Introduction to Graphs and the Graphing Calculator
Contents
1. given in the table and draw the graph In the equation y 4x 1 the value of y depends on the value chosen for x so x is said to be the independent variable and y the de pendent variable We can also graph an equation on a grapher Be sure that the stat plots are turned off as described on p 4 before graphing equations like those below Failure to do this could affect the viewing window and pre vent the desired graph from being seen EXAMPLE 6 Graph using a grapher y 5x 1 Solution We enter y x Fd y tx 1 on the equation editor screen in 10 the form y 1 2 x 1 select a viewing window and draw the graph The graph is shown in the standard window 10 10 8 INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR EXAMPLE 7 Graph y x 5 Solution We select values for x and find the corresponding y values Then we plot the points and connect them with a smooth curve 1 Select values for x 2 Compute values for y We can also graph this equation using a grapher as shown below y x 5 Plot Plot2 Plot3 Y1x2 5 Vos Y3 Y4 o Y6 Y7 EXAMPLE 8 Graph y x 9x 12 Solution 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 ee INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR 9 We can also graph the equation using a grapher We enter y x 9x 12 on the equa2. marks on each of the axes Generally in this text Xscl and Yscl are both assumed to be 1 unless different values are given Some ex ceptions will occur when the scaling factor is different from 1 but readily apparent EXAMPLE 2 Use a grapher to graph the points 3 5 4 3 3 4 4 3 4 0 4 3 0 and 0 0 Solution We note that the x coordinates of the given points range from 4 to 4 and the y coordinates range from 4 to 5 Thus one good choice for the viewing window is the standard window because x and y values both range from 10 to 10 in this window The coordinates of the points are entered into the grapher in lists using the EDIT operation from the STAT menu Note that the key rather than the key must be used when we are entering negative num bers We enter the first coordinates in one list and then the correspond ing second coordinates in the same order in a second list Then we turn 4 INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR on and set up a STAT PLOT to draw a scatterplot using the data from these two lists _ i Plot2 Plot3 Ol Off Type BA L dh Hh HOH ret tt Xlist L1 Ylist L2 Mark E Some graphers have ZOOMSTAT a feature that automatically selects a viewing window that displays all the data points in the lists for a given STAT PLOT This feature is activated after the data have been entered 6 53 MEMORY 3TZoom Out 4 ZDecimal 5
3. of some procedures Keep in mind that a grapher cannot be used effectively without a firm mathematical foundation upon which to build For ex ample expressions cannot be entered correctly nor can results be inter preted well if the relevant concepts are not understood and applied correctly The Use of the Grapher A grapher is a tool that can be used in the process of learning and understanding mathematics It should be used to enhance the understanding of concepts not to replace the learning of skills 2 INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR WORK PERMITS ISSUED to Indiana Teens 0 40 000 80 000 Source Indiana Department of Labor 120 000 Graphs Graphs provide a means of displaying interpreting and analyzing data in a visual format It is not uncommon to open a newspaper or magazine and encounter graphs Shown below are examples of bar circle and line graphs SOURCES OF WATER PERCENT UNINSURED vs How we consume water PERCENT UNEMPLOYED 18 ae 16 Soft drinks Drinking Dairy 10 water a products 12 11 10 Vegetables 11 oe 6 z 4 Tea and coffee 30 2 0 88 90 O2 94 96 Source U S Department of Agriculture Sources U S Bureau of the Census U S Bureau of Labor Statistics Many real world situations can be modeled or described mathe matically using equations in which two variables appear We use a plane to graph a pair of numbers To locate po
4. Introduction to Graphs and the Graphing Calculator e Plot points by hand and using a grapher e Graph equations by hand and using a grapher e Find the point s of intersection of two graphs Graphing calculators and computers equipped with graphing software are useful tools in applying and understanding mathematical concepts All such graphing utilities will be referred to as graphers in this text although the emphasis will be on graphing calculators When we think of the ways in which a grapher can be used graphing equations might come to mind first Although the grapher will be used extensively in this text to graph equations many of its other uses will also be explored These include performing calculations evaluating ex pressions solving equations analyzing the graphs of equations creating tables of data finding mathematical models of real data analyzing those models and using them to make estimates and predictions Keystrokes and features vary among different brands and models of graphers We will use features that are commonly found on many graph ing calculators Specific keystrokes and instructions for using these features can be found in the Graphing Calculator Manual that ac companies this text You can also consult either your instructor or the user s manual for your particular grapher The goal of this text is to use the grapher as a tool to enhance the learning and understanding of mathematics and to relieve the tedium
5. ZSquare 6 ZStandard 7 ZTrig 8 Zinteger EMZoomStat 5 93 To turn off the plot we return to the plot screen and select Off Solutions of Equations Equations in two variables like 2x 3y 18 have solutions x y that are ordered pairs such that when the first coordinate is substituted for x and the second coordinate is substituted for y the result is a true equation EXAMPLE 3 Determine whether each ordered pair is a solution of 2 T oy Id a 5 7 b 3 4 Solution We substitute the ordered pair into the equation and determine whether the resulting equation is true a 2x 3y 18 SEE 2 5 3 7 18 We substitute 5 for x and 7 for y alphabetical order SP F 2l l1 18 FALSE The equation 11 18 is false so 5 7 is not a solution Suggestions for Hand Drawn Graphs Use graph paper Draw axes and label them with the variables Use arrows on the axes to indicate positive directions Scale the axes that is mark numbers on the axes Calculate solutions and list the ordered pairs in a table Plot the ordered pairs look for patterns and complete the graph Label the graph with the equation being graphed INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR 5 b 2x 3y 18 E 2 3 3 4 18 We substitute 3 for x and 4 for y 6 F12 18 18 TRUE The equation 18 18 is true so 3 4 is a solution We can also perform these substit
6. e keystrokes to use when entering points in lists setting up the STAT PLOT and graphing the points INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR 3 5 units up from the x axis To graph the other points we proceed in a similar manner See the graph at left Note that the origin 0 0 is lo cated at the intersection of the axes and that the point 4 3 is different from the point 3 4 A graph of a set of points like the one in Example 1 is called a scat terplot A grapher can also be used to make a scatterplot This involves among other things determining the portion of the xy plane that will appear on the grapher s screen That portion of the plane is called the viewing window The notation used in this text to denote a window setting consists of four numbers L R B T which represent the Left and Right endpoints of the x axis and the Bottom and Top endpoints of the y axis respec tively The window with the settings 10 10 10 10 is the standard viewing window shown in the following figure On some graphers the standard window can be selected quickly using the ZSTANDARD feature from the ZOOM menu 10 Xmin and Xmax are used to set the left and right endpoints of the x axis respectively Ymin and Ymax are used to set the bottom and top endpoints of the y axis The settings Xscl and Yscl give the scales for the axes For example Xscl 1 and Yscl 1 means that there is 1 unit be tween tick
7. e use the INTERSECT feature from the CALC menu to find the coordinates of the point of intersection See the Graphing Calculator Manual that accompanies this text for the keystrokes y x t5 y 4x 10 10 y210 y Plot Plot2 Plot3 wip x 5 Y2E4Xx 10 oa 10 10 Y4 Y5 Y6 Y7 y Intersection X 1 666667 Y 3 3333333 N2 10 10 The graphs intersect at the point 1 666667 3 3333333 which is a decimal approximation for the point of intersection If the coordinates are rational numbers their exact values can be found using the FRAC feature from the MATH menu The keystrokes for doing these conversions are given in the Graphing Calculator Manual that accompanies this text X gt Frac Y gt Frac The point of intersection is 2 2 If the coordinates in Example 9 had not been rational numbers the gt FRAC operation would have returned the original decimal approxima tion rather than a fraction If a pair of equations has more than one point of intersection we use the INTERSECT feature repeatedly to find the coordinates of all the points
8. ints on a plane we use two per pendicular number lines called axes which intersect at 0 0 We call this point the origin The horizontal axis is called the x axis and the vertical axis is called the y axis Other variables such as a and b can also be used The axes divide the plane into four regions called quad rants denoted by Roman numerals and numbered counterclockwise from the upper right Arrows show the positive direction of each axis Each point x y in the plane is called an ordered pair The first num ber x indicates the point s horizontal location with respect to the y axis and the second number y indicates the point s vertical location with respect to the x axis We call x the first coordinate x coordinate or abscissa We call y the second coordinate y coordinate or ordinate Such a representation is called the Cartesian coordinate system in honor of the great French mathematician and philosopher Ren Descartes 1596 1650 EXAMPLE 1 Graph and label the points 3 5 4 3 3 4 4 2 3 4 0 4 3 0 and 0 0 Solution To graph or plot 3 5 we note that the x coordinate tells us to move from the origin 3 units to the left of the y axis Then we move MEMORY 1 ZBox 2 Zoom In 3 Zoom Out 4 ZDecimal 5 ZSquare FA ZStandard 7 ZTrig The Graphing Calculator Manual that accompanies this text gives keystrokes for selected examples See this manual for th
9. tion editor screen and graph the equation in the standard window y x D 10 ee 10 Note that this window does not give a good picture of the graph Be cause the graph is cut off to the right of x 10 and below y 10 it appears as though Xmax should be larger than 10 and Ymin should be less than 10 Since the graph rises steeply in the second quadrant we could also let Xmin be 5 rather than 10 We try 5 15 15 10 with Yscl 2 y x 9x 12 10 Yscl 2 The graph is still cut off below Ymin or 15 We try other settings for Ymin until we find one that shows the lower portion of the graph One good window is 5 15 35 10 with Yscl 5 y oR 12 10 35 Yscl 5 ae Finding Points of Intersection There are many situations in which we want to determine the point s of intersection of two graphs l 0 INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR EXAMPLE 9 Use a grapher to find the point of intersection of the graphs of x y 5 and y 4x 10 Solution Since equations must be entered in the form y on many graphers we solve the first equation for y x Yu C y Adding y on both sides Ko Adding 5 on both sides Now we can enter y x 5 and y 4x 10 on the equation editor screen and graph the equations We begin by using the standard window and see that it is a good choice because it shows the point of intersection of the graphs Next w
10. utions on a grapher When we sub stitute 5 for x and 7 for y we get 11 Since 11 18 5 7 is not a so lution of the equation When we substitute 3 for x and 4 for y we get 18 so 3 4 is a solution 2 5 3 7 2 3 3 4 Graphs of Equations The equation considered in Example 3 actually has an infinite number of solutions Since we cannot list all the solutions we will make a draw ing called a graph that represents them To Graph an Equation To graph an equation is to make a drawing that represents the solutions of that equation Shown at left are some suggestions for making hand drawn graphs EXAMPLE 4 Graph 2x 3y 18 Solution To find ordered pairs that are solutions of this equation we can replace either x or y with any number and then solve for the other variable For instance if x is replaced with 0 then 2 0 3y 18 3y 18 yo Dividing by 3 Thus 0 6 is a solution If x is replaced with 5 then 2 S 1 Sy 18 10 3y 18 3y 8 Subtracting 10 y Dividing by 3 INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR Thus s s is a solution If y is replaced with 0 then 2 FO 18 2x 15 x 9 Dividing by 2 Thus 9 0 is a solution We continue choosing values for one variable and finding the corre sponding values of the other We list the solutions in a table and then plot the points Note that the points appear to lie on a straight line 6 8 3 0 20 3 In fact were
11. we to graph additional solutions of 2x 3y 18 they 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 That line represents all solutions of the equation EXAMPLE 5 Use a grapher to create a table of ordered pairs that are so lutions of the equation y x 1 Then graph the equation by hand Solution We use the TABLE feature on a grapher to create the table of ordered pairs We must first enter the equation on the equation editor or y screen Many graphers require an equation to be entered in the form y as this one is written If an equation is not initially given in this form it must be solved for y before it is entered in the grapher Plot Plot2 Plot3 wA 1 2 X 1 2 Y3 Y4 Y5 Y6 Y7 On a grapher we enter y 1 2 x 1 Since 1 2x is interpreted as 1 2 x by some graphers and as 1 2x by others we use parentheses to INTRODUCTION TO GRAPHS AND THE GRAPHING CALCULATOR 7 ensure that the equation is interpreted correctly Then we set up a table in AUTO mode by designating a value for TBLSTART and a value for ATBL The grapher will produce a table starting with the value of TBLSTART and continuing by adding ATBL to supply succeeding x values For the equa tion y x 1 we let TBLSTART 3 and ATBL 1 TABLE SETUP TblStart 3 ATbl 1 Indpnt Ask Depend AWK Ask Next we plot some of the points
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