART and TBL (see Fig. ). We can scroll up and down in the table to find values other than those shown in Fig.. We can also graph this equation on the graphing calculator, as shown in Fig. 4. 9 X 0 X Y 4 0 6 0 Figure Figure 4
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 7. Eercise Set Graph and label the given points b hand.. 4, 0,,,, 4, 0,,,., 4, 4,,, 0,, 4, 4, 0.,,,,,,,, 0, 4. 4, 0, 4,,,,, 0,, Epress the data pictured in the graph as ordered pairs, letting the first coordinate represent the ear and the second coordinate the amount.. 6. Consumption of Bottled Water (U.S.) Number of gallons (in billions) Gift Card Sales (U.S.) Sales (in billions) 7 6 4 $40 0 0 4. $9 999 Source: Bain and Co. 4.6 $ 000 4.9 $ 00.4 $6 00 6.0 $4 00 estimated 6.4 Use substitution to determine whether the given ordered pairs are solutions of the given equation. 7.,, 0, ; Yes; no 8.,,, ; Yes; no 9.,,, 4 ; 6 4 Yes; no..,.6,, 0 ; 9 No; es., 0,, 4 ; a b No; es. 0,,, ; m 4n 6 Yes; es. 0.7,.7,, ; No; es 4., 4, 4, ; 70 No; es Use a graphing calculator to create a table of values with TBLSTART and TBL. Then graph the equation b hand.. 6. 7. 8. 4 6 9. 4 0.. 4.. 4. 4. 4 6. 6 4 7. 4 8. 6 9. 0. 4... 4. 4. 6. In Eercises 7 40, use a graphing calculator to match the equation with one of the graphs (a) (d), which follow. a) b) 98 99 00 0 0 0 Source: Beverage Marketing Corp. Answers to Eercises 6 and 6 can be found on p. IA-.
Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 7 Chapter Graphs, Functions, and Models c) 7. (b) 8. 6 (d) 9. (a) 40. 4 (c) Use a graphing calculator to graph the equation in the standard window. 4. 4. 4 4. 4 7 44. 8 4. 46. 4 47. 48. 4 49. 6 0. 8... 4 4. Graph the equation in the standard window and in the given window. Determine which window better shows the shape of the graph and the - and -intercepts.. 6 4, 4, 4, 4 Standard window d) 6. 4,,, 0,with Xscl and Yscl,,, 0 7. 6,, 0., 0.,with Xscl 0. and Yscl 0.,, 0., 0. 8. 8,,, Standard window Find the distance between the pair of points. Give an eact answer and, where appropriate, an approimation to three decimal places. 4, 6.40 9. 4, 6 and, 9 60., 7 and,,.6 6. 6, and 9, 6. 4, 7 and, 4, 6.708 9,.440 6. 4., and., 6.4 8.0,.6 4 64. and,, 4 6.,4 and,4 66. 0.6,. and 8.,. 8.7 67., and 6,0 4 6, 4.74 68., and 0, 7 7,.70 69. 0, 0 and a, b 70. r, s and r, s a b r s 7. The points, and 9, 4 are the endpoints of the diameter of a circle. Find the length of the radius of the circle. 6. 7. The point 0, is on a circle that has center,. Find the length of the diameter of the circle. The converse of the Pthagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pthagorean theorem to determine whether the set of points could be vertices of a right triangle. 7. 4,, 6,,and 8, Yes 74.,,,,and 6, 9 Yes 7. 4,, 0,,and, 4 No 76. The points, 4,,,,, and 0, 7 are vertices of a quadrilateral. Show that the quadrilateral is a rectangle. (Hint: Show that the quadrilateral s opposite sides are the same length and that the two diagonals are the same length.) Find the midpoint of the segment having the given endpoints. 77. 4, 9 and, 4, 6 78. 7, and 9, 8, 79. 6.,.8 and.8, 6. 4.9, 4.9 80. 0.,.7 and 4.8, 0..,. 8. 6, and 6, 8 6, 8., and, 0, 0 8. and, 6, 4, 40 84. and, 4 9, 4 4, 7 0 8., and,4, 7 86., and, 7 0, 87. Graph the rectangle described in Eercise 76. Then determine the coordinates of the midpoint of each Answers to Eercises 4 4 and 76 can be found on pp. IA- and IA-.