Section 2-3 Quadratic Functions

Transcription

1 118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the Celsius range in temperature that corresponds to the Fahrenheit range of 6 F to 8 F. 72. Celsius/Fahrenheit. A formula for converting Fahrenheit degrees to Celsius degrees is given by the linear function C (F 32) 9 Determine to the nearest degree the Fahrenheit range in temperature that corresponds to a Celsius range of 2 C to 3 C. 73. Earth Science. In 1984, the Soviets led the world in drilling the deepest hole in the Earth s crust more than 12 kilometers deep. They found that below 3 kilometers the temperature T increased 2. C for each additional 1 meters of depth. (A) If the temperature at 3 kilometers is 3 C and is the depth of the hole in kilometers, write an equation using that will give the temperature T in the hole at any depth beyond 3 kilometers. (B) What would the temperature be at 1 kilometers? [The temperature limit for their drilling equipment was about 3 C.] (C) At what interval of depths will the temperature be between 2 C and 3 C, inclusive? 74. Aeronautics. Because air is not as dense at high altitudes, planes require a higher ground speed to become airborne. A rule of thumb is 3% more ground speed per 1, feet of elevation, assuming no wind and no change in air temperature. (Compute numerical answers to 3 significant digits.) (A) Let V s Takeoff ground speed at sea level for a particular plane (in miles per hour) A Altitude above sea level (in thousands of feet) V Takeoff ground speed at altitude A for the same plane (in miles per hour) Write a formula relating these three quantities. (B) What takeoff ground speed would be required at Lake Tahoe airport (6,4 feet), if takeoff ground speed at San Francisco airport (sea level) is 12 miles per hour? (C) If a landing strip at a Colorado Rockies hunting lodge (8, feet) requires a takeoff ground speed of 12 miles per hour, what would be the takeoff ground speed in Los Angeles (sea level)? (D) If the takeoff ground speed at sea level is 13 miles per hour and the takeoff ground speed at a mountain resort is 1 miles per hour, what is the altitude of the mountain resort in thousands of feet? Section 2-3 Quadratic Functions Quadratic Functions Completing the Square Properties of Quadratic Functions and Their Graphs Applications FIGURE 1 Square function h() 2. h() Quadratic Functions The graph of the square function, h() 2, is shown in Figure 1. Notice that the graph is symmetric with respect to the y ais and that (, ) is the lowest point on the graph. Let s eplore the effect of applying a sequence of basic transformations to the graph of h. (A brief review of Section 1- might prove helpful at this point.)

2 2-3 Quadratic Functions 119 Eplore/Discuss 1 Indicate how the graph of each function is related to the graph of h() 2. Discuss the symmetry of the graphs and find the highest or lowest point, whichever eists, on each graph. (A) f() ( 3) (B) g().( 2) (C) m() ( 4) (D) n() 3( 1) Graphing the functions in Eplore/Discuss 1 produces figures similar in shape to the graph of the square function in Figure 1. These figures are called parabolas. The functions that produced these parabolas are eamples of the important class of quadratic functions, which we now define. QUADRATIC FUNCTIONS If a, b, and c are real numbers with a, then the function f() a 2 b c is a quadratic function and its graph is a parabola.* Since the epression a 2 b c represents a real number for all real number replacements of, the domain of a quadratic function is the set of all real numbers. FIGURE 2 Graphs of quadratic functions. We will discuss methods for determining the range of a quadratic function later in this section. Typical graphs of quadratic functions are illustrated in Figure (a) f() 2 9 (b) g() (c) h() Completing the Square In Eplore/Discuss 1 we wrote each function as two different, but equivalent, epressions. For eample, f() ( 3) *A more general definition of a parabola that is independent of any coordinate system is given in Section 7-1.

3 12 2 LINEAR AND QUADRATIC FUNCTIONS It is easy to verify that these two epressions are equivalent by epanding the first epression. The first epression is more useful than the second for analyzing the graph of f. If we are given only the second epression, how can we determine the first? It turns out that this is a routine process, called completing the square, that is another useful tool to be added to our mathematical toolbo. Eplore/Discuss 2 Replace? in each of the following with a number that makes the equation valid. (A) ( 1) 2 2 2? (B) ( 2) 2 2 4? (C) ( 3) 2 2 6? (D) ( 4) 2 2 8? Replace? in each of the following with a number that makes the epression a perfect square of the form ( h) 2. (E) 2 1? (F) 2 12? (G) 2 b? Given the quadratic epression 2 b what must be added to this epression to make it a perfect square? To find out, consider the square of the following epression: ( m) 2 2 2m m 2 m 2 is the square of one-half the coefficient of. We see that the third term on the right side of the equation is the square of onehalf the coefficient of in the second term on the right; that is, m 2 is the square 1 of (2m). This observation leads to the following rule: 2 COMPLETING THE SQUARE To complete the square of the quadratic epression 2 b add the square of one-half the coefficient of ; that is, add b 2 2 or b 2 4 The resulting epression can be factored as a perfect square: b 2 b 2 2 b 2 2

4 2-3 Quadratic Functions 121 EXAMPLE 1 Solutions Completing the Square Complete the square for each of the following: (A) 2 3 (B) 2 6b (A) Add 3 that is, ; (B) 2 6b 2 6b 9b 2 ( 3b) 2 Add 6b that is, 9b ; MATCHED PROBLEM 1 Complete the square for each of the following: (A) 2 (B) 2 4m It is important to note that the rule for completing the square applies to only quadratic epressions in which the coefficient of 2 is 1. This causes little trouble, however, as you will see. Properties of Quadratic Functions and Their Graphs We now use the process of completing the square to transform the quadratic function f() a 2 b c into the standard form f() a( h) 2 k Many important features of the graph of a quadratic function can be determined by eamining the standard form. We begin with a specific eample and then generalize the results. Consider the quadratic function given by f() (1) We use completing the square to transform this function into standard form: f() ( 2 4) 4 2( 2 4?) 4 2( 2 4 4) 4 8 2( 2) 2 4 Factor the coefficient of 2 out of the first two terms. We add 4 to complete the square inside the parentheses. But because of the 2 outside the parentheses, we have actually added 8, so we must subtract 8. The transformation is complete and can be checked by epanding.

5 122 2 LINEAR AND QUADRATIC FUNCTIONS Thus, the standard form is f() 2( 2) 2 4 (2) If 2, then 2( 2) 2 and f(2) 4. For any other value of, the positive number 2( 2) 2 is added to 4, making f() larger. Therefore, f(2) 4 is the minimum value of f() for all a very important result! Furthermore, if we choose any two values of that are equidistant from 2, we will obtain the same value for the function. For eample, 1 and 3 are each one unit from 2 and their functional values are f(1) 2( 1) f(3) 2(1) Thus, the vertical line 2 is a line of symmetry if the graph of equation (1) is drawn on a piece of paper and the paper folded along the line 2, then the two sides of the parabola will match eactly. The above results are illustrated by graphing equation (1) or (2) and the line 2 in a suitable viewing window (Fig. 3). FIGURE 3 Graph of a quadratic function. f() ( 2) Minimum: f(2) 4 Ais of symmetry: 2 From the analysis of equation (2), illustrated by the graph in Figure 3, we conclude that f() is decreasing on (, 2] and increasing on [2, ). Furthermore, f() can assume any value greater than or equal to 4, but no values less than 4. Thus, Range of f: y 4 or [ 4, ) In general, the graph of a quadratic function is a parabola with line of symmetry parallel to the vertical ais. The lowest or highest point on the parabola, whichever eists, is called the verte. The maimum or minimum value of a quadratic function always occurs at the verte of the graph. The vertical line of symmetry through the verte is called the ais of the parabola. Thus, for f() , the vertical line 2 is the ais of the parabola and (2, 4) is its verte.

6 2-3 Quadratic Functions 123 From equation (2), we can see that the graph of f is simply the graph of g() 2 2 translated to the right 2 units and down 4 units, as shown in Figure 4. FIGURE 4 Graph of f is the graph of g translated. g() f() ( 2) 2 4 Notice the important results we have obtained from the standard form of the quadratic function f: The verte of the parabola The ais of the parabola The minimum value of f() The range of f A relationship between the graph of f and the graph of g Eplore/Discuss 3 Eplore the effect of changing the constants a, h, and k on the graph of f() a( h) 2 k. (A) Let a 1 and h. Graph function f for k 4,, and 3 simultaneously in the same viewing window. Eplain the effect of changing k on the graph of f. (B) Let a 1 and k 2. Graph function f for h 4,, and simultaneously in the same viewing window. Eplain the effect of changing h on the graph of f. (C) Let h and k 2. Graph function f for a.2, 1, and 3 simultaneously in the same viewing window. Graph function f for a 1, 1, and.2 simultaneously in the same viewing window. Eplain the effect of changing a on the graph of f. (D) Can all quadratic functions of the form y a 2 b c be rewritten as a( h) 2 k? We generalize the above discussion in the following bo:

7 124 2 LINEAR AND QUADRATIC FUNCTIONS PROPERTIES OF A QUADRATIC FUNCTION AND ITS GRAPH Given a quadratic function and the standard form obtained by completing the square f() a 2 b c a( h) 2 k a we summarize general properties as follows: 1. The graph of f is a parabola: f() Ais h f() k Ais h Verte (h, k) Verte (h, k) Ma f() k h Min f() h a Opens upward a Opens downward 2. Verte: (h, k) (parabola increases on one side of the verte and decreases on the other). 3. Ais (of symmetry): h (parallel to y ais). 4. f(h) k is the minimum if a and the maimum if a.. Domain: all real numbers; range: (, k] if a or [k, ) if a. 6. The graph of f is the graph of g() a 2 translated horizontally h units and vertically k units. EXAMPLE 2 Analyzing a Quadratic Function Find the standard form for the following quadratic function, analyze the graph, and check your results with a graphing utility: f(). 2 Solution We complete the square to find the standard form: f(). 2.( 2 2?).( 2 2 1)..( 1) 2.

8 2-3 Quadratic Functions 12 6 FIGURE 6 6 From the standard form we see that h 1 and k.. Thus, the verte is ( 1,.), the ais of symmetry is 1, the maimum value is f( 1)., and the range is (,.]. The function f is increasing on (, 1] and decreasing on [ 1, ). The graph of f is the graph of g(). 2 shifted to the left 1 unit and upward. units. To check these results, we graph f and g simultaneously in the same viewing window, use the built-in maimum routine to locate the verte, and add the graph of the ais of symmetry (Fig. ). 6 MATCHED PROBLEM 2 Find the standard form for the following quadratic function, analyze the graph, and check your results with a graphing utility: f() EXAMPLE 3 FIGURE 6 Finding the Equation of a Parabola Find an equation for the parabola whose graph is shown in Figure (a) (b) Solution Figure 6(a) shows that the verte of the parabola is (h, k) (3, 2). Thus, the standard equation must have the form f() a( 3) 2 2 (3) Figure 6(b) shows that f(4). Substituting in equation (3) and solving for a, we have f(4) a(4 3) 2 2 a 2 Thus, the equation for the parabola is f() 2( 3) MATCHED PROBLEM 3 Find the equation of the parabola with verte (2, 4) and y intercept (, 2).

9 126 2 LINEAR AND QUADRATIC FUNCTIONS Applications We now look at several applications that can be modeled using quadratic functions. EXAMPLE 4 Maimum Area A dairy farm has a barn that is 1 feet long and 7 feet wide. The owner has 24 ft of fencing and wishes to use all of it in the construction of two identical adjacent outdoor pens with the long side of the barn as one side of the pens and a common fence between the two (Fig. 7). The owner wants the pens to be as large as possible. FIGURE 7 1 feet 7 feet y Solutions (A) Since y 24 3, (A) Construct a mathematical model for the combined area of both pens in the form of a function A() (see Fig. 7) and state the domain of A. (B) Find the value of that produces the maimum combined area. (C) Find the dimensions and the area of each pen. A() (24 3) The distances and y must be nonnegative. Since y 24 3, it follows that cannot eceed 8. Thus, a model for this problem is FIGURE 8 A() , A() , 8 (B) Omitting the details, the standard form for A is A() 3( 4) 2 4,8 8 Thus, the maimum combined area of 4,8 ft 2 occurs at 4. This result is confirmed in Figure 8. (C) Each pen is by y/2 or 4 ft by 6 ft. The area of each pen is 4 ft 6 ft 2,4 ft 2.

10 2-3 Quadratic Functions 127 MATCHED PROBLEM 4 Repeat Eample 4 with the owner constructing three identical adjacent pens instead of two. Now that we have added quadratic functions to our mathematical toolbo, we can use this new tool in conjunction with another tool discussed previously regression analysis. In the net eample, we use both of these tools to investigate the effect of recycling efforts on solid waste disposal. EXAMPLE Solid Waste Disposal Franklin Associates Ltd. of Prairie Village, Kansas, reported the data in Table 1 to the U.S. Environmental Protection Agency. Year T A B L E 1 Municipal Solid Waste Disposal Annual Landfill Disposal (millions of tons) Per Person Per Day (pounds) (A) Let represent time in years with corresponding to 196, and let y represent the corresponding annual landfill disposal. Use regression analysis on a graphing utility to find a quadratic function of the form y a 2 b c that models this data. (Round the constants a, b, and c to three significant digits* when reporting your results.) (B) If landfill disposal continues to follow the trend ehibited in Table 1, when (to the nearest year) would the annual landfill disposal return to the 197 level? (C) Is it reasonable to epect the annual landfill disposal to follow this trend indefinitely? Eplain. Solutions (A) Since the values of y increase from 197 to 1987 and then begin to decrease, a quadratic model seems a better choice than a linear one. Figure 9 shows the details of constructing the model on a graphing utility. *For those not familiar with the meaning of significant digits, see Appendi C for a brief discussion of this concept.

11 128 2 LINEAR AND QUADRATIC FUNCTIONS 1 6 (a) Data FIGURE 9 (b) Regression equation (c) Regression equation transferred to equation editor (d) Graph of data and regression equation Rounding the constants to three significant digits, a quadratic regression equation for this data is y The graph in Figure 9(d) indicates that this is a reasonable model for this data. It is, in fact, the best quadratic equation for this data. (B) To determine when the annual landfill disposal returns to the 197 level, we add the graph of y to the graph [Fig. 1(a)]. The graphs of y 1 and y 2 intersect twice, once at 1 (197), and again at a later date. Using a built-in intersection routine [Fig. 1(b)] shows that the coordinate of the second intersection point (to the nearest integer) is 42. Thus, the annual landfill disposal returns to the 197 level of 88.2 million tons in 22. [Note: You will obtain slightly different results if you round the constants a, b, and c before finding the intersection point. As we stated before, we will always use the unrounded constants in calculations and only round the final answer.] FIGURE 1 1 y (a) (b) (C) The graph of y 1 continues to decrease and reaches somewhere between 211 and 211. It is highly unlikely that the annual landfill disposal will ever reach. As time goes by and more data becomes available, new models will have to be constructed to better predict future trends. MATCHED PROBLEM Refer to Table 1. (A) Let represent time in years with corresponding to 196, and let y represent the corresponding landfill disposal per person per day. Use regression analysis on a graphing utility to find a quadratic function of the form y a 2 b c that models this data. (Round the constants a, b, and c to three significant digits when reporting your results.)

12 2-3 Quadratic Functions 129 (B) If landfill disposal per person per day continues to follow the trend ehibited in Table 1, when (to the nearest year) would it fall below 1. pounds per person per day? (C) Is it reasonable to epect the landfill disposal per person per day to follow this trend indefinitely? Eplain. Answers to Matched Problems 1. (A) 2 2 (B) 2 4m 4m 2 ( 2m) Standard form: f() ( 1.) The verte is (1., 1.2), the ais of symmetry is 1., the maimum value of f() is 1.2, and the range of f is (, 1.2]. The function f is increasing on (, 1.] and decreasing on [1., ). The graph of f is the graph of g() 2 shifted 1. units to the right and 1.2 units upward. 3. f().( 2) (A) A() (24 4), 6 (B) The maimum combined area of 3,6 ft 2 occurs at 3 ft. (C) Each pen is 3 ft by 4 ft with area 1,2 ft 2.. (A) y (B) 23 EXERCISE A In Problems 1 6, complete the square and find the standard form of each quadratic function. 1. f() g() h() k() m() n() In Problems 7 12, write a brief verbal description of the relationship between the graph of the indicated function (from Problems 1 6) and the graph of y f() g() h() k() m() n() In Problems 13 18, match each graph with one of the functions in Problems

13 13 2 LINEAR AND QUADRATIC FUNCTIONS B For each quadratic function in Problems 19 24, sketch a graph of the function and label the ais and the verte. 19. f() f() f() f() f() f() In Problems 2 28, find the intervals where f is increasing, the intervals where f is decreasing, and the range. Epress answers in interval notation. 2. f() f() f() f() In Problems 29 32, use the graph of the parabola to find the equation of the corresponding quadratic function In Problems 33 38, find the equation of a quadratic function whose graph satisfies the given conditions. 33. Verte: (4, 8); intercept: Verte: ( 2, 12); intercept: 4 3. Verte: ( 4, 12); y intercept: Verte: (, 8); y intercept: Verte: (, 2); additional point on graph: ( 2, 2) 38. Verte: (6, 4); additional point on graph: (3, ) 39. Graph the line y. 3. Choose any two distinct points on this line and find the linear regression model for the data set consisting of the two points you chose. Eperiment with other lines of your choosing. Discuss the relationship between a linear regression model for two points and the line that goes through the two points. 4. Graph the parabola y 2. Choose any three distinct points on this parabola and find the quadratic regression model for the data set consisting of the three points you chose. Eperiment with other parabolas of your choice. Discuss the relationship between a quadratic regression model for three noncollinear points and the parabola that goes through the three points. 41. Let f() ( 1) 2 k. Discuss the relationship between the values of k and the number of intercepts for the graph of f. Generalize your comments to any function of the form f() a( h) 2 k, a 42. Let f() ( 2) 2 k. Discuss the relationship between the values of k and the number of intercepts for the graph of f. Generalize your comments to any function of the form f() a( h) 2 k, a C Recall that the standard equation of a circle with radius r and center (h, k) is ( h) 2 (y k) 2 r 2 In Problems 43 46, use completing the square twice to find the center and radius of the circle with the given equation.

14 2-3 Quadratic Functions y 2 6 4y y 2 2 1y 4. 2 y 2 8 2y y y Let f() a( h) 2 k. Compare the values of f(h r) and f(h r) for any real number r. Interpret the results in terms of the graph of f. 48. Let f() a 2 b c, a. Epress each of the following in terms of a, b, and c: (A) The ais of symmetry (B) The verte (C) The maimum or minimum value of f, whichever eists. 2. Repeat Problem 1 for f() Find the minimum product of two numbers whose difference is 3. Is there a maimum product? Eplain. 4. Find the maimum product of two numbers whose sum is 6. Is there a minimum product? Eplain. APPLICATIONS. Construction. A horse breeder wants to construct a corral net to a horse barn feet long, using all of the barn as one side of the corral (see the figure). He has 2 feet of fencing available and wants to use all of it. Horse barn Problems 49 2 are calculus-related. In geometry, a line that intersects a circle in two distinct points is called a secant line, as shown in figure (a). In calculus, the line through the points ( 1, f( 1 )) and ( 2, f( 2 )) is called a secant line for the graph of the function f, as shown in figure (b). feet Corral y P Q Secant line for a circle (a) In Problems 49 and, find the equation of the secant line through the indicated points on the graph of f. Graph f and the secant line on the same coordinate system. 49. f() 2 4; ( 1, 3), (3, ). f() 9 2 ; ( 2, ), (4, 7) ( 1, f( 1 )) f() ( 2, f( 2 )) Secant line for the graph of a function (b) 1. Let f() 2 3. If h is a nonzero real number, then (2, f(2)) and (2 h, f(2 h)) are two distinct points on the graph of f. (A) Find the slope of the secant line through these two points. (B) Evaluate the slope of the secant line for h 1, h.1, h.1, and h.1. What value does the slope seem to be approaching? (A) Epress the area A() of the corral as a function of and indicate its domain. (B) Find the value of that produces the maimum area. (C) What are the dimensions of the corral with the maimum area? 6. Construction. Repeat Problem if the horse breeder has only 14 feet of fencing available for the corral. Does the maimum value of the area function still occur at the verte? Eplain. 7. Projectile Flight. An arrow shot vertically into the air from a cross bow reaches a maimum height of 484 feet after. seconds of flight. Let the quadratic function d(t) represent the distance above ground (in feet) t seconds after the arrow is released. (If air resistance is neglected, a quadratic model provides a good approimation for the flight of a projectile.) (A) Find d(t) and state its domain. (B) At what times (to two decimal places) will the arrow be 2 feet above the ground? 8. Projectile Flight. Repeat Problem 7 if the arrow reaches a maimum height of 324 feet after 4. seconds of flight. 9. Engineering. The arch of a bridge is in the shape of a parabola 14 feet high at the center and 2 feet wide at the base (see the figure).

15 132 2 LINEAR AND QUADRATIC FUNCTIONS (A) Find a quadratic regression model for the revenue data using as the independent variable. 2 ft h() 14 ft (B) Find a linear regression model for the cost data using as the independent variable. (C) Use the regression models from parts A and B to estimate the coordinates (to the nearest integer) of the break-even points. (A) Epress the height of the arch h() in terms of and state its domain. (B) Can a truck that is 8 feet wide and 12 feet high pass through the arch? (C) What is the tallest 8-foot-wide truck that can pass through the arch? (D) What (to two decimal places) is the widest 12-foothigh truck that can pass through the arch? 6. Engineering. The roadbed of one section of a suspension bridge is hanging from a large cable suspended between two towers that are 2 feet apart (see the figure). The cable forms a parabola that is 6 feet above the roadbed at the towers and 1 feet above the roadbed at the lowest point. (A) Epress the vertical distance d() (in feet) from the roadbed to the suspension cable in terms of and state the domain of d. (B) The roadbed is supported by seven equally spaced vertical cables (see the figure). Find the combined total length of these supporting cables. 61. Break-Even Analysis. Table 1 contains revenue and cost data for the production of lawn mowers where R is the total revenue (in dollars) from the sale of lawn mowers and C is the total cost (in dollars) of producing lawn mowers. T A B L E 1 2 ft d() ft 6 ft 62. Profit Analysis. Use the regression models computed in Problem 61 to estimate the indicated quantities. (A) How many lawn mowers (to the nearest integer) must be produced and sold to realize a profit of $,? (B) How many lawn mowers (to the nearest integer) must be produced and sold to realize the maimum profit? What is the maimum profit (to the nearest dollar)? 63. Water Consumption. Table 2 contains data related to the water consumption in the United States for selected years from 196 to 199. This data is based on U.S. Geological Survey, Estimated Use of Water in the United States in 199, circular 181, and previous quinquennial issues. Year T A B L E 2 Daily Water Consumption Total (billion gallons) Irrigation (billion gallons) (A) Let the independent variable represent years since 196. Find a quadratic regression model for the total daily water consumption. (B) If daily water consumption continues to follow the trend ehibited in Table 2, when (to the nearest year) would the total consumption return to the 196 level? 2 6 1, 1,3 1,7 R ($) 9, 27, 29, 26, 14, C ($) 14, 16, 21, 23, 27, 64. Water Consumption. Refer to Problem 63. (A) Let the independent variable represent years since 196. Find a quadratic regression model for the daily water consumption for irrigation. (B) If daily water consumption continues to follow the trend ehibited in Table 2, when (to the nearest year) would the consumption for irrigation return to the 196 level?