2 Analysis of Graphs of


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1 ch.pgs116 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr, the circle in the plane and the sphere in space were considered b the earl Greeks to be the most perfect geometric figures. Aristotle assumed a spherical shape for the celestial bodies because an other would detract from their heavenl perfection. Smmetr has been an important characteristic of art from the earliest times. The art of M.C. Escher ( ) is composed of smmetries and translations. Leonardo da Vinci s sketches indicate a superior understanding of smmetr. Almost all nature ehibits smmetr from the heagons of snowflakes to the diatom, a microscopic sea plant. Perhaps the most striking eamples of smmetr in nature are crstals. The photos on this page show a diatom and a cross section of a tourmaline. This chapter introduces, among other things, how smmetr is found in the graphs of certain functions. CHAPTER OUTLINE.1 Graphs of Basic Functions and Relations; Smmetr. Vertical and Horizontal Shifts of Graphs.3 Stretching, Shrinking, and Ref lecting Graphs.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications.5 PiecewiseDefined Functions.6 Operations and Composition (Source: Mathematics, Life Science Librar, Time Inc., New York, 1963.) 1
2 CHAPTER Analsis of Graphs of Functions.1 GRAPHS OF BASIC FUNCTIONS AND RELATIONS; SYMMETRY Continuit Increasing and Decreasing Functions The Identit Function The Squaring Function and Smmetr with Respect to the Ais The Cubing Function and Smmetr with Respect to the Origin The Square Root and Cube Root Functions The Absolute Value Function The Relation and Smmetr with Respect to the Ais Even and Odd Functions Continuit In Chapter 1, our work dealt mainl with linear functions. The graph of a linear function, a straight line, ma be drawn b hand over an interval of its domain without picking the pencil up from the paper. In mathematics we sa that a function with this propert is continuous over an interval. The formal definition of continuit requires concepts from calculus, but we can give an informal definition at the precalculus level. Continuit A function is continuous over an interval of its domain if its handdrawn graph over that interval can be sketched without lifting the pencil from the paper. If a function is not continuous at a point, then it ma have a point of discontinuit (Figure 1(a)), or it ma have a vertical asmptote (a vertical line that the graph does not intersect, as in Figure 1(b)). Asmptotes will be discussed further in Chapter 4. FIGURE Both functions are discontinuous at =. 3 FIGURE 3 (a) FIGURE 1 EXAMPLE 1 Determining Intervals of Continuit Figures, 3, and 4 show graphs of functions. Indicate the intervals of the domain over which the are continuous. (b) FIGURE 4 Solution The function in Figure is continuous over the entire domain of real numbers,,. The function in Figure 3 has a point of discontinuit at 3. It is continuous over the interval,3 and the interval 3,. Finall, the function in Figure 4 has a vertical asmptote at, as indicated b the dashed line. It is continuous over the intervals, and,.
3 .1 Graphs of Basic Functions and Relations; Smmetr 3 Increasing and Decreasing Functions If a continuous function is not constant over an interval, then its graph will either rise from left to right or fall from left to right or some combination of both. We use the words increasing and decreasing to describe this behavior. For eample, a linear function with positive slope is increasing over its entire domain, while one with negative slope is decreasing. See Figure 5. f() = m + b, m > (a) f() = m + b, m < f is increasing, since it rises from left to right. f is decreasing, since it falls from left to right. (b) FIGURE 6 Looking Ahead to Calculus In calculus, determining where a function increases, decreases, or is constant is important in curvesketching. The derivative of a function provides a formula for determining the slope of a line tangent to the curve. If the slope is positive for a given domain value, the function is increasing at that point; if the slope is negative, the function is decreasing; and if the slope is, the function is constant. For eample, look at Figure 7(a). Choose a point on the curve between 1, f 1 and, f, and draw a line tangent to the curve at that point. The line rises from left to right, so its slope is positive. This positive slope indicates that the function is increasing at that point. (Etend this idea to Figures 7(b) and 7(c).) FIGURE 5 The table feature of a graphing calculator can be used to illustrate the concepts of increasing and decreasing functions. As seen in Figure 6(a), the graph of Y 1 1 has positive slope m 1, and as values increase, range values increase. The graph of Y 1 has negative slope m 1, and as values increase, range values decrease, as shown in Figure 6(b). Informall speaking, a function increases on an interval of its domain if its graph rises from left to right. A function decreases on an interval of its domain if its graph falls from left to right. A function is constant on an interval of its domain if its graph is horizontal on the interval. (While some functions are increasing, decreasing, or constant over their entire domains, man are not. Hence, we speak of increasing, decreasing, or constant over an interval of the domain.) Criteria for Increasing, Decreasing, and Constant Suppose that a function f is defined over an interval I. (a) f increases on I if, whenever 1, f 1 f ; (b) f decreases on I if, whenever 1, f 1 f ; (c) f is constant on I if, for ever and, f 1 f. Figure 7 illustrates these ideas. f ( ) f ( 1 ) When 1 <, f ( 1 ) < f ( ). f is increasing. f ( 1 ) 1 f ( ) When 1 <, f ( 1 ) > f ( ). f is decreasing. (a) (b) (c) f ( 1 )=f ( ) For 1 and, f ( 1 ) = f ( ). f is constant. FIGURE 7
4 4 CHAPTER Analsis of Graphs of Functions NOTE To decide whether a function is increasing, decreasing, or constant on an interval, ask ourself What does do as goes from left to right? 6 FIGURE 8 FOR DISCUSSION 1. In the standard viewing window of our calculator, enter an linear function defined b m b with m. Now, trace the graph from left to right. Watch the values as the values get larger. What is happening to? How does this reinforce the concepts presented so far in this section?. Repeat this eercise, but with m. 3. Repeat this eercise, but with m. EXAMPLE Determining Intervals over Which a Function Is Increasing, Decreasing, or Constant Figure 8 shows the graph of a function. Determine the intervals over which the function is increasing, decreasing, or constant. Solution We should ask, What is happening to the values as values are getting larger? Moving from left to right on the graph, we see that on the interval,1, the values are decreasing; on the interval 1, 3, the values are increasing; and on the interval 3,, the values are constant (and equal to 6). Therefore, the function is decreasing on,1, increasing on 1, 3, and constant on 3,. CAUTION A common error involves writing range values ( values) when determining intervals like those in Eample. Remember that we are determining intervals of the domain, and thus are interested in values for our interval designations. Range values are not written when designating such intervals. The net part of this section introduces several important basic functions and relations. The Identit Function If we let m 1 and b in the general form of the linear function defined b f m b, we get the identit function, or f. This function pairs ever real number with itself. See Figure 9. FUNCTION CAPSULE IDENTITY FUNCTION Domain:, f() Range:, f () = 1 1 f() = 1 1 FIGURE 9 f is increasing on its entire domain,,. It is continuous on its entire domain,,. The Squaring Function and Smmetr with Respect to the Ais The degree of a polnomial function in the variable is the greatest eponent on in the polnomial. The identit function is a degree 1 function. The squaring function,
5 .1 Graphs of Basic Functions and Relations; Smmetr 5 Looking Ahead to Calculus Calculus allows us to find areas of regions in the plane. For eample, we can find the area of the region below the graph of, above the ais, bounded on the left b the line, and bounded on the right b. Draw a sketch of this region, and notice that due to the smmetr of the graph of, the desired area is twice that of the area to the right of the ais. Thus, smmetr can be used to reduce the original problem to an easier one b simpl finding the area to the right of the ais, and then doubling the answer. defined b f, is the simplest degree (or quadratic) function. (The word quadratic refers to degree.) See Figure 1. The squaring function pairs ever real number with its square. Its graph is called a parabola. The point at which the graph changes from decreasing to increasing (the point, ) is called the verte of the graph. (For a parabola that opens downward, the verte is the point at which the graph changes from increasing to decreasing.) FUNCTION CAPSULE SQUARING FUNCTION f() = Domain:, 1 1 Range:, f () = f() 1 1 FIGURE 1 f decreases on the interval, and increases on the interval,. It is continuous on its entire domain,,. If we were able to fold the graph of f along the ais, the two halves would coincide eactl. In mathematics we refer to this propert as smmetr, and we sa that the graph of f is smmetric with respect to the ais. Smmetr with Respect to the Ais If a function f is defined so that f( ) f() for all in its domain, then the graph of f is smmetric with respect to the ais. This table supports, but does not prove, that the graph of the squaring function is smmetric with respect to the ais. To illustrate that the graph of f is smmetric with respect to the ais, note that f 4 f 4 16 f 3 f 3 9 f f 4 f 1 f 1 1. This pattern holds for an real number, since f f. In general, if a graph is smmetric with respect to the ais, the following is true: If a, b is on the graph, so is a, b.
6 6 CHAPTER Analsis of Graphs of Functions The Cubing Function and Smmetr with Respect to the Origin The cubing function, defined b f 3, is the simplest degree 3 function. It pairs with each real number the third power, or cube, of the number. See Figure 11. The point at which the graph of the cubing function changes from opening downward to opening upward (the point, ) is called an inflection point. FUNCTION CAPSULE CUBING FUNCTION f() 3 Domain:, Range:, 8 f() = 3 1 f() = FIGURE 11 f 3 increases on its entire domain,,. It is continuous on its entire domain,,. If we were able to fold the graph of f 3 along the ais and then along the ais, forming a corner at the origin, the two parts of the graph would coincide eactl. We sa that the graph of f 3 is smmetric with respect to the origin. Smmetr with Respect to the Origin If a function f is defined so that f( ) f() for all in its domain, then the graph of f is smmetric with respect to the origin. This table supports, but does not prove, that the graph of the cubing function is smmetric with respect to the origin. To illustrate that the graph of f 3 is smmetric with respect to the origin, note that f f 8 f 1 f 1 1. This pattern holds for an real number, since f f. In general, if a graph is smmetric with respect to the origin, the following is true: If a, b is on the graph, so is a, b. EXAMPLE 3 Determining Smmetr Show that (a) f has a graph that is smmetric with respect to the ais; (b) f 3 4 has a graph that is smmetric with respect to the origin.
7 .1 Graphs of Basic Functions and Relations; Smmetr 7 Analtic Solution (a) We must show that f f for an. f Substitute for f This proves that the graph is smmetric with respect to the ais. (b) We must show that f f for an. f 3 4 Substitute for f In the line denoted *, note that the signs of the coefficients are all opposites of those in f. We completed the solution b factoring out 1, showing that the final result is f. * Graphing Calculator Solution (a) The graph in Figure 1 supports the analtic proof. 1 FIGURE 1 (b) The graph in Figure 13 supports the analtic proof. 1 f() = This graph is smmetric with respect to the ais. f() = This graph is smmetric with respect to the origin. FIGURE The Square Root and Cube Root Functions Some functions involve epressions with radicals. One of these is the square root function, defined b f. See Figure 14. Notice that for the function value to be a real number, we must have. Thus, the domain is restricted to,. FUNCTION CAPSULE SQUARE ROOT FUNCTION f() Domain:, Range:, 1 f() = f() = 1 1 FIGURE 14 f increases on its entire domain,,. It is continuous on its entire domain,,.
8 8 CHAPTER Analsis of Graphs of Functions TECHNOLOGY NOTE The definition of a rational eponent allows us to enter as 1/ and 3 as 1/3 when using a calculator. Further discussion of rational eponents can be found in Chapter R. The cube root function, defined b f 3 (Figure 15), differs from the square root function in that an real number positive,, or negative has a real cube root, and thus the domain is,. When when,, 3, 3, and when, 3. As a result, the range is also,. FUNCTION CAPSULE CUBE ROOT FUNCTION f() Domain:, Range:, f() = f() = 1 1 FIGURE 15 f 3 increases on its entire domain,,. It is continuous on its entire domain,,. TECHNOLOGY NOTE You should become familiar with the command on our particular calculator that allows ou to graph the absolute value function. Consult the graphing calculator manual that accompanies this tet or our owner s manual. The Absolute Value Function On a number line, the absolute value of a real number, denoted, represents its undirected distance from the origin,. The absolute value function, which pairs ever real number with its absolute value, is defined as follows. Absolute Value f() if > if < Notice that this function is defined in two parts. We use if is positive or, and we use if is negative. Since can be an real number, the domain of the absolute value function is,, but since cannot be negative, the range is,. See Figure 16.
9 .1 Graphs of Basic Functions and Relations; Smmetr 9 FUNCTION CAPSULE ABSOLUTE VALUE FUNCTION f() Domain:, Range:, f() = 1 f() = FIGURE 16 f decreases on the interval, and increases on,. It is continuous on its entire domain,,. FOR DISCUSSION Based on the functions discussed so far in this section, answer each question. 1. Which functions have graphs that are smmetric with respect to the ais?. Which functions have graphs that are smmetric with respect to the origin? 3. Which functions have graphs with neither of these smmetries? 4. Wh is it not possible for the graph of a nonzero function to be smmetric with respect to the ais? The Relation and Smmetr with Respect to the Ais Recall that a function is a relation where ever domain value is paired with one and onl one range value. However, there are cases where we are interested in graphing relations that are not functions, and one of the simplest of these is the relation defined b the equation. Notice that the table of selected ordered pairs below indicates that this relation has two different values for each positive value of. If we plot these points and join them with a smooth curve, we find that the graph of is a parabola opening to the right. See Figure 17. Selected Ordered Pairs for Two different values for the same value = FIGURE 17
10 1 CHAPTER Analsis of Graphs of Functions TECHNOLOGY NOTE You should be aware of how function mode (as opposed to parametric mode) is activated for our model. If a graphing calculator is set in function mode, it is not possible to graph directl. (However, if it is set in parametric mode, such a curve is possible with direct graphing.) To overcome this problem, we begin with and take the square root on each side, remembering to choose both the positive and negative square roots of. Given equation Transform so that is on the left. Take square roots. Now, we have defined b two functions, Y 1 and Y. Entering both of these into a calculator gives the graph shown in Figure 18(a). Y 1 = 1 Y = The graph of the relation = is smmetric with respect to the ais. (a) This screen shows selected ordered pairs for the relation =, where Y 1 = and Y =. (b) FIGURE 18 If we were to fold the graph of along the ais, the two halves of the parabola would coincide. This graph ehibits smmetr with respect to the ais. A table supports this smmetr, as seen in Figure 18(b). Smmetr with Respect to the Ais If replacing with in an equation results in the same equation, then the graph is smmetric with respect to the ais. To illustrate this, begin with and replace with. 1 The result is the same equation. In general, if a graph is smmetric with respect to the ais, the following is true: If a, b is on the graph, so is a, b.
11 .1 Graphs of Basic Functions and Relations; Smmetr 11 A summar of the tpes of smmetr just discussed follows. Basic Fact about Points Tpe of Smmetr Eample on the Graph ais smmetr If a, b is on the graph, so is a, b. ( a, b) (a, b) FOR DISCUSSION The three functions discussed in Eample 4 on the net page are graphed in Figure 19, but not necessaril in the same order as in the eample. Without actuall using our calculator, identif each function. 1 Origin smmetr If a, b is on the graph, so is a, b. (a, b) ( a, b) ais smmetr If a, b is on the graph, so (not possible is a, b. (a, b) for a nonzero function) (a, b) 4 4 Even and Odd Functions 1 (a) The concepts of smmetr with respect to the ais and smmetr with respect to the origin are closel associated with the concepts of even and odd functions. 3 Even and Odd Functions 4 1 (b) 4 A function f is called an even function if f f for all in the domain of f. (Its graph is smmetric with respect to the ais.) A function f is called an odd function if f f for all in the domain of f. (Its graph is smmetric with respect to the origin.) 15 As an illustration, observe the following. f is an even function because f 3 is an odd function because f 4 5 (c) FIGURE 19 4 f f. f. A function ma be neither even nor odd; for eample, nor odd. f is neither even
12 1 CHAPTER Analsis of Graphs of Functions EXAMPLE 4 Determining Whether Functions are Even, Odd, or Neither Decide if each function defined is even, odd, or neither. (a) f (b) f (c) f 3 5 Solution (a) Replacing with gives Since f f for each in the domain of the function, f is even. (b) Here The function is odd. (c) f f f. f f. Since f f and f f, f is neither even nor odd..1 EXERCISES Concept Check Fill in each blank with the correct response. 1. The domain and the range of the identit function are both.. The domain of the squaring function is, while its range is. 3. The graph of the cubing function changes from opening downward to opening upward at the point. 4. The domain of the square root function is, and its range is. 5. The cube root function on its entire (increases/decreases) domain. 6. The absolute value function decreases on the interval and increases on the interval. 7. The graph of the relation given b is smmetric with respect to the. 8. The function defined b f 4 is an (even/odd) function. 9. The function defined b f 3 is an (even/odd) function. 1. If a function is even, its graph is smmetric with respect to the ; if it is odd, its graph is smmetric with respect to the. Determine the intervals of the domain over which the given function is continuous (, 3) (1, ) (, 1) 3
13 .1 Graphs of Basic Functions and Relations; Smmetr 13 Determine the intervals of the domain over which the given function is (a) increasing, (b) decreasing, and (c) constant. Then give the (d) domain and (e) range (3, ) ( 1, 3) (4, 3) (1, 3) (4, 3). 1.. (, ) (3, 1.5) ( 3, 3) (3, 3) (3, 1) (, ) (3, ) In Eercises 3 8, graph each function in the standard viewing window of our calculator, and trace from left to right along a representative portion of the specified interval. Then, fill in the blank of this sentence with either increasing or decreasing: OVER THE INTERVAL SPECIFIED, THIS FUNCTION IS. 3. f 5 ;, 4. f 3 ;, 5. f 4 ;, 6. f 4 ;, 7. f ;, 8. f ;, Using visual observation, determine whether each graph is smmetric with respect to the (a) ais, (b) ais, (c) origin
14 14 CHAPTER Analsis of Graphs of Functions Concept Check In Eercises 37 4, ou are given a calculatorgenerated graph of a relation that ehibits a tpe of smmetr. Based on the given information at the bottom of each figure, determine the coordinates of another point that must also lie on the graph. 37. smmetric with respect 38. smmetric with respect 39. smmetric with respect to the ais to the ais to the ais smmetric with respect 41. smmetric with respect 4. smmetric with respect to the ais to the origin to the origin Concept Check Complete the table if f is an even function. f() Concept Check Complete the left half of the graph of f in the figure for each of the following conditions: (a) f f (b) f f Concept Check Complete the table if g is an odd function. g() Concept Check Complete the right half of the graph of f in the figure for each of the following conditions: (a) f is odd (b) f is even. Based on the ordered pairs seen in each pair of tables, make a conjecture about whether the function defined in neither even nor odd Y 1 is even, odd, or
15 .1 Graphs of Basic Functions and Relations; Smmetr 15 Prove analticall that each function is even. 49. f Prove analticall that each function is odd. 51. f f 6 8 f Use the analtic method of Eample 3 to determine whether the graph of the given function is smmetric with respect to the ais, smmetric with respect to the origin, or neither. Use our calculator to support our conclusion, using the standard window. 53. f f f f f f Refer to Eercises 53 58, and determine which eercises have functions that are: (a) even (b) odd (c) neither even nor odd. 6. Write an eplanation of the connections between smmetr with respect to the ais, smmetr with respect to the origin, even functions, and odd functions. In Eercises 61 and 6, ou are given a portion of the graph of a function near the origin, along with information regarding smmetr of the graph of the complete function. Sketch b hand a simulated calculator graph, using the window given, to show what each graph would look like. 61. Smmetric with respect to the ais; 6. Smmetric with respect to the origin; window: 5, 5 b, 5 window: 5, 5 b, RELATING CONCEPTS For individual or group investigation (Eercises 63 66) The line in the sketch and the line on the screen are said to be tangent to the curve at point P. 5 P Sketch b hand the graph of, and choose a point at which the function is decreasing. Draw a tangent line at that point. Is the slope of this line positive, negative, or zero? 64. Repeat Eercise 63, but choose a point at which the function is increasing Repeat Eercise 63, but choose the point at which the function changes from decreasing to increasing. 66. Based on our answers to Eercises 63 65, what conclusions can be drawn? These conclusions can be verified using methods discussed in calculus.
16 16 CHAPTER Analsis of Graphs of Functions Graph the function defined b 9 3 in the standard window of our calculator. Tr to locate the point for which 3. What happens? Wh do ou think this happens? (Functions of this kind, called rational functions, will be studied in detail in Chapter 4.) Graph the function defined b 9 3 in the standard window of our calculator. Do ou think that determining continuit strictl b observation of a calculatorgenerated graph is foolproof?. VERTICAL AND HORIZONTAL SHIFTS OF GRAPHS Vertical Shifts Horizontal Shifts Combinations of Vertical and Horizontal Shifts Effects of Shifts on Domain and Range Horizontal Shifts Applied to Equations for Modeling In this section we eamine how the graphs of the basic functions introduced in the previous section can be shifted verticall and horizontall in the plane. Vertical Shifts TECHNOLOGY NOTE This screen shows how to minimize kestrokes in the activit in the For Discussion bo. The entr for Y 1 can be altered as needed. FOR DISCUSSION In each group, we give four related functions. Graph the four functions in the first group (Group A), and then answer the questions regarding those functions. Repeat the process for Group B, Group C, and Group D. Use the standard viewing window in each case. A B C D How does the graph of compare to the graph of 1?. How does the graph of 3 compare to the graph of 1? 3. How does the graph of 4 compare to the graph of 1? 4. If c, how do ou think the graph of f c would compare to the graph of f? 5. If c, how do ou think the graph of f c would compare to the graph of f? Choosing our own value of c, support our answers to Items 4 and 5 graphicall. (Be sure that our choice is appropriate for the standard window.) The objective of the preceding discussion activit was to make conjectures about how adding or subtracting a positive constant c affects the graph of a function. In each case, we obtained a vertical shift, or translation, of the graph of the basic function with which we started. Although our observations were based on the graphs of four different basic functions, the can be generalized to an function.
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