Solved: Maximizing Revenue Suppose that the manufacturer

Graph y = 2x - 3. (pp. 173-( 85)

Find the slope of the line joining the points (2,5) and ( -1, 3). (pp. 1 73-174)

Find the average rate of change of f(x) = 3x2 - 2, from 2 to 4. (pp. 231 -238)

Solve: 60x - 900 = - 15x + 2850. (pp. 86-93)

If f(x) = x2 - 4, find f( -2). (pp. 208-219)

True or False The graph of the function f(x) = x2 is increasing on the interval (0, oc ) . (pp. 231-238)

For the graph of the linear function f(x) = mx + b, m is the ____ and b is the _____.

For the graph of the linear function H(z) = -4z + 3, the slope is __ and the y-intercept is __ .

If the slope m of the graph of a linear function is __ , the function is increasing over its domain.

True or False The slope of a line is the average rate of change of the linear function.

True or False If the average rate of change of a linear function iS, then if y increases by 3,x will increase by 2.

True or False The average rate of change of f(x) = 2x + 8 is 8.

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. f(x) = 2x + 3

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. g(x) = 5x - 4

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. h(x) = -3x + 4

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. p( x) = -x + 6

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. f(X) = X - 3

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. h(x) = --x + 4 3

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. F(x) = 4

In Problems 13-20, a linear function is given. (a) Determine the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph the linear function. (c) Determine the average rate of change of each function. (d) Determine whether the linear function is increasing, decreasing, or constant. G(x) = -2

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) 22. x y = f (x) -2 4 -1 0 -2 -5 2 -8

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) --2 1 /4 -1 1 /2 0 2 2 4

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) 24. x y =f (x) -2 -8 - 1 -3 0 0 2 0

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y =f (x) -2 -4 - 1 0 0 4 8 2 12

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) 26. x y = f (x) -2 -26 -1 -4 0 2 -2 2 - 10

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) -2 -4 -1 -3.5 0 -3 -2.5 2 -2

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) 28. x y = f (x) -2 8 - 1 8 0 8 8 2 8

In Problems 21-28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. x y = f (x) -2 0 - 1 1 0 4 9 2 16

Suppose that f(x) = 4x - 1 and g(x) = -2x + 5. (a) Solve f(x) = O. (b) Solve f(x) > O. (c) Solve f(x) = g(x). (d) Solve f(x) :s g(x). (e) Graph y = f(x) and y = g(x) and label the point that represents the solution to the equation f(x) = g(x).

Suppose that f(x) = 3x + 5 and g(x) = -2x + 15. (a) Solve f(x) = O. (b) Solve f(x) < O. (c) Solve f(x) = g(x). (d) Solve f(x) 2: g(x). (e) Graph y = f(x) and y = g(x) and label the point that represents the solution to the equation f(x) = g(x).

n parts (a)-(f), use the following figure. Y x (a) Solve f(x) = 50. (c) SOlve f(x) = O. (e) Solve f(x) :5 80. (b) Solve f(x) = 80. (d) Solve f(x) > 50. (f) Solve 0 < f(x) < 80.

In parts (a)-(f), use the following figure. Y= g(x) Y x (a) Solve g(x) = 20. (c) Solve g(x) = O. (e) Solve g(x) :5 60. (b) Solve g(x) = 60. (d) Solve g(x) > 20. (f) Solve 0 < g(x) < 60.

In parts (a) and (b) use the following figure. Y = f(x) Y (a) Solve the equation: f(x) = g(x). (b) Solve the inequality: f(x) > g(x).

In parts (a) and (b), use the following figure. Y Y= f(x) (a) Solve the equation: f(x) = g(x). (b) Solve the inequality: f(x) :5 g(x).

In parts (a) and (b), use the following figure. Y = f(x) x Y= h(x) x Y= g(x) (a) Solve the equation: f(x) = g(x). (b) Solve the inequality: g(x) :5 f(x) < hex).

In parts (a) and (b), use the following figure. (a) Solve the equation: f(x) = g(x). Y= h(x) x Y= g(x) Y= f(x) (b) Solve the inequality: g(x) < f(x) :5 hex).

Car Rentals The cost C, in dollars, of renting a moving truck for a day is given by the function C(x) = 0.25x + 35, where x is the number of miles driven. (a) What is the cost if you drive x = 40 miles? (b) If the cost of renting the moving truck is $80, how many miles did you drive? (c) Suppose that you want the cost to be no more than $100. What is the maximum number of miles that you can drive?

Phone Charges The monthly cost C, in dollars, for international calls on a certain cellular phone plan is given by the function C(x) = 0.38x + 5, where x is the number of minutes used. (a) What is the cost if you talk on the phone for x = 50 minutes? (b) Suppose that your monthly bill is $29.32. How many minutes did you use the phone? (c) Suppose that you budget yourself $60 per month for the phone. What is the maximum number of minutes that you can talk?

Disability Benefits The average monthly benefit B, in dollars, for individuals on disability is given by the function B(t) = 19.25t + 585.72 where t is the number of years since January 1 , 1990. (a) What was the average monthly benefit in 2000 (t = 10)? (b) In what year will the average monthly benefit be $893.72? [Hint: use a table with 6t = 1 .] (c) In what year will the average monthly benefit exceed $1000?

Health Expenditures The total private health expenditures H, in billions of dollars, is given by the function H(t) = 26t + 411 where t is the number of years since January 1, 1990. (a) What were total private health expenditures in 2000 (t = 1O)? (b) In what year will total private health expenditures be $879 billion? [Hint: use a table with 6t = 1.] (c) In what year will total private health expenditures exceed $1 trillion ($1,000 billion)?

Supply and Demand Suppose that the quantity supplied S and quantity demanded D of T-shirts at a concert are given by the following functions: S(p) = -200 + SOp D(p) = 1000 - 25p where p is the price of a T-shirt. (a) Find the equilibrium price for T-shirts at this concert. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is greater than quantity supplied. (c) What do you think will eventually happen to the price of T-shirts if quantity demanded is greater than quantity supplied?

Supply and Demand Suppose that the quantity supplied S and quantity demanded D of hot dogs at a baseball game are given by the following functions: S(p) = -2000 + 3000p D(p) = 10,000 - 1000p where p is the price of a hot dog. (a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is less than quantity supplied. (c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied?

Taxes The function T(x) = 0.15(x - 7300) + 730 represents the tax bill T of a single person whose adjusted gross income is x dollars for income between $7300 and $29,700, inclusive in 2005. Source: Internal Revenue Service (a) What is the domain of this linear function? (b) What is a single filer's tax bill if adjusted gross income is $18,000? (c) Which variable is independent and which is dependent? (d) Graph the linear function over the domain specified in part (a). (e) What is a single filer's adjusted gross income if the tax bill is $2860?

Luxury Tax In 2002, major league baseball signed a labor agreement with the players. In this agreement, any team whose payroll exceeds $128 million in 2005 will have to pay a luxury tax of 22.5% (for first-time offenses). The linear function T(p) = 0.225(p - 128) describes the luxury tax T of a team whose payroll is p (in millions of dollars). (a) What is the implied domain of this linear function? (b) What is the luxury tax for a team whose payroll is $160 million? (c) Graph the linear function. (d) What is the payroll of a team that pays a lUxury tax of $11.7 million?

The point at which a company's profits equal zero is called the company 's break-even point. For Problems 45 and 46, let R represent a company's revenue, let C represent the company's costs, and let x represent the number of units produced and sold each day. (a) Find the flrm 's break-even point; that is,find x so that R = C. (b) Find the values ofx such that R(x) > C(x). This represents the number of units that the company must sell to earn a profit. R(x) = 8x C(x) = 4.5x + 17,500

The point at which a company's profits equal zero is called the company 's break-even point. For Problems 45 and 46, let R represent a company's revenue, let C represent the company's costs, and let x represent the number of units produced and sold each day. (a) Find the flrm 's break-even point; that is,find x so that R = C. (b) Find the values ofx such that R(x) > C(x). This represents the number of units that the company must sell to earn a profit. R(x) = 12x C(x) = lOx + 15,000

Straight-line Depreciation Suppose that a company has just purchased a new computer for $3000. The company chooses to depreciate the computer using the straight-line method over 3 years. (a) Write a linear function that expresses the book value V of the computer as a function of its age x. (b) Graph the linear function. (c) What is the book value of the computer after 2 years? (d) When will the computer have a book value of $2000?

Straight-line Depreciation Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The company chooses to depreciate the machine using the straight-line method over 10 years. (a) Write a linear function that expresses the book value V of the machine as a function of its age x. (b) Graph the linear function. (c) What is the book value of the machine after 4 years? (d) When will the machine have a book value of $72,000?

Cost Function The simplest cost function is the linear cost function, C( x) = mx + b, where the y-intercept b represents the fixed costs of operating a business and the slope m represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of $1800 and each bicycle costs $90 to manufacture. (a) Write a linear function that expresses the cost C of manufacturing x bicycles in a day. (b) Graph the linear function. (c) What is the cost of manufacturing 14 bicycles in a day? (d) How many bicycles could be manufactured for $3780?

Cost Function Refer to Problem 49. Suppose that the landlord of the building increases the bicycle manufacturer's rent by $100 per month. (a) Assuming that the manufacturer is open for business 20 days per month, what are the new daily fixed costs? (b) Write a linear function that expresses the cost C of manufacturing x bicycles in a day with the higher rent. (c) Graph the linear function. (d) What is the cost of manufacturing 14 bicycles in a day? (e) How many bicycles could be manufactured for $3780?

Truck Rentals A truck rental company rents a truck for one day by charging $29 plus $0.07 per mile. (a) Write a linear function that relates the cost C, in dollars, of renting the truck to the number x of miles driven. (b) What is the cost of renting the truck if the truck is driven 110 miles? 230 miles?

Long Dishll1ce A phone company offers a domestic long distance package by charging $5 plus $0.05 per minute. (a) Write a linear function that relates the cost C, in dollars, of talking x minutes. (b) What is the cost of talking 105 minutes? 180 minutes?

Which of the following functions might have the graph shown? (More than one answer is possible.) (a) f(x) = 2x - 7 Y (b) g(x) = -3x + 4 (c) H(x) = 5 (d) F(x) = 3x + 4 1 (e) G(x) = 2 x + 2 x

Which of the following functions might have the graph shown? (More than one answer is possible.)(a) f(x) = 3x + 1 Y (b) g(x) = -2x + 3 (c) H(x) = 3 (d) F(x) = -4x - 1 2 (e) G(, x:) = - -x + 3 3

Under what circumstances is a linear function f(x) = mx + b odd? Can a linear function ever be even?

A(n) ____ is used to help us to see the type of relation, if any, that may exist between two variables.

True or False The correlation coefficient is a measure of the strength of a linear relation between two variables and must lie between -1 and 1, inclusive.

In Problems 3-8, examine the scatter diagram and determine whether the type of relation that may exist is linear or nonlinear. y 4. Y ---.--------, 35 30 25 ., 20 . . . ' 15 10 . . .' 5 0 5 1015 20 2530 3540 x

In Problems 3-8, examine the scatter diagram and determine whether the type of relation that may exist is linear or nonlinear. Y 5. 14 .---.--------, 12 ., 10 . . 8 . ' 6 . . .' 4 -2 . . x o 2 4 6 8 1 0121416 x

In Problems 3-8, examine the scatter diagram and determine whether the type of relation that may exist is linear or nonlinear. 22 22 ., . . . ' . . -2 12 . . 0

In Problems 3-8, examine the scatter diagram and determine whether the type of relation that may exist is linear or nonlinear. 50 a a a B B ,po. .,pP'. . B 5 30 20

In Problems 3-8, examine the scatter diagram and determine whether the type of relation that may exist is linear or nonlinear. 25 a a a B B ,po. .,pP'. . B 0 10 0

In Problems 3-8, examine the scatter diagram and determine whether the type of relation that may exist is linear or nonlinear. 35 a a a B B ,po. .,pP'. . B 0 0 45

In Problems 9-14 (a) Draw a scatter diagram. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. * c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. x 3 4 5 6 7 8 9 Y 4 6 7 10 12 14 16

In Problems 9-14 (a) Draw a scatter diagram. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. * c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. x 3 5 7 9 11 13 y 0 2 3 6 9 11

In Problems 9-14 (a) Draw a scatter diagram. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. * c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. x -2 -1 0 2 Y -4 0 4 5

In Problems 9-14 (a) Draw a scatter diagram. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. * c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. x -2 -1 0 2 Y 7 6 3 2 0

In Problems 9-14 (a) Draw a scatter diagram. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. * c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. x -20 - 17 - 15 - 14 - 10 Y 1 00 1 20 118 130 140

In Problems 9-14 (a) Draw a scatter diagram. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. * c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. x -30 -27 -25 -20 -14 Y 10 12 13 13 18

Consumption and Disposable Income An economist wants to estimate a line that relates personal consumption expenditures C and disposable income 1. Both C and I are in thousands of dollars. She interviews eight heads of households for families of size 3 and obtains the data shown below. Let I represent the independent variable and C the dependent variable. (a) Draw a scatter diagram. (b) Find a line that fits the data. * (c) Interpret the slope. The slope of this line is called the marginal propensity to consume. (d) Predict the consumption of a family whose disposable income is $42,000. (e) Use a graphing utility to find the line of best fit to the data. What is the correlation coefficient? It 1(000) e(OOO) 20 16 20 18 18 13 27 21 36 27 37 26 45 36 50 39

Marginal Propensity to save The same economist as in Problem 15 wants to estimate a line that relates savings S and disposable income I. Let S = I - C be the dependent variable and I the independent variable. (a) Draw a scatter diagram. (b) Find a line that fits the data. ':' (c) Interpret the slope. The slope of this line is called the marginal propensity to save. (d) Predict the savings of a family whose income is $42,000. (c) Use a graphing utility to find the line of best fit. What is the correlation coefficient?

Candy The following data represent the weight (in grams) of various candy bars and the corresponding number of calories. (a) Draw a scatter diagram of the data treating weight as the independent variable. (b) What type of relation appears to exist between the weight of a candy bar and the number of calories? 7 3 SECTION 4.2 Building Linear Functions from Data 291 (c) Graph the line found in part (b) on the scatter diagram. I,: (Ii) Use a graphing utility to find the line of best fit. 9 6 - 15 118 (e) Use a graphing utility to draw the scatter diagram an.d graph the line of best fit on it. 1 11. 1 11 13 x -2 -1 0 2 9 11 Y -4 0 4 5 14. - 14 - 10 x -30 -27 -25 -20 -14 1 30 140 Y 10 12 13 13 18 Weight, x Calories, y Hershey's Milk Chocolate 44.28 230 Nestle's Crunch 44.84 230 Butterfinger 61.30 270 Baby Ruth 66.45 280 Almond Joy 47.33 220 Twix (with Caramel) 58.00 280 Snickers 61.12 280 Heath 39.52 210 Source: Megan Pocius, Student at Joliet Junior College (c) Select two points and find an equation of the line containing the points.* (d) Graph the line on the scatter diagram drawn in part (a). (e) Predict the number of calories in a candy bar that weighs 62.3 grams. (f) Interpret the slope of the line found in part (c).

Raisins The following data represent the weight w (in grams) of a box of raisins and the number N of raisins in the box. Number of Raisins, N 42.3 87 42.7 91 42.8 93 42.4 87 42.6 89 42.4 90 42.3 82 42.5 86 42.7 86 42.5 86 Source: Jennifer Maxwell, Student at Joliet Junior College (a) Does the relation defined by the set of ordered pairs (w, N) represent a function? (b) Draw a scatter diagram of the data treating weight as the independent variable. (c) Select two points and find the equation of the line containing the points.* (d) Graph the line on the scatter diagram drawn in part (b). (e) Express the relationship found in part (c) using function notation. (f) Predict the number of raisins in a box that weighs 42.5 grams. (g) Interpret the slope of the line found in part (c).

Height versus Head Circumference A pediatrician wanted to estimate a linear function that relates a child's height, H, to his or her head circumference, C. She randomly selects nine children from her practice, measures their height and head circumference, and obtains the data shown below. Let H represent the independent variable and C the dependent variable. 8 . _'0 . (a) Use a graphing utility to draw a scatter diagram. (b) Use a graphing utility to find the line of best fit to the data. Express the solution using function notation. (c) Interpret the slope. (d) Predict the head circumference of a child that is 26 inches tall. ( e) Predict the height of a child whose head circumference is 1 7.4 inches. Height, H Head Circumference, C (inches) (inches) 25.25 16.4 25.75 16.9 25 16.9 27.75 17.6 26.5 17.3 27 17.5 26.75 17.3 26.75 1 7.5 27.5 17.5 Source: Denise Slucki, Student at Joliet Junior College

Gestation Period versus Life Expectancy A researcher would like to estimate the linear function relating the gestation period of an animal, G, and its life expectancy, L. She collects the following data. ..... . ..--- Y Gestation (or incubation) Life Expectancy, L Animal Period, G (days) (years) , Cat 63 11 Chicken 22 7.5 Dog 63 11 Duck 28 10 Goat 151 12 Lion 1 08 10 Para keet 18 8 Pig 115 10 Rabbit 31 7 Squirrel 44 9 Source: Tllne Almanac 2000 Let G represent the independent variable and L the dependent variable. (a) Use a graphing utility to draw a scatter diagram. (b) Use a graphing utility to find the line of best fit to the data. Express the solution using function notation. (c) Interpret the slope. (d) Predict the life expectancy of an animal whose gestation period is 89 days.

Demand for Jeans The marketing manager at Levi-Strauss wishes to find a function that relates the demand D for men's jeans and p, the price of the jeans. The following data were obtained based on a price history of the jeans. Demand (Pairs of Jeans Price (S/Pair), p Sold per Day), 0 20 22 23 23 27 29 30 60 57 56 53 52 49 44 (a) Does the relation defined by the set of ordered pairs (p, D) represent a function? (b) Draw a scatter diagram of the data. " (c) Using a graphing utility, find the line of best fit relating price and quantity demanded. What is the correlation coefficient? 22. (d) Interpret the slope. (e) Express the relationship found in part (c) using function notation. (f) What is the domain of the function? (g) How many jeans will be demanded if the price is $28 a pair?

Advertising and Sales Revenue A marketing firm wishes to find a function that relates the sales S of a product and A, the amount spent on advertising the product. The data are obtained from past experience. Advertising and sales are measured in thousands of dollars. i Advertising Expenditures, A Sales, S !o 20 335 22 339 22.5 338 24 343 24 341 27 350 28.3 351 (a) Does the relation defined by the set of ordered pairs (A, S) represent a function? (b) Draw a scatter diagram of the data. ;, (c) Using a graphing utility, find the line of best fit relating advertising expenditures and sales. What is the correlation coefficient? d) Interpret the slope. (e) Express the relationship found in part (c) using function notation. (f) What is the domain of the function? (g) Predict sales if advertising expenditures are $25,000.

Maternal Age \'erSlIS Down Syndrome A biologist would like to know how the age of the mother affects the incidence rate of Down syndrome. The following data represent the age of the mother and the incidence rate of Down syndrome per 1000 pregnancies. Draw a scatter diagram treating age of the mother as the independent variable. Explain why it would not make sense to find the line of best fit for these data. '.' Incidence of Down Syndrome. y 33 2.4 34 3.1 35 4 36 5 37 6.7 38 8.3 39 10 40 1 3.3 41 1 6.7 42 22.2 43 28.6 44 33.3 45 50 Source: E. B. Hook, P. K. Cross, and D. M. Schreinemachers, " Chromosomal abnormality rates at amniocentesis and in live-born infants," Journal of the American Medical Association, 249(15), 1983, pp. 2034-2038.

Find the line of best fit for the ordered pairs (1,5) and (3, 8). What is the correlation coefficient for these data? Why is this result reasonable?

What does a correlation coefficient of 0 imply?

List the intercepts of the equation y = x2 - 9. (pp. 165-166)

Find the real solutions of the equation 2x2 + 7 x - 4 = O.

To complete the square of x2 - 5x, you add the number ___ (pp. 99-100)

To __ graph y = (x - 4)2, you shift the graph of y = x2 to the a distance of __ units. (pp. 252-260)

The graph of a quadratic function is called a(n) __ .

The vertical line passing through the vertex of a parabola is called the ___.

The x-coordinate of the vertex of f(x) = ax2 + bx + c, a #- 0, is __.

True or False The graph of f(x) = 2x2 + 3x - 4 opens up.

True or False The y-coordinate of the vertex of f(x) = -x2 + 4x + 5 is f(2).

True or False If the discriminant b2 - 4ac = 0, the graph of f(x) = ax2 + bx + c, a #- 0, will touch the x-axis at its vertex.

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f ( x) = x2 - 1

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f ( x) = -x2 - 1

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f ( x) = x2 - 2x + 1

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f(x) = x2 + 2x + 1

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f(x) = x2 - 2x + 2

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f(x) = x2 + 2x

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f(x) = x2 - 2x

In Problems 11-18, match each graph to one the following functions. A Y B y e y 3 2 2 2 x -2 2 x 2 x I -2 (0, - 1 ) (-1, -1) -2 E y F y G y 1 -2 -1 (1 , 0) 3 x -1 3 x -2 -1 1 D y -2 -1 H y 2 -2 -2 2 x x (1 , -1) f(x) = x2 + 2x + 2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = -x2 4

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = 2x2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = "4 X2 - 2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = 2X2 - 3

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f ( x) = "4 x2 + 2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = 2x2 + 4

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = X2 + 1

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = _2x2 - 2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = x2 + 4x + 2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f ( x) = x2 - 6x - 1

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = 2X2 - 4x + 1

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = 3x2 + 6x

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = -x2 - 2x

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f(x) = _2X2 + 6x + 2

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f (x) = -x2 + x-I

In Problems 19-34, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write fin the form f(x) = a(x - hl + k.] f( x) = -x2 + -x-I

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = x2 + 2x

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = x2 - 4x

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = -x2 - 6x

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = -x2 + 4x

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = 2x2 - 8x

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = 3x2 + 18x

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = x2 + 2x - 8

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = x2 - 2x - 3

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f ( x) = x2 + 2x + 1

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f ( x) = x2 + 6x + 9

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f ( x) = 2x2 - X + 2

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = 4x2 - 2x + 1

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = -2x2 + 2x - 3

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = -3x2 + 3x - 2

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = 3x2 + 6x + 2

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = 2x2 + 5x + 3

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = -4x2 - 6x + 2

In Problems 35-52, (a) graph each quadratic function by determining whether its graph apens up ar down and by finding its vertex, axis af symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range af the functian. (c) Determine where the functian is increasing and where it is decreasing. f(x) = 3x2 - 8x + 2

In Problems 53-58, determine the quadratic function whose graph is given. y Vertex: (1 , -2)

In Problems 53-58, determine the quadratic function whose graph is given. y -1 234 5 x

In Problems 53-58, determine the quadratic function whose graph is given. Vertex: (-3, 5) Y \ 6

In Problems 53-58, determine the quadratic function whose graph is given. y Vertex: (2 , 3)

In Problems 53-58, determine the quadratic function whose graph is given. x -4 Vertex: (1 , -3)

In Problems 53-58, determine the quadratic function whose graph is given. Y Vertex: (-2, 6) 6

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 2X2 + 12x

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = _2x2 + 12x

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 2x2 + 12x - 3

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 4x + 8x + 3 2 - 8x + 3

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = -x2 + lax - 4

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = _2x2 + 8x + 3

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = -3x2 + 12x + 1

In Problems 59-66, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 4x2 - 4x

The graph of the function f(x) = ax2 + bx + c has vertex at (0, 2) and passes through the point (1, 8). Find a, b, and c.

The graph of the function f(x) = ax2 + bx + c has vertex at (1,4) and passes through the point (-1, -8). Find a, b, and c.

In Problems 69-74, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve f(x) = g(x). (c) Use the result of part (b) to label the points of intersection of the graphs off and g (d) Shade the region for which f(x) > g(x), that is, the region below f and above g. f(x) = 2x - 1; g(x) = x2 - 4

In Problems 69-74, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve f(x) = g(x). (c) Use the result of part (b) to label the points of intersection of the graphs off and g (d) Shade the region for which f(x) > g(x), that is, the region below f and above g. f(x) = -2x - 1; g(x) = x2 - 9

In Problems 69-74, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve f(x) = g(x). (c) Use the result of part (b) to label the points of intersection of the graphs off and g (d) Shade the region for which f(x) > g(x), that is, the region below f and above g. f(x) = -x2 + 4; g(x) = -2x + 1

In Problems 69-74, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve f(x) = g(x). (c) Use the result of part (b) to label the points of intersection of the graphs off and g (d) Shade the region for which f(x) > g(x), that is, the region below f and above g. f(x) = -x2 + 9; g(x) = 2x + 1

In Problems 69-74, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve f(x) = g(x). (c) Use the result of part (b) to label the points of intersection of the graphs off and g (d) Shade the region for which f(x) > g(x), that is, the region below f and above g. f(x) = -x2 + 5x; g(x) = x2 + 3x - 4

In Problems 69-74, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve f(x) = g(x). (c) Use the result of part (b) to label the points of intersection of the graphs off and g (d) Shade the region for which f(x) > g(x), that is, the region below f and above g. f(x) = -x2 + 7x - 6; g(x) = x2 + X - 6

Answer Problems 75 and 76 using the following: A quadratic function of the form f(x) = ax2 + bx + c with b2 - 4ac > a may a/so be written in the form f(x) = a(x - r1 ) (x - r2 ), where r1 and r 2 are the x-intercepts of the graph of the quadratic function. (a) Find a quadratic function whose x-intercepts are -3 and 1 with a = 1; a = 2; a = -2; a = 5. (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts. What might you conclude?

Answer Problems 75 and 76 using the following: A quadratic function of the form f(x) = ax2 + bx + c with b2 - 4ac > a may a/so be written in the form f(x) = a(x - r1 ) (x - r2 ), where r1 and r 2 are the x-intercepts of the graph of the quadratic function. (a) Find a quadratic function whose x-intercepts are -5 and 3 with a = 1; a = 2; a = -2; a = 5. (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts. What might you conclude?

Suppose that f(x) = x2 + 4x - 21 (a) What is the vertex of f? (b) What are the x-intercepts of the graph of f? (c) Solve f(x) = -21 for x. What points are on the graph of f? (d) Use the information obtained in parts (a)-(c) to graph f(x) = x2 + 4x - 2l.

Suppose that f(x) = x2 + 2x - 8 (a) What is the vertex of f? (b) What are the x-intercepts of the graph of f? (c) Solve f(x) = -8 for x. What points are on the graph off? (d) Use the information obtained in parts (a)-(c) to graph f(x) = x2 + 2x - 8.

Find the point on the line y = x that is closest to the point (3, 1). [Hint: Express the distance d from the point to the line as a function of x, and then find the minimum value of [d(x)f

Find the point on the line y = x + 1 that is closest to the point (4, 1).

Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is R(p) = -4p2 + 4000p What unit price should be established for the dryer to maximize revenue? What is the maximum revenue?

Maximizing Revenue The John Deere company has found that the revcnue, in dollars, from sales of heavy-duty tractors is a function of the unit price p, in dollars, that it charges. If the revenue R is 1 R(p) = - Z p2 + 1900p what unit price p should be charged to maximize revenue? What is the maximum revenue?

Minimizing Marginal Cost The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is $6.20, it cost $6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C (in dollars) to produce x thousand mp3 players is given by the function C(x) = x2 - 140x + 7400 (a) How many players should be produced to minimize the marginal cost? (b) What is the minimum marginal cost?

Minimizing Marginal Cost (See Problem 83.) The marginal cost C (in dollars) of manufacturing x cell phones (in thousands) is given by C(x) = 5x2 - 200x + 4000. (a) How many cell phones should be manufactured to minimize the marginal cost? (b) What is the minimum marginal cost?

Hunting The function H(x) = -3.24x2 + 242.1x - 738.4 models the number of individuals whose age is x and engage in hunting activities. (a) What is the age at which there are the most hunters? Approximately how many hunters are this age? (b) Are the number of hunters increasing or decreasing for individuals who are between 40 and 45 years of age? Source: National Sporting Goods Association

Advanced Degrees The function P(x) = -0.008x2 + 0.815x - 9.983 models the percentage of the U.S. population whose age is x that have earned an advanced degree (more than a bachelor's degree) in March 2003. (a) What is the age for which the highest percentage of Americans have earned an advanced degree? What is the highest percentage? (b) Is the percentage of Americans who have earned an advanced degree increasing or decreasing for individuals between the ages of 40 and 50? Source: Us. Census Bureau

Male Murder V ictims The function M(x) = l.00x2 - 136.74x + 4764.89 models the number of male murder victims who are x years of age (20 :=; x < 90). (a) Use the model to approximate the number of male murder victims who are x = 23 years of age. (b) At what age is the number of male murder victims 1456? (c) Describe what happens to the number of male murder victims as age increases from 20 to 65. Source: Federal Bureau of Investigation

Health Care Expenditures The function H(x) = 0.004x2 - 0.197x + 5.406 models the percentage of total income that an individual who is x years of age spends on health care. (a) Use the model to approximate the percentage of total income an individual who is x = 45 years of age spends on health care. (b) At what age is the percentage of income spent on health care 1O%? . (c) Using a graphing utility, graph H = H(x). (d) Based on the graph drawn in part ( c), describe what happens to the percentage of income spent on health care as individuals age. Source: Bureau of Labor Statistics

Business The monthly revenue R achieved by selling x wristwatches is figured to be R(x) = 75x - 0.2x2 . The monthly cost C of selling x wristwatches is C(x) = 32x + 1750. (a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as P(x) = R(x) - C(x). What is the profit function? (c) How many wristwatches must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.

Business The daily revenue R achieved by selling x boxes of candy is figured to be R(x) = 9.5x - 0.04x2. The daily cost C of selling x boxes of candy is C (x) = 1.25x + 250. (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue. (b) Profit is given as P(x) = R(x) - C(x). What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit?

Let f(x) = ax2 + bx + c, where a, b, and c are odd integers. What is the maximum profit? If x is an integer, show that f(x) must be an odd integer. [Hint: x is either an even integer or an odd integer.]

Make up a quadratic function that opens down and has only one x-intercept. Compare yours with others in the class. What are the similarities? What are the differences?

On one set of coordinate axes, graph the family of parabolas and down if a < O? f(x) = x 2 + 2x + c for c = -3, c = 0, and c = 1. Describe the characteristics of a member of this family.

On one set of coordinate axes, graph the family of parabolas f(x) = x 2 + bx + 1 for b = -4, b = 0, and b = 4. Describe the general characteristics of this family.

State the circumstances that cause the graph of a quadratic function f(x) = ax2 + bx + c to have no x-intercepts.

Why does the graph of a quadratic function open up if a > 0?

Can a quadratic function have a range of ( - 00, oo )? Justify your answer.

What are the possibilities for the number of times the graphs of two different quadratic functions intersect?

Translate the following sentence to a mathematical equation: The total revenue R from selling x hot dogs is $3 times the number of hot dogs sold. (pp. 139-1 45)

Use a graphing utility to find the line of best fit for the following data: (pp. 287-290) x I 3 y 10 5 13 5 12 6 7 8 15 16 19

Demand E(luation The price p (in dollars) and the quantity x sold of a certain product obey the demand equation 1 P = - -x + 100 6 a :S X :S 600 (a) Express the revenue R as a function of x. (Remember, R = xp.) (b) What is the revenue if 200 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue?

Demand Equation The price p (in dollars) and the quantity x sold of a certain product obey the demand equation 1 P = - -x + 1 00 3 a :S X :S 300 (a) Express the revenue R as a function of x. (b) What is the revenue if 100 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue?

Demand Equation The price p (in dollars) and the quantity x sold of a certain product obey the demand equation x = -5p + 100, 0 :5 P :5 20 (a) Express the revenue R as a function of x. (b) What is the revenue if 15 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue?

Demand Equation The price p (in dollars) and the quantity x sold of a certain product obey the demand equation x = -20p + 500, 0 :5 P :5 25 (a) Express the revenue R as a function of x. (b) What is the revenue if 20 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue?

Enclosing a Rectangular Field David has 400 yards of fencing and wishes to enclose a rectangular area. (a) Express the area A of the rectangle as a function of the width w of the rectangle. (b) For what value of w is the area largest? (c) What is the maximum area?

Enclosing a Rectangular Field Beth has 3000 feet of fencing available to enclose a rectangular field. (a) Express the area A of the rectangle as a function of x, where x is the length of the rectangle. (b) For what value of x is the area largest? (c) What is the maximum area?

Enclosing the Most Area with a Fence A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? (See the figure.) x

Enclosing the Most Area with a Fence A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

Anal)'zing the Motion of a Projectile A projectile is fired from a cliff 200 feet above the water at an inclination of 45 to the horizontal, with a muzzle velocity of 50 feet per second. The height h of the projectile above the water is given by -32x2 hex) = --, + x + 200 (50)- where x is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Using a graphing utility, graph the function h, o :5 X :5 200. (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

Analyzing the Motion of a Projectile A projectile is fired at an inclination of 45 to the horizontal, with a muzzle velocity of 100 feet per second. The height h of the projectile is given by -32x2 hex) = (100)2 + x where x is the horizontal distance of the projectile from the firing point. (a) At what horizontal distance from the firing point is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the firing point will the projectile strike the ground? (d) Using a graphing utility, graph the function h, o :5 X :5 350. (e) Use a graphing utility to verify the results obtained in parts (b) and (c). (f) When the height of the projectile is 50 feet above the ground, how far has it traveled horizontally?

Suspension Bridge A suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (Assume that the road is level.)

Architecture A parabolic arch has a span of 120 feet and a maximum height of 25 feet. Choose suitable rectangular coordinate axes and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center.

Constructing Rain Gutters A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90. See the illustration. What depth will provide maximum cross-sectional area and hence allow the most water to flow?

Norman Windows A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle. See the figure. If the perimeter of the window is 20 feet, what dimensions will admit the most light (maximize the area)? [Hint: Circumference of a circle = 27fr; area of a circle = 7fr2, where r is the radius of the circle.]

Constructing a Stadium A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be 1500 meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

Architecture A special window has the shape of a rectangle surmounted by an equilateral triangle. See the figure. If the perimeter of the window is 16 feet, what dimensions will admit the most light? [Hint: Area of an equilateral triangle = ()x2, where x is the length of a side of the triangle.]

Stopping Distance An accepted relationship between stopping distance, d (in feet), and the speed of a car, v (in mph), is d = 1.1v + 0.06v2 on dry, level concrete. (a) How many feet will it take a car traveling 45 mph to stop on dry, level concrete? (b) If an accident occurs 200 feet ahead of you, what is the maximum speed you should be traveling to avoid being involved? (c) What might the term 1.1v represent? Source: www2.nsta.orgIEnergylfn_braking.html

Insurance Claims The years 1999 to 2005 were particularly costly for insurance companies, with 7 of the 10 most costly catastrophes in U.S. history (as of 2005). For the United States Automobile Association (USAA) and its affiliates, the total cost of claims for catastrophic losses, in millions, can be approximated by C(x) = 34.87x2 - 98.1x + 258.3, where x = 0 for 1999, x = 1 for 2000, x = 2 for 2001, and so on. (a) Estimate the total cost of claims for the year 2003. (b) According to the model, during which year were catastrophic loss claims at a minimum? ( c) Would C(x) be useful for predicting total catastrophic loss claims for the year 2015? Why or why not? Source: USAA Report to Members 2005

Chemical Reactions A self-catalytic chemical reaction results in the formation of a compound that causes the formation ratio to increase. If the reaction rate V is given by Vex) = kx(a - x), where k is a positive constant, a is the initial amount of the compound, and x is the variable amount of the compound, for what value of x is the reaction rate a maximum?

Calculus: Siml )SOn'S Rule The figure shows the graph of y = ax2 + bx + c. Suppose that the points ( -h, Yo), (0, Yl ), and (h, Y2 ) are on the graph. It can be shown that the area enclosed by the parabola, the x-axis, and the lines x = -h and x = h is h Area = -(2ah2 + 6c) 3 Show that this area may also be given by y x

Use the result obtained in Problem 22 to find the area enclosed by f (x) = -5x2 + 8, the x-axis, and the lines x = -1 and x = 1.

Use the result obtained in Problem 22 to find the area enclosed by f (x) = 2x2 + 8, the x-axis, and the lines x = -2 and x = 2.

Use the result obtained in Problem 22 to find the area enclosed by f(x) = x2 + 3x + 5, the x-axis, and the lines x = -4 and x = 4.

Use the result obtained in Problem 22 to find the area enclosed by f (x) = -x2 + X + 4, the x-axis, and the lines x = -1 and x = 1.

Life Cycle Hypothesis An individual's income varies with his or her age. The following table shows the median income I of individuals of different age groups within the United States for 2003. For each age group, let the class midpoint represent the independent variable x. For the class "65 years and older," we will assume that the class midpoint is 69.5. Age (yearsl Class Midpoint, x Median Income (Sl, I 1 5-24 1 9.5 8,614 25-34 29.5 26,21 2 35-44 39.5 30,914 45-54 49.5 32,583 55-64 59.5 28,068 65 and older 69.5 1 4,664 Source: U.S. Census Bureau, 2003 Annual Social and Economic Supplement (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) The quadratic function of best fit to these data is I(x) = -34.3x2 + 3157x - 39,115 Use this function to determine the age at which an individual can expect to earn the most income. (c) Use the function to predict the peak income earned. (d) Use a graphing utility to verify that the function given in part (b) is the quadratic function of best fit. (e) With a graphing utility, draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.

Life Cycle Hypothesis An individual's income varies with his or her age. The following table shows the median income I of individuals of different age groups within the United States for 2004. For each age group, let the class midpoint represent the independent variable x. For the class "65 years and older," we will assume that the class midpoint is 69.5. Age (yearsl Class Midpoint, x Median Income (Sl, I 1 5-24 1 9.5 8,782 25-34 29.5 26,642 35-44 39.5 31,629 45-54 49.5 32,908 55-64 59.5 28,518 65 and older 69.5 1 5,193 Source: U.S. Census B ureau, 2004 Annual Social and Economic Supplement (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) The quadratic function of best fit to these data is I(x) = -34.5x2 + 3186x - 39,335 Use this function to determine the age at which an individual can expect to earn the most income. (c) Use the function to predict the peak income earned. (d) Use a graphing utility to verify that the function given in part (b) is the quadratic function of best fit. (e) With a graphing utility, draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.

Height of a Ball A shot-putter throws a ball at an inclination of 45 to the horizontal. The following data represent the height h of the ball at the instant that it has traveled x feet horizontally. Distance, x Height, h 20 25 40 40 60 55 80 65 1 00 71 1 20 77 1 40 77 1 60 75 1 80 71 200 64 (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) The quadratic function of best fit to these data is hex) = -O.0037x2 + 1 .03x + 5.7 Use this function to determine the horizontal distance the ball will travel before it reaches its maximum height. (c) Use the function to find the maximum height of the ball. (d) Use a graphing utility to verify that the function given in part (b) is the quadratic function of best fit. (e) With a graphing utility, draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.

Miles per Gallon An engineer collects data showing the speed s of a Ford Taurus and its average miles per gallon, M. See the table. Speed, s Miles per Gallon, M 30 18 35 20 40 23 40 25 45 25 50 28 55 30 60 29 65 26 65 25 70 25 31 4 CHAPTER 4 Linear and Quadratic Functions (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (c) Use the function to predict miles per gallon for a speed of 63 miles per hour. ,; (d) Use a graphing utility to verify that the function given (b) The quadratic function of best fit to these data is in part (b) is the quadratic function of best fit. M(s) = -0.017s2 + 1.93s - 25.34 Use this function to determine the speed that maximizes miles per gallon. (e) With a graphing utility, draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.

Refer to Example 1 on page 306. Notice that if the price charged for the calculators is $0 or $140 the revenue is $0. It is easy to explain why revenue would be $0 if the price charged is $0, but how can revenue be $0 if the price charged is $140?

Solve the inequality -3x - 2 < 7 (p. 1 31)

Write ( -2, 7] using inequality notation. (pp. 1 25-1 26)

In Problems 3-6, use the figure to solve each inequality. Y= f(x) (2, 0) (a) f(x) > 0 (b) f(x) :s 0

In Problems 3-6, use the figure to solve each inequality. Y= g(x) (4, 0) (a) g(x) < 0 (b) g(x) 2: 0

In Problems 3-6, use the figure to solve each inequality. Y= f(x) ( 1, 2 ) (a) g(x) 2: f(x) (b) f(x) > g(x)

In Problems 3-6, use the figure to solve each inequality. Y g(x) ( 1, -3 ) (a) f(x) < g(x) (b) f(x) 2: g(x)

In Problems 7-22, solve each inequality. x2 - 3x - 10 < 0

In Problems 7-22, solve each inequality. x2 + 3x - 10 > 0

In Problems 7-22, solve each inequality. x2 - 4x > 0

In Problems 7-22, solve each inequality. x2 + 8x > 0

In Problems 7-22, solve each inequality. x2 - 9 < 0

In Problems 7-22, solve each inequality. x2 - 1 < 0

In Problems 7-22, solve each inequality. x2 + X > 12

In Problems 7-22, solve each inequality. x2 + 7x < -12

In Problems 7-22, solve each inequality. 2X2 < 5x + 3

In Problems 7-22, solve each inequality. 6x2 < 6 + 5x

In Problems 7-22, solve each inequality. x(x - 7) > 8

In Problems 7-22, solve each inequality. x(x + 1) > 20

In Problems 7-22, solve each inequality. 4x2 + 9 < 6x

In Problems 7-22, solve each inequality. 25x2 + 16 < 40x

In Problems 7-22, solve each inequality. 6(x2 - 1) > 5x

In Problems 7-22, solve each inequality. 2 (2x2 - 3x) > -9

What is the domain of the function lex) = V x2 - 167

What is the domain of the function f(x) = V x - 3x27

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = x2 - 1 g(x) = 3x + 3

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = -x2 + 3 g(x) = -3x + 3

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = -x2 + 1 g(x) = 4x + 1

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = -x2 + 4 g(x) = -x - 2

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = x2 - 4 g(x) = -x2 +4

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f ( x) = x2 - 2x + 1 g(x) = -x2 + 1

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = x2 - X - 2 g(x) = x2 + X - 2

In Problems 25-32, use the given functions f and g. (a) Solve f(x) = O. (b) Solve g(x) = O. (e) Solve g(x) o. (f) Solve f(x) > g(x). (c) Solve f(x) = g(x). (g) Solve f(x) l. (d) Solve f(x) > O. f(x) = -x2 - X + 1 g(x) = -x2 + X + 6

Physics A ball is thrown vertically upward with an initial velocity of 80 feet per second. The distance s (in feet) of the ball from the ground after t seconds is set) = 80t - 16t2 I 96 ft 1 I s=80t - 1 6t 2 j (a) At what time t will the ball strike the ground? (b) For what time t is the ball more than 96 feet above the ground?

Physics A ball is thrown vertically upward with an initial velocity of 96 feet per second. The distance s (in feet) of the ball from the ground after t seconds is set) = 96t - 16t2. (a) At what time t will the ball strike the ground? (b) For what time t is the ball more than 128 feet above the ground?

Revenue Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is R(p) = -4p2 + 4000p (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed $800,000?

Revenue The John Deere company has found that the revenue from sales of heavy-duty tractors is a function of the unit price p, in dollars, that it charges. If the revenue R, in dollars, is 1 ? R(p) = 2P + 1900p (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed $1,200,000?

Artillery A projectile fired from the point (0, 0) at an angle to the positive x-axis has a trajectory given by where x = horizontal distance in meters y = height in meters v = initial muzzle velocity in meters per second (m/sec) g = acceleration due to gravity = 9.81 meters per second squared (m/sec2) c > 0 is a constant determined by the angle of elevation. A howitzer fires an artillery round with a muzzle velocity of 897 m/sec. (a) If the round must clear a hill 200 meters high at a distance of 2000 meters in front of the howitzer, what c values are permitted in the trajectory equation? (b) If the goal in part (a) is to hit a target on the ground 75 kilometers away, is it possible to do so? If so, for what values of c? If not, what is the maximum distance the round will travel? Source: www.answers.com

Runaway Car Using Hooke's Law, we can show that the work done in compressing a spring a distance of x feet from its 1 at-rest position is W = 2 kx2, where k is a stiffness constant depending on the spring. It can also be shown that the work done by a body in motion before it comes to rest is given by - w W = -v2, where w = weight of the object (lb), g = acceler2g ation due to gravity (32.2 ftlsec2), and v = object's velocity (in ftlsec). A parking garage has a spring shock absorber at the end of a ramp to stop runaway cars. The spring has a stiffness constant k = 9450 Ib/ft and must be able to stop a 4000-lb car traveling at 25 mph. What is the least compression required of the spring? Express your answer using feet to the nearest tenth. [Hint: Solve W > Hi, x 0]. Source: www.sciforums.com

Show that the inequality (x - 4)2 0 has exactly one solution.

Show that the inequality (x - 2)2 > 0 has one real number that is not a solution.

Explain why the inequality x2 + x + 1 > 0 has all real numbers as the solution set.

Explain why the inequality x2 - x + 1 < 0 has the empty set as solution set.

In Problems 1-6: (a) Determine the slope and y-intercept of each linear fimction. (b) Graph each function. Label the intercepts. (c) Determine whether the function is increasing, decreasing, or constant. f(x) = 2x - 5

In Problems 1-6: (a) Determine the slope and y-intercept of each linear fimction. (b) Graph each function. Label the intercepts. (c) Determine whether the function is increasing, decreasing, or constant. g(x) = -4x + 7

In Problems 1-6: (a) Determine the slope and y-intercept of each linear fimction. (b) Graph each function. Label the intercepts. (c) Determine whether the function is increasing, decreasing, or constant. h(x) = S x - 6

In Problems 1-6: (a) Determine the slope and y-intercept of each linear fimction. (b) Graph each function. Label the intercepts. (c) Determine whether the function is increasing, decreasing, or constant. F(x) = - 3 x+1

In Problems 1-6: (a) Determine the slope and y-intercept of each linear fimction. (b) Graph each function. Label the intercepts. (c) Determine whether the function is increasing, decreasing, or constant. G(x) = 4

In Problems 1-6: (a) Determine the slope and y-intercept of each linear fimction. (b) Graph each function. Label the intercepts. (c) Determine whether the function is increasing, decreasing, or constant. H(x) = -3

In Problems 7 and 8, determine whether the function is linear or non lineal: If the function is lineal; state its slope. x x y = g (x) -1 -2 0 3 8 2 13 3 18

In Problems 7 and 8, determine whether the function is linear or non lineal: If the function is lineal; state its slope. x y = g (x) -1 -3 0 4 7 2 6 3

In Problems 9-14, graph each quadratic function using transformations (shifting , compressing , stretching, and/or reflecting). f(x) = (x - 2f + 2

In Problems 9-14, graph each quadratic function using transformations (shifting , compressing , stretching, and/or reflecting). f(x) = (x + 1 )2 - 4

In Problems 9-14, graph each quadratic function using transformations (shifting , compressing , stretching, and/or reflecting). f(x) = - (x - 4)2

In Problems 9-14, graph each quadratic function using transformations (shifting , compressing , stretching, and/or reflecting). f(x) = (x - If - 3

In Problems 9-14, graph each quadratic function using transformations (shifting , compressing , stretching, and/or reflecting). f(x) = 2(x + 1)2 + 4

In Problems 9-14, graph each quadratic function using transformations (shifting , compressing , stretching, and/or reflecting). f(x) = -3 (x + 2f + 1

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f ( x) = (x - 2 f + 2

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f ( x) = (x + 1 f - 4

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f(x) = "4 x2 - 16

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f(x) - - 2 "2x + 2

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f(x) = -4x2 + 4x

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f(x) = 9x2 - 6x + 3

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f(x) = x2 + 3x + 1

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f ( x) = -x2 + X + - 2

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f(x) = 3x2 + 4x - 1

In Problems 15-24, (a ) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex , axis of symmetry, y-intercept , and x-intercepts, if any. (b ) Determine the do main and the range of the function. (c ) Determine where the func tion is increasing and where it is decreasin g. f ( x) = -2x2 - X + 4

In Problems 25-30, determine whether the given quadratic function has a maximum value or a minimu m value, and then find the value. f(x) = 3x2 - 6x + 4

In Problems 25-30, determine whether the given quadratic function has a maximum value or a minimu m value, and then find the value. f(x) = 2x2 + 8x + 5

In Problems 25-30, determine whether the given quadratic function has a maximum value or a minimu m value, and then find the value. f(x) = -x2 + 8x - 4

In Problems 25-30, determine whether the given quadratic function has a maximum value or a minimu m value, and then find the value. f(x) = -x2 - lOx - 3

In Problems 25-30, determine whether the given quadratic function has a maximum value or a minimu m value, and then find the value. f(x) = -3x2 + 12x + 4

In Problems 25-30, determine whether the given quadratic function has a maximum value or a minimu m value, and then find the value. f(x) = _2x2 + 4

In Problems 31-34, solve each quadratic ine quality. x2 + 6x - 16 < 0

In Problems 31-34, solve each quadratic ine quality. 3x2 - 2x - 1 2: 0

In Problems 31-34, solve each quadratic ine quality. 3x2 2: 14x + 5

In Problems 31-34, solve each quadratic ine quality. 4x2 < 13x - 3

Comparing Phone Companies Marissa must decide between one of two companies as her long-distance phone provider. Company A charges a monthly fee of $7.00 plus $0.06 per minute, while Company B does not have a monthly fee, but charges $0.08 per minute. (a) Find a linear function that relates cost, C, to total minutes on the phone, x, for each company. (b) Determine the number of minutes x for which the bi.1I from Company A will equal the bill from Company B. (c) Over what interval of minutes x will the bill from Company B be less than the bill from Company A?

Sales Commissions Bill was just offered a sales position for a computer company. His salary would be $15 ,000 per year plus 1 % of his total annual sales. (a) Find a linear function that relates Bill's annual salary, S, to his total annual sales, x. (b) In 2005, Bill had total annual sales of $1 ,000 ,000. What was Bill's salary? (c) What would Bill have to sell to earn $100 ,000? (d) Determine the sales required of Bill for his salary to exceed $1 50 ,000.

Demand Equation The price p (in dollars) and the quantity x sold of a certain product obey the demand equation 1 p = - 10 x + 150 0 :S x :S 1500 (a) Express the revenue R as a function of x. (b) What is the revenue if 100 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue?

Landscaping A landscape engineer has 200 feet of border to enclose a rectangular pond. What dimensions will result in the largest pond?

Enclosing the Most Area with a Fence A farmer with 10,000 meters of fencing wants to enclose a rectangular field and then divide it into two plots with a fence parallel to one of the sides. See the figure. What is the largest area that can be enclosed?

A rchitecture A special window in the shape of a rectangle with semicircles at each end is to be constructed so that the outside dimensions are 1 00 feet in length. See the illustration. Find the dimensions of the rectangle that maximizes its area.

Minimizing Marginal Cost Callaway Golf Company has determined that the marginal cost C of manufacturing x Big Bertha golf clubs may be expressed by the quadratic function C(x) = 4. 9x2 - 617.4x + 1 9,600 (a) How many clubs should be manufactured to minimize the marginal cost? (b) At this level of production, what is the marginal cost?

Violent Crimes The function V(t) = 5.0 P - 87.3t + 1761.1 models the number V (in thousands) of violent crimes committed in the United States t years after 1 995 , based on data obtained from the Federal Bureau of Investigation. So t = 0 represents 1 995 , t = 1 represents 1 996 , and so on. (a) Determine the year in which the fewest violent crimes were committed. (b) Approximately how many violent crimes were committed during this year? ; (c) Using a graphing utility, graph V = Vet). Were the number of violent crimes increasing or decreasing during the years 1995 to 2003?

A rectangle has one vertex on the line y = 10 - x, x > 0, another at the origin, one on the positive x-axis, and one on the positive y-axis. Express the area A of the rectangle as a function of x. Find the largest area A that can be enclosed by the rectangle.

Parabolic Arch Bridge A horizontal bridge is in the shape of a parabolic arch. Given the information shown in the figure, what is the height h of the arch 2 feet from shore?

Bone Length Research performed at NASA, led by Dr. Emily R. Morey-Holton, measured the lengths of the right humerus and right tibia in 11 rats that were sent to space on Spacelab Life Sciences 2. The following data were collected. Right Humerus Right Tibia (mm).x (mm),y 24.80 36.05 24.59 35.57 24.59 35.57 24.29 34.58 23.81 34.20 24.87 34.73 25.90 37.38 26.11 37.96 26.63 37.46 26.31 37.75 26.84 38.50 Source: NASA Life Sciences Data Archive (a) Draw a scatter diagram of the data treating length of the right humerus as the independent variable. (b) Based on the scatter diagram, do you think that there is a linear relation between the length of the right humerus and the length of the right tibia? [ij (c) Use a graphing utility to find the line of best fit relating length of the right humerus and length of the right tibia. (d) Predict the length of the right tibia on a rat whose right humerus is 26.5 millimeters (mm).

Ad"ertising A small manufacturing firm collected the following data on advertising expenditures A (in thousands of dollars) and total revenue R (in thousands of dollars). Advertising Total Revenue 20 $6101 22 $6222 25 $6350 25 $6378 27 $6453 28 $6423 29 $6360 31 $6231 (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) The quadratic function of best fit to these data is R(A) = -7.76A2 + 411.88A + 942.72 Use this function to determine the optimal level of advertising. (c) Use the function to predict the total revenue when the optimal level of advertising is spent. (d) Use a graphing utility to verify that the function given in part (b) is the quadratic function of best fit. (e) Use a graphing utility to draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.