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1 Section.: Relations, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 0, Carl Stitz.

2 0 Relations and Functions. Relations From one point of view, all of Precalculus can be thought of as studing sets of points in the plane. With the Cartesian Plane now fresh in our memor we can discuss those sets in more detail and as usual, we begin with a definition. Definition.. A relation is a set of points in the plane. Since relations are sets, we can describe them using the techniques presented in Section... That is, we can describe a relation verball, using the roster method, or using set-builder notation. Since the elements in a relation are points in the plane, we often tr to describe the relation graphicall or algebraicall as well. Depending on the situation, one method ma be easier or more convenient to use than another. As an eample, consider the relation R = {(, ), (, ), (0, )}. As written, R is described using the roster method. Since R consists of points in the plane, we follow our instinct and plot the points. Doing so produces the graph of R. (, ) (, ) (0, ) The graph of R. In the following eample, we graph a variet of relations. Eample... Graph the following relations.. A = {(0, 0), (, ), (, ), (, )}. HLS = {(, ) }. HLS = {(, ) < }. V = {(, ) is a real number}. H = {(, ) = } 6. R = {(, ) < } Carl s, of course.

3 . Relations Solution.. To graph A, we simpl plot all of the points which belong to A, as shown below on the left.. Don t let the notation in this part fool ou. The name of this relation is HLS, just like the name of the relation in number was A. The letters and numbers are just part of its name, just like the numbers and letters of the phrase King George III were part of George s name. In words, {(, ) } reads the set of points (, ) such that. All of these points have the same -coordinate,, but the -coordinate is allowed to var between and, inclusive. Some of the points which belong to HLS include some friendl points like: (, ), (, ), (0, ), (, ), (, ), (, ), and (, ). However, HLS also contains the points (0.89, ), ( 6, ), ( π, ), and so on. It is impossible to list all of these points, which is wh the variable is used. Plotting several friendl representative points should convince ou that HLS describes the horizontal line segment from the point (, ) up to and including the point (, ). The graph of A The graph of HLS. HLS is hauntingl similar to HLS. In fact, the onl difference between the two is that instead of we have <. This means that we still get a horizontal line segment which includes (, ) and etends to (, ), but we do not include (, ) because of the strict inequalit <. How do we denote this on our graph? It is a common mistake to make the graph start at (, ) end at (, ) as pictured below on the left. The problem with this graph is that we are forgetting about the points like (., ), (., ), (.9, ), (.99, ), and so forth. There is no real number that comes immediatel before, so to describe the set of points we want, we draw the horizontal line segment starting at (, ) and draw an open circle at (, ) as depicted below on the right. Reall impossible. The interested reader is encouraged to research countable versus uncountable sets.

4 Relations and Functions This is NOT the correct graph of HLS The graph of HLS. Net, we come to the relation V, described as the set of points (, ) such that is a real number. All of these points have an -coordinate of, but the -coordinate is free to be whatever it wants to be, without restriction. Plotting a few friendl points of V should convince ou that all the points of V lie on the vertical line =. Since there is no restriction on the -coordinate, we put arrows on the end of the portion of the line we draw to indicate it etends indefinitel in both directions. The graph of V is below on the left.. Though written slightl differentl, the relation H = {(, ) = } is similar to the relation V above in that onl one of the coordinates, in this case the -coordinate, is specified, leaving to be free. Plotting some representative points gives us the horizontal line =. The graph of H The graph of V 6. For our last eample, we turn to R = {(, ) < }. As in the previous eample, is free to be whatever it likes. The value of, on the other hand, while not completel free, is permitted to roam between and ecluding, but including. After plotting some friendl elements of R, it should become clear that R consists of the region between the horizontal We ll revisit the concept of a free variable in Section 8.. Don t worr, we ll be refreshing our memor about vertical and horizontal lines in just a moment! The word some is a relative term. It ma take, 0, or 0 points until ou see the pattern.

5 . Relations lines = and =. Since R requires that the -coordinates be greater than, but not equal to, we dash the line = to indicate that those points do not belong to R. The graph of R The relations V and H in the previous eample lead us to our final wa to describe relations: algebraicall. We can more succinctl describe the points in V as those points which satisf the equation =. Most likel, ou have seen equations like this before. Depending on the contet, = could mean we have solved an equation for and arrived at the solution =. In this case, however, = describes a set of points in the plane whose -coordinate is. Similarl, the relation H above can be described b the equation =. At some point in our mathematical upbringing, ou probabl learned the following. Equations of Vertical and Horizontal Lines ˆ The graph of the equation = a is a vertical line through (a, 0). ˆ The graph of the equation = b is a horizontal line through (0, b). Given that the ver simple equations = a and = b produced lines, it s natural to wonder what shapes other equations might ield. Thus our net objective is to stud the graphs of equations in a more general setting as we continue to unite Algebra and Geometr... Graphs of Equations In this section, we delve more deepl into the connection between Algebra and Geometr b focusing on graphing relations described b equations. The main idea of this section is the following. The Fundamental Graphing Principle The graph of an equation is the set of points which satisf the equation. That is, a point (, ) is on the graph of an equation if and onl if and satisf the equation. Here, and satisf the equation means and make the equation true. It is at this point that we gain some insight into the word relation. If the equation to be graphed contains both and, then the equation itself is what is relating the two variables. More specificall, in the net two eamples, we consider the graph of the equation + =. Even though it is not specificall

6 Relations and Functions spelled out, what we are doing is graphing the relation R = {(, ) + = }. The points (, ) we graph belong to the relation R and are necessaril related b the equation + =, since it is those pairs of and which make the equation true. Eample... Determine whether or not (, ) is on the graph of + =. Solution. We substitute = and = into the equation to see if the equation is satisfied. () + ( )? = Hence, (, ) is not on the graph of + =. We could spend hours randoml guessing and checking to see if points are on the graph of the equation. A more sstematic approach is outlined in the following eample. Eample... Graph + =. Solution. To efficientl generate points on the graph of this equation, we first solve for + = = = = We now substitute a value in for, determine the corresponding value, and plot the resulting point (, ). For eample, substituting = into the equation ields = = ( ) = 8 =, so the point (, ) is on the graph. Continuing in this manner, we generate a table of points which are on the graph of the equation. These points are then plotted in the plane as shown below. (, ) (, ) (, ) 0 (, 0) 0 (0, ) 0 (, 0) (, ) (, ) Remember, these points constitute onl a small sampling of the points on the graph of this equation. To get a better idea of the shape of the graph, we could plot more points until we feel comfortable

7 . Relations connecting the dots. Doing so would result in a curve similar to the one pictured below on the far left. Don t worr if ou don t get all of the little bends and curves just right Calculus is where the art of precise graphing takes center stage. For now, we will settle with our naive plug and plot approach to graphing. If ou feel like all of this tedious computation and plotting is beneath ou, then ou can reach for a graphing calculator, input the formula as shown above, and graph. Of all of the points on the graph of an equation, the places where the graph crosses or touches the aes hold special significance. These are called the intercepts of the graph. Intercepts come in two distinct varieties: -intercepts and -intercepts. The are defined below. Definition.. Suppose the graph of an equation is given. ˆ ˆ A point on a graph which is also on the -ais is called an -intercept of the graph. A point on a graph which is also on the -ais is called an -intercept of the graph. In our previous eample the graph had two -intercepts, (, 0) and (, 0), and one -intercept, (0, ). The graph of an equation can have an number of intercepts, including none at all! Since -intercepts lie on the -ais, we can find them b setting = 0 in the equation. Similarl, since -intercepts lie on the -ais, we can find them b setting = 0 in the equation. Keep in mind, intercepts are points and therefore must be written as ordered pairs. To summarize, Finding the Intercepts of the Graph of an Equation Given an equation involving and, we find the intercepts of the graph as follows: ˆ -intercepts have the form (, 0); set = 0 in the equation and solve for. ˆ -intercepts have the form (0, ); set = 0 in the equation and solve for. Another fact which ou ma have noticed about the graph in the previous eample is that it seems to be smmetric about the -ais. To actuall prove this analticall, we assume (, ) is a generic point on the graph of the equation. That is, we assume + = is true. As we learned in Section., the point smmetric to (, ) about the -ais is (, ). To show that the graph is

8 6 Relations and Functions smmetric about the -ais, we need to show that (, ) satisfies the equation + =, too. Substituting (, ) into the equation gives ( ) + ()? = + = Since we are assuming the original equation + = is true, we have shown that (, ) satisfies the equation (since it leads to a true result) and hence is on the graph. In this wa, we can check whether the graph of a given equation possesses an of the smmetries discussed in Section.. We summarize the procedure in the following result. Testing the Graph of an Equation for Smmetr To test the graph of an equation for smmetr ˆ about the -ais substitute (, ) into the equation and simplif. If the result is equivalent to the original equation, the graph is smmetric about the -ais. ˆ about the -ais substitute (, ) into the equation and simplif. If the result is equivalent to the original equation, the graph is smmetric about the -ais. ˆ about the origin - substitute (, ) into the equation and simplif. If the result is equivalent to the original equation, the graph is smmetric about the origin. Intercepts and smmetr are two tools which can help us sketch the graph of an equation analticall, as demonstrated in the net eample. Eample... Find the - and -intercepts (if an) of the graph of ( ) + =. Test for smmetr. Plot additional points as needed to complete the graph. Solution. To look for -intercepts, we set = 0 and solve ( ) + = ( ) + 0 = ( ) = ( ) = etract square roots = ± = ± =, We get two answers for which correspond to two -intercepts: (, 0) and (, 0). Turning our attention to -intercepts, we set = 0 and solve

9 . Relations 7 ( ) + = (0 ) + = + = = Since there is no real number which squares to a negative number (Do ou remember wh?), we are forced to conclude that the graph has no -intercepts. Plotting the data we have so far, we get (, 0) (, 0) Moving along to smmetr, we can immediatel dismiss the possibilit that the graph is smmetric about the -ais or the origin. If the graph possessed either of these smmetries, then the fact that (, 0) is on the graph would mean (, 0) would have to be on the graph. (Wh?) Since (, 0) would be another -intercept (and we ve found all of these), the graph can t have -ais or origin smmetr. The onl smmetr left to test is smmetr about the -ais. To that end, we substitute (, ) into the equation and simplif ( ) + = ( ) + ( )? = ( ) + = Since we have obtained our original equation, we know the graph is smmetric about the -ais. This means we can cut our plug and plot time in half: whatever happens below the -ais is reflected above the -ais, and vice-versa. Proceeding as we did in the previous eample, we obtain

10 8 Relations and Functions A couple of remarks are in order. First, it is entirel possible to choose a value for which does not correspond to a point on the graph. For eample, in the previous eample, if we solve for as is our custom, we get = ± ( ). Upon substituting = 0 into the equation, we would obtain = ± (0 ) = ± = ±, which is not a real number. This means there are no points on the graph with an -coordinate of 0. When this happens, we move on and tr another point. This is another drawback of the plug-and-plot approach to graphing equations. Luckil, we will devote much of the remainder of this book to developing techniques which allow us to graph entire families of equations quickl. 6 Second, it is instructive to show what would have happened had we tested the equation in the last eample for smmetr about the -ais. Substituting (, ) into the equation ields ( ) + = ( ) +? = (( )( + )) +? = ( + ) +? =. This last equation does not appear to be equivalent to our original equation. However, to actuall prove that the graph is not smmetric about the -ais, we need to find a point (, ) on the graph whose reflection (, ) is not. Our -intercept (, 0) fits this bill nicel, since if we substitute (, 0) into the equation we get This proves that (, 0) is not on the graph. ( ) +? = ( ) Without the use of a calculator, if ou can believe it!

11 . Relations 9.. Eercises In Eercises - 0, graph the given relation.. {(, 9), (, ), (, ), (0, 0), (, ), (, ), (, 9)}. {(, 0), (, ), (, ), (0, ), (0, ), (, ), (, )}. {(m, m) m = 0, ±, ±}. {( 6 k, k) k = ±, ±, ±, ±, ±, ±6 }. {( n, n ) n = 0, ±, ± } 6. {( j, j ) j = 0,,, 9 } 7. {(, ) > } 8. {(, ) } 9. {(, ) > } 0. {(, ) }. {(, ) < }. {(, ) < }. {(, ) < }. {(, ) < }. {(, ) > } 6. {(, ) } 7. {(, ) < } 8. {(, ), < } 9. {(, ) > 0, < } 0. {(, ), π < 9 } In Eercises - 0, describe the given relation using either the roster or set-builder method... Relation B Relation A

12 0 Relations and Functions.. Relation C Relation D. 6. Relation E Relation F Relation G Relation H

13 . Relations Relation I Relation J In Eercises - 6, graph the given line.. =. =. =. =. = 0 6. = 0 Some relations are fairl eas to describe in words or with the roster method but are rather difficult, if not impossible, to graph. Discuss with our classmates how ou might graph the relations given in Eercises 7-0. Please note that in the notation below we are using the ellipsis,..., to denote that the list does not end, but rather, continues to follow the established pattern indefinitel. For the relations in Eercises 7 and 8, give two eamples of points which belong to the relation and two points which do not belong to the relation. 7. {(, ) is an odd integer, and is an even integer.} 8. {(, ) is an irrational number } 9. {(, 0), (, ), (, ), (8, ), (6, ), (, ),...} 0. {..., (, 9), (, ), (, ), (0, 0), (, ), (, ), (, 9),...} For each equation given in Eercises - : ˆ Find the - and -intercept(s) of the graph, if an eist. ˆ Follow the procedure in Eample.. to create a table of sample points on the graph of the equation. ˆ Plot the sample points and create a rough sketch of the graph of the equation. ˆ Test for smmetr. If the equation appears to fail an of the smmetr tests, find a point on the graph of the equation whose reflection fails to be on the graph as was done at the end of Eample..

14 Relations and Functions. = +. = 8. =. =. = 6. = + 7. = 7 8. = 0 9. ( + ) + = 6 0. =. 9 = 6. = The procedures which we have outlined in the Eamples of this section and used in Eercises - all rel on the fact that the equations were well-behaved. Not everthing in Mathematics is quite so tame, as the following equations will show ou. Discuss with our classmates how ou might approach graphing the equations given in Eercises - 6. What difficulties arise when tring to appl the various tests and procedures given in this section? For more information, including pictures of the curves, each curve name is a link to its page at For a much longer list of fascinating curves, click here.. + = 0 Folium of Descartes. = + Kample of Eudous. = + Tschirnhausen cubic 6. ( + ) = + Crooked egg 7. With the help of our classmates, find eamples of equations whose graphs possess ˆ ˆ ˆ ˆ smmetr about the -ais onl smmetr about the -ais onl smmetr about the origin onl smmetr about the -ais, -ais, and origin Can ou find an eample of an equation whose graph possesses eactl two of the smmetries listed above? Wh or wh not?

15 . Relations.. Answers

16 Relations and Functions

17 . Relations

18 6 Relations and Functions. A = {(, ), (, ), (0, ), (, )}. B = {(, ) }. C = {(, ) > }. D = {(, ) < }. E = {(, ) < } 6. F = {(, ) 0} 7. G = {(, ) > } 8. H = {(, ) < } 9. I = {(, ) 0, 0} 0. J = {(, ) < <, < < }.. The line = The line =.. The line = The line =. 6. The line = 0 is the -ais The line = 0 is the -ais

19 . Relations 7. = + The graph has no -intercepts -intercept: (0, ) (, ) (, ) (, ) 0 (0, ) (, ) (, ) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is smmetric about the -ais The graph is not smmetric about the origin (e.g. (, ) is on the graph but (, ) is not). = 8 -intercepts: (, 0), (, 0) -intercept: (0, 8) (, ) 7 (, 7) 0 (, 0) (, ) 0 8 (0, 8) 9 (, 9) 8 (, 8) (, ) 0 (, 0) 7 (, 7) The graph is not smmetric about the -ais (e.g. (, 7) is on the graph but (, 7) is not) The graph is not smmetric about the -ais (e.g. (, 7) is on the graph but (, 7) is not) The graph is not smmetric about the origin (e.g. (, 7) is on the graph but (, 7) is not)

20 8 Relations and Functions. = -intercepts: (, 0), (0, 0), (, 0) -intercept: (0, 0) (, ) 6 (, 6) 0 (, 0) 0 0 (0, 0) 0 (, 0) 6 (, 6) 6 6 The graph is not smmetric about the -ais. (e.g. (, 6) is on the graph but (, 6) is not) The graph is not smmetric about the -ais. (e.g. (, 6) is on the graph but (, 6) is not) The graph is smmetric about the origin.. = -intercepts: ( ±, 0 ), (0, 0) -intercept: (0, 0) (, ) (, ) ( ) 9, 9 (, ) (, 0 0 (0, 0) ) ( ), (, ) 9 ( ), 9 (, ) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is smmetric about the origin

21 . Relations 9. = -intercept: (, 0) The graph has no -intercepts (, ) 0 (, 0) (, ) 6 (6, ) (, ) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the origin (e.g. (, ) is on the graph but (, ) is not) 6. = + -intercept: (, 0) -intercept: (0, ) (, ) (, ) 0 (, 0) (, ) (, ) 0 (0, ) (, ) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the origin (e.g. (, ) is on the graph but (, ) is not)

22 0 Relations and Functions 7. = 7 Re-write as: = 7. -intercept: ( 7, 0) -intercept: (0, 7) (, ) (, ) 0 (, 0) 0 7 (0, 7) (, ) (, ) (, ) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the origin (e.g. (, ) is on the graph but (, ) is not) 8. = 0 Re-write as: = 0. -intercepts: ( 0, 0) -intercept: (0, ) (, ) 8 (, 8) ( ), 0 (0, ) 7 ( ), 7 (, ) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the origin (e.g. (, ) is on the graph but (, ) is not)

23 . Relations 9. ( + ) + = 6 Re-write as = ± 6 ( + ). -intercepts: ( 6, 0), (, 0) -intercepts: ( 0, ± ) (, ) 6 0 ( 6, 0) ± (, ± ) ± (, ±) 0 ± ( ) 0, ± 0 (, 0) 7 6 The graph is smmetric about the -ais The graph is not smmetric about the -ais (e.g. ( 6, 0) is on the graph but (6, 0) is not) The graph is not smmetric about the origin (e.g. ( 6, 0) is on the graph but (6, 0) is not) 0. = Re-write as: = ±. -intercepts: (, 0), (, 0) The graph has no -intercepts (, ) ± 8 (, ± 8) ± (, ± ) 0 (, 0) 0 (, 0) ± (, ± ) ± 8 (, ± 8) The graph is smmetric about the -ais The graph is smmetric about the -ais The graph is smmetric about the origin

24 Relations and Functions. 9 = 6 Re-write as: = ± The graph has no -intercepts -intercepts: (0, ±) (, ) ± (, ± ) ± (, ± ) 0 ± (0, ±) ± ( ), ± ± ( ), ± = Re-write as: =. The graph has no -intercepts The graph has no -intercepts (, ) (, ) (, ) (, ) (, ) (, ) (, ) The graph is smmetric about the -ais The graph is smmetric about the -ais The graph is smmetric about the origin The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is not smmetric about the -ais (e.g. (, ) is on the graph but (, ) is not) The graph is smmetric about the origin

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