Functions and Graphs. Chapter 2

Transcription

1 BLITMC0A.099_7- /0/0 :07 AM Page 7 Functions and Graphs Chapter T he cost of mailing a package depends on its weight. The probabilit that ou and another person in a room share the same birthda depends on the number of people in the room. In both these situations, the relationship between variables can be described b a function. Understanding this concept will give ou a new perspective on man ordinar situations. Tis the season and ou ve waited until the last minute to mail our holida gifts. Your onl option is overnight epress mail. You realize that the cost of mailing a gift depends on its weight, but the mailing costs seem somewhat odd. Your packages that weigh. pounds,. pounds, and pounds cost $.7 each to send overnight. Packages that weigh.0 pounds and pounds cost ou $8.0 each. Finall, our heaviest gift is barel over pounds and its mailing cost is $.. What sort of sstem is this in which costs increase b $.7, stepping from $.7 to $8.0 and from $8.0 to $.? 7

2 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs SECTION. Lines and Slopes Objectives Figure. Is there a relationship between literac and child mortalit? As the percentage of adult females who are literate increases, does the mortalit of children under five decrease? Figure., based on data from the United Nations, indicates that this is, indeed, the case. Each point in the figure represents one countr Source: United Nations Literac and Child Mortalit 0 Under-Five Mortalit (per thousand). Compute a line s slope.. Write the point-slope equation of a line.. Write and graph the slope-intercept equation of a line.. Recognize equations of horizontal and vertical lines.. Recognize and use the general form of a line s equation. 6. Find slopes and equations of parallel and perpendicular lines. 7. Model data with linear equation Percentage of Adult Females Who Are Literate Data presented in a visual form as a set of points is called a scatter plot. Also shown in Figure. is a line that passes through or near the points. A line that best fits the data points in a scatter plot is called a regression line. B writing the equation of this line, we can obtain a model of the data and make predictions about child mortalit based on the percentage of adult females in a countr who are literate. Data often fall on or near a line. In this section we will use equations to model such data and make predictions. We begin with a discussion of a line s steepness. The Slope of a Line Calculate a line s slope. Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the run) when moving from one fied point to another along the line. To calculate the slope of a line, we use a ratio that compares the change in (the rise) to the corresponding change in (the run). Run Rise Definition of Slope (, ) (, ) The slope of the line through the distinct points (, ) and (, ) is Change in Rise = Change in Run - = - where - Z 0.

3 BLITMC0A.099_7- /0/0 :07 AM Page 77 Section. Lines and Slope 77 Slope and the Streets of San Francisco It is common notation to let the letter m represent the slope of a line. The letter m is used because it is the first letter of the French verb monter, meaning to rise, or to ascend. EXAMPLE Using the Definition of Slope Find the slope of the line passing through each pair of points: a. (-, -) and (-, ) b. (-, ) and (, -). Solution a. Let (, ) = (-, -) and (, ) = (-, ). We obtain a slope of m = - (-) Change in - = = = =. - Change in - - (-) The situation is illustrated in Figure.(a).The slope of the line is, indicating that there is a vertical change, a rise, of units for each horizontal change, a run, of unit. The slope is positive, and the line rises from left to right. Stud Tip San Francisco s Filbert Street has a slope of 0.6, meaning that for ever horizontal distance of 00 feet, the street ascends 6. feet verticall. With its. angle of inclination, the street is too steep to pave and is onl accessible b wooden stairs. When computing slope, it makes no difference which point ou call (, ) and which point ou call (, ). If we let (, ) = (-, ) and (, ) = (-, -), the slope is still : m = = = = (-) - However, ou should not subtract in one order in the numerator ( - ) and then in a different order in the denominator ( - ). The slope is not = = -. Incorrect - - (-) b. We can let (, ) = (-, ) and (, ) = (, - ). The slope of the line shown in Figure.(b) is computed as follows: m = = = -. - (-) The slope of the line is - 6. For ever vertical change of - 6 units (6 units down), there is a corresponding horizontal change of units. The slope is negative and the line falls from left to right. (, ) Rise: units Rise: 6 units (, ) Figure. Visualizing slope (a) (, ) Run: unit Run: units (b) (, )

4 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs Find the slope of the line passing through each pair of points: a. (-, ) and (-, - ) b. (, - ) and (-, ). Eample illustrates that a line with a positive slope is rising from left to right and a line with a negative slope is falling from left to right. B contrast, a horizontal line neither rises nor falls and has a slope of zero. A vertical line has no horizontal change, so - = 0 in the formula for slope. Because we cannot divide b zero, the slope of a vertical line is undefined. This discussion is summarized in Table.. Table. Possibilities for a Line s Slope Positive Slope Negative Slope Zero Slope Undefined Slope m > 0 m < 0 m = 0 m is undefined. Line rises from left to right. Line falls from left to right. Line is horizontal. Line is vertical. Write the point-slope equation of a line. (, ) Slope is m. This point is fied. This point is arbitrar. (, ) Figure. A line passing through (, ) with slope m The -Slope Form of the Equation of a Line We can use the slope of a line to obtain various forms of the line s equation. For eample, consider a nonvertical line with slope m that contains the point (, ). Now, let (, ) represent an other point on the line, shown in Figure.. Keep in mind that the point (, ) is arbitrar and is not in one fied position. B contrast, the point (, ) is fied. Regardless of where the point (, ) is located, the shape of the triangle in Figure. remains the same. Thus, the ratio for slope stas a constant m.this means that for all points along the line, We can clear the fraction b multipling both sides b -. ma - B = - - m( - ) = - m = - -, Z. - Simplif. Now, if we reverse the two sides, we obtain the point-slope form of the equation of a line. -Slope Form of the Equation of a Line The point-slope equation of a nonvertical line with slope m that passes through the point (, ) is - = m( - ).

5 BLITMC0A.099_7- /0/0 :07 AM Page 79 For eample, an equation of the line passing through (, ) with slope (m = ) is After we obtain the point-slope form of a line, it is customar to epress the equation with isolated on one side of the equal sign. Eample illustrates how this is done. EXAMPLE Writing the -Slope Equation of a Line Write the point-slope form of the equation of the line passing through with slope. Then solve the equation for. Solution We use the point-slope equation of a line with m =, = -, and =. - = m( - ) This is the point-slope form of the equation. - = [ - (-)] Substitute the given values. - = ( + ) We now have the point-slope form of the equation for the given line. We can solve this equation for b appling the distributive propert on the right side. Finall, we add to both sides. - = ( - ). - = + = + 7 Section. Lines and Slope 79 (-, ) Write the point-slope form of the equation of the line passing through (, - ) with slope 6. Then solve the equation for. EXAMPLE Writing the -Slope Equation of a Line (, 6) 6 (, ) Figure. Write the point-slope equation of this line. Write the point-slope form of the equation of the line passing through the points (, - ) and (-, 6). (See Figure..) Then solve the equation for. Solution To use the point-slope form, we need to find the slope. The slope is the change in the -coordinates divided b the corresponding change in the -coordinates. m = 6 - (-) - - = 9-6 =- This is the definition of slope using (, ) and (, 6). We can take either point on the line to be (, ). Let s use (, ) = (, - ). Now, we are read to write the point-slope equation. - = m( - ) This is the point-slope form of the equation. - (- ) = - ( - ) Substitute: (, and m =-. ) = (, ) + = - ( - ) Simplif. We now have the point-slope form of the equation of the line shown in Figure.. Now, we solve this equation for.

6 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs Discover You can use either point for (, ) when ou write a line s point-slope equation. Rework Eample using (-, 6) for (, ). Once ou solve for, ou should still obtain = =- ( - ) + = = - + This is the point-slope form of the equation. Use the distributive propert. Subtract from both sides. Write the point-slope form of the equation of the line passing through the points (-, - ) and (-, - 6). Then solve the equation for. Write (0, b) and graph the slope-intercept equation of a line. -intercept is b. Slope is m. Figure. A line with slope m and -intercept b The Slope-Intercept Form of the Equation of a Line Let s write the point-slope form of the equation of a nonvertical line with slope m and -intercept b.the line is shown in Figure.. Because the -intercept is b, the line passes through (0, b). We use the point-slope form with = 0 and = b. We obtain Simplifing on the right side gives us - = m( - ) Let = b. Let = 0. - b = m( - 0). - b = m. Finall, we solve for b adding b to both sides. = m + b Thus, if a line s equation is written with isolated on one side, the -coefficient is the line s slope and the constant term is the -intercept. This form of a line s equation is called the slope-intercept form of a line. Slope-Intercept Form of the Equation of a Line The slope-intercept equation of a nonvertical line with slope m and -intercept b is = m + b. EXAMPLE Graph the line whose equation is Graphing b Using the Slope and -Intercept = +. Solution The equation of the line is in the form = m + b. We can find the slope, m, b identifing the coefficient of. We can find the -intercept, b, b identifing the constant term. = + The slope is. The -intercept is.

7 BLITMC0A.099_7- /0/0 :07 AM Page 8 Section. Lines and Slope 8 Rise = Run = -intercept: Figure.6 The graph of = + We need two points in order to graph the line. We can use the -intercept,, to obtain the first point (0, ). Plot this point on the -ais, shown in Figure.6. We know the slope and one point on the line. We can use the slope,, to determine a second point on the line. B definition, m = = Rise Run. We plot the second point on the line b starting at (0, ), the first point. Based on the slope, we move units up (the rise) and units to the right (the run).this puts us at a second point on the line, (, ), shown in Figure.6. We use a straightedge to draw a line through the two points. The graph of = + is shown in Figure.6. Graphing = m + b b Using the Slope and -Intercept. Plot the -intercept on the -ais. This is the point (0, b).. Obtain a second point using the slope, m.write m as a fraction, and use rise over run, starting at the point containing the -intercept, to plot this point.. Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitel in both directions. Graph the line whose equation is = +. Recognize equations of horizontal and vertical lines. Equations of Horizontal and Vertical Lines Some things change ver little. For eample, Figure.7 shows that the percentage of people in the United States satisfied with their lives remains relativel constant for all age groups. Shown in the figure is a horizontal line that passes near most tops of the si bars. 00 Percentage of People in the U.S. Satisfied with Their Lives Percentage Satisfied with Life Age 6 6 We can use = m + b, the slope-intercept form of a line s equation, to write the equation of the horizontal line in Figure.7. We need the line s slope, m, and its -intercept, b. Because the line is horizontal, m = 0. The line intersects the -ais at (0, 80), so its -intercept is 80: b = 80. Figure.7 Source: Culture Shift in Advanced Industrial Societ, Princeton Universit Press

8 BLITMC0A.099_7- /0/0 :07 AM Page 8 8 Chapter Functions and Graphs Thus, an equation in the form = m + b that models the percentage,, of people at age satisfied with their lives is = , or = 80. The percentage of people satisfied with their lives remains relativel constant in the United States for all age groups, at approimatel 80%. In general, if a line is horizontal, its slope is zero: m = 0. Thus, the equation = m + b becomes = b, where b is the -intercept. All horizontal lines have equations of the form = b. (, ) 6 -intercept is. (, ) (0, ) EXAMPLE Graphing a Horizontal Line Graph =-in the rectangular coordinate sstem. Solution All points on the graph of =-have a value of that is alwas -. No matter what the -coordinate is, the -coordinate for ever point on the line is -. Let us select three of the possible values for : -, 0, and. So, three of the points on the graph =-are (-, -), (0, -), and (, -). Plot each of these points. Drawing a line that passes through the three points gives the horizontal line shown in Figure.8. Figure.8 The graph of =- Graph = in the rectangular coordinate sstem. -intercept is. (, ) 6 7 (, ) (, 0) Net, let s see what we can discover about the graph of an equation of the form = a b looking at an eample. EXAMPLE 6 Graphing a Vertical Line Graph = in the rectangular coordinate sstem. Solution All points on the graph of = have a value of that is alwas. No matter what the -coordinate is, the corresponding -coordinate for ever point on the line is. Let us select three of the possible values of : -, 0, and. So, three of the points on the graph of = are (, ), (, 0), and (, ). Plot each of these points. Drawing a line that passes through the three points gives the vertical line shown in Figure.9. Figure.9 The graph = Horizontal and Vertical Lines The graph of = b is a horizontal The graph of = a is a vertical line. The -intercept is b. line. The -intercept is a. (0, b) -intercept: b -intercept: a (a, 0) 6 Graph =-in the rectangular coordinate sstem.

9 BLITMC0A.099_7- /0/0 :07 AM Page 8 Recognize and use the general form of a line s equation. Section. Lines and Slope 8 The General Form of the Equation of a Line The vertical line whose equation is = cannot be written in slope-intercept form, = m + b, because its slope is undefined. However, ever line has an equation that can be epressed in the form A + B + C = 0. For eample, = can be epressed as = 0, or - = 0. The equation A + B + C = 0 is called the general form of the equation of a line. General Form of the Equation of a Line Ever line has an equation that can be written in the general form A + B + C = 0 where A, B, and C are real numbers, and A and B are not both zero. If the equation of a line is given in general form, it is possible to find the slope, m, and the -intercept, b, for the line. We solve the equation for, transforming it into the slope-intercept form = m + b. In this form, the coefficient of is the slope of the line, and the constant term is its -intercept. EXAMPLE 7 Finding the Slope and the -Intercept Find the slope and the -intercept of the line whose equation is = 0. Solution The equation is given in general form. We begin b rewriting it in the form = m + b. We need to solve for = 0 This is the given equation. + 6 = To isolate the -term, add to both sides. = + 6 Reverse the two sides. (This step is optional.) = + Divide both sides b., The coefficient of, is the slope and the constant term,, is the -intercept. This is the form of the equation that we graphed in Figure.6 on page 8. 7 Find the slope and the -intercept of the line whose equation is = 0. Then use the -intercept and the slope to graph the equation. We ve covered a lot of territor. Let s take a moment to summarize the various forms for equations of lines. Equations of Lines. -slope form: - = m( - ). Slope-intercept form: = m + b. Horizontal line: = b. Vertical line: = a. General form: A + B + C = 0

10 BLITMC0A.099_7- /0/0 :07 AM Page 8 8 Chapter Functions and Graphs 6 Find slopes and equations of parallel and perpendicular lines. Parallel and Perpendicular Lines Two nonintersecting lines that lie in the same plane are parallel. If two lines do not intersect, the ratio of the vertical change to the horizontal change is the same for each line. Because two parallel lines have the same steepness, the must have the same slope. Slope and Parallel Lines. If two nonvertical lines are parallel, then the have the same slope.. If two distinct nonvertical lines have the same slope, then the are parallel.. Two distinct vertical lines, both with undefined slopes, are parallel. EXAMPLE 8 Writing Equations of a Line Parallel to a Given Line Write an equation of the line passing through (-, ) and parallel to the line whose equation is = +. Epress the equation in point-slope form and slope-intercept form. Solution The situation is illustrated in Figure.0. We are looking for the equation of the line shown on the left. How do we obtain this equation? Notice that the line passes through the point (-, ). Using the point-slope form of the line s equation, we have = - and =. - = m( - ) = = The equation of this line is given: = (, ) Rise = We must write the equation of this line. Slope = = + Run = -intercept = Now, the onl thing missing from the equation is m, the slope of the line on the left. Do we know anthing about the slope of either line in Figure.0? The answer is es; we know the slope of the line on the right, whose equation is given. = + Figure.0 Writing equations of a line parallel to a given line The slope of the line on the right in Figure.0 is.

11 BLITMC0A.099_7- /0/0 :07 AM Page 8 Section. Lines and Slope 8 Parallel lines have the same slope. Because the slope of the line with the given equation is, m = for the line whose equation we must write. - = m( - ) = m = = The point-slope form of the line s equation is - = [ - (-)] or - = ( + ). Solving for, we obtain the slope-intercept form of the equation. - = + 6 Appl the distributive propert. = + 8 Add to both sides. This is the slope-intercept form, = m + b, of the equation. 8 Write an equation of the line passing through (-, ) and parallel to the line whose equation is = +. Epress the equation in point-slope form and slope-intercept form. Two lines that intersect at a right angle (90 ) are said to be perpendicular, shown in Figure.. There is a relationship between the slopes of perpendicular lines. 90 Figure. Perpendicular lines Slope and Perpendicular Lines. If two nonvertical lines are perpendicular, then the product of their slopes is -.. If the product of the slopes of two lines is -, then the lines are perpendicular.. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. An equivalent wa of stating this relationship is to sa that one line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other. For eample, if a line has slope, an line having slope - is perpendicular to it. Similarl, if a line has slope -, an line having slope is perpendicular to it. EXAMPLE 9 Finding the Slope of a Line Perpendicular to a Given Line Find the slope of an line that is perpendicular to the line whose equation is = 0.

12 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs Solution We begin b writing the equation of the given line, = 0, in slope-intercept form. Solve for = 0 This is the given equation. = + 8 To isolate the -term, subtract and add 8 on both sides. = - + Divide both sides b. Slope is. The given line has slope -. An line perpendicular to this line has a slope that is the negative reciprocal of -. Thus, the slope of an perpendicular line is. 9 Find the slope of an line that is perpendicular to the line whose equation is + - = 0. 7 Model data with linear equations. Applications Slope is defined as the ratio of a change in to a corresponding change in. Our net eample shows how slope can be interpreted as a rate of change in an applied situation. EXAMPLE 0 Slope as a Rate of Change Number Living Alone (millions) Number of People in the U.S Living Alone (99, ) Women (99, 0) Men (00, 7) (00, ) A best guess at the look of our nation in the net decade indicates that the number of men and women living alone will increase each ear. Figure. shows line graphs for the number of U.S. men and women living alone, projected through 00. Find the slope of the line segment for the women. Describe what the slope represents. Solution We let represent a ear and the number of women living alone in that ear. The two points shown on the line segment for women have the following coordinates: (99, ) and (00, 7) Year Figure. Source: Forrester Research In 99, million U.S. women lived alone. Now we compute the slope: m = Change in Change in = = = In 00, 7 million U.S. women are projected to live alone. = 0. million people ear. The unit in the numerator is million people. The unit in the denominator is ear. The slope indicates that the number of U.S. women living alone is projected to increase b 0. million each ear.the rate of change is 0. million women per ear. 0 Use the graph in Eample 0 to find the slope of the line segment for the men. Epress the slope correct to two decimal places and describe what it represents. If an equation in slope-intercept form models relationships between variables, then the slope and -intercept have phsical interpretations. For the equation = m + b, the -intercept, b, tells us what is happening to when is 0. If represents time,the -intercept describes the value of at the beginning,or when time equals 0.The slope represents the rate of change in per unit change in.

13 BLITMC0A.099_7- /0/0 :07 AM Page 87 Technolog U.S. Population (millions) You can use a graphing utilit to obtain a model for a scatter plot in which the data points fall on or near a straight line. The line that best fits the data is called the regression line. After entering the data in Table., a graphing utilit displas a scatter plot of the data and the regression line Section. Lines and Slope 87 Using these ideas, we can develop a model for the data for women living alone, shown in Figure. on the previous page. We let = the number of ears after 99. At the beginning of our data, or 0 ears after 99, million women lived alone. Thus, b =. In Eample 0, we found that m = 0. (rate of change is 0. million women per ear).an equation of the form = m + b that models the data is = 0. +, where is the number, in millions, of U.S. women living alone ears after 99. Linear equations are useful for modeling data in scatter plots that fall on or near a line. For eample, Table. gives the population of the United States, in millions, in the indicated ear. The data are displaed in a scatter plot as a set of si points in Figure.. 0 (0, 0) (0, 6.) Years after 960 Also shown in Figure. is a line that passes through or near the si points. B writing the equation of this line, we can obtain a model of the data and make predictions about the population of the United States in the future. EXAMPLE 0 Table. (U.S. Population) Year (Years after 960) (in millions) Figure. Modeling U.S. Population Write the slope-intercept equation of the line shown in Figure.. Use the equation to predict U.S. population in 00. Solution The line in Figure. passes through (0, 6.) and (0, 0). We start b finding the slope. [,, ] b [70, 8, 0] Also displaed is the regression line s equation. m = Change in Change in = =. 0 =. The slope indicates that the rate of change in the U.S. population is. million people per ear. Now we write the line s slope-intercept equation. - = m( - ) - 0 =.( - 0) - 0 = = Begin with the point-slope form. Either ordered pair can be (, ). Let (, ) = (0, 0). From above, m =.. Appl the distributive propert on the right. Add 0 to both sides and solve for.

14 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs A linear equation that models U.S. population,, in millions, ears after 960 is = Now, let s use this equation to predict U.S. population in 00. Because 00 is 0 ears after 960, substitute 0 for and compute. =.(0) = 97 Our equation predicts that the population of the United States in the ear 00 will be 97 million. (The projected figure from the U.S. Census Bureau is million.) Use the data points (0, 0.) and (0, 6.) from Table. to write an equation that models U.S. population ears after 960. Use the equation to predict U.S. population in 00. Deaths per Million Males from Lung Cancer Cigarettes and Lung Cancer Switzerland Holland 00 Denmark Australia Canada Sweden 00 Norwa Iceland 0 Great Britain Finland Annual Cigarette Consumption for Each Adult Male U.S This scatter plot shows a relationship between cigarette consumption among males and deaths due to lung cancer per million males. The data are from countries and date back to a 96 report b the U.S. Surgeon General. The scatter plot can be modeled b a line whose slope indicates an increasing death rate from lung cancer with increased cigarette consumption. At that time, the tobacco industr argued that in spite of this regression line, tobacco use is not the cause of cancer. Recent data do, indeed, show a causal effect between tobacco use and numerous diseases. Source: Smoking and Health, Washington, D.C., 96 EXERCISE SET. Practice Eercises In Eercises 0, find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.. (, 7) and (8, 0). (, ) and (, ). (-, ) and (, ). (-, ) and (, ). (, - ) and (, - ) 6. (, - ) and (, - ) 7. (-, ) and (-, - ) 8. (6, - ) and (, - ) 9. (, ) and (, - ) 0. (, - ) and (, ) In Eercises 8, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.. Slope =, passing through (, ). Slope =, passing through (, ). Slope = 6, passing through (-, ). Slope = 8, passing through (, - ). Slope = -, passing through (-, - )

15 BLITMC0A.099_7- /0/0 :07 AM Page 89. =- 6. = In Eercises 7, graph each equation in the rectangular coordinate sstem. 7. = - 8. = 9. = - 0. =. = 0. = 0 In Eercises 60, a. Rewrite the given equation in slope-intercept form. b. Give the slope and -intercept. c. Graph the equation = = = = = = = = 0 In Eercises 6 68, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. 6. Passing through (-8, -0) and parallel to the line whose equation is = Passing through (-, -7) and parallel to the line whose equation is =- + Source: National Ski Areas Association Eercise Set Slope = -, passing through (-, - ) 6. Passing through (, -) and perpendicular to the line 7. Slope = -, passing through (-, 0) whose equation is = Slope = -, passing through (0, - ) 6. Passing through (-, ) and perpendicular to the line 9. passing through A-, - B whose equation is = Slope = -, passing through A-, - 6. Passing through (-, ) and parallel to the line whose B. Slope =, passing through the origin equation is = 0. Slope =, passing through the origin 66. Passing through (-, ) and parallel to the line whose. Slope = -, equation is - - = 0 passing through (6, - ). Slope = -, 67. Passing through (, -7) and perpendicular to the line passing through (0, - ) whose equation is - - = 0. Passing through (, ) and (, 0) 68. Passing through (, -9) and perpendicular to the line 6. Passing through (, ) and (8, ) whose equation is = 0 7. Passing through (-, 0) and (0, ) 8. Passing through (-, 0) and (0, ) 9. Passing through (-, - ) and (, ) Application Eercises 0. Passing through (-, - ) and (, - ) 69. The scatter plot shows that from 98 to 00, the. Passing through (-, - ) and (, 6) number of Americans participating in downhill skiing. Passing through (-, 6) and (, - ) remained relativel constant. Write an equation that. Passing through (-, - ) and (, - ). Passing through (-, - ) and (6, - ) models the number of participants in downhill skiing,, in millions, for this period.. Passing through (, ) with -intercept = - 6. Passing through (, - ) with -intercept =- Number of U.S. Participants 7. -intercept = - and -intercept = in Downhill Skiing 8. -intercept = and -intercept = - 0 In Eercises 9 6, give the slope and -intercept of each line whose equation is given. Then graph the line. 9. = + 0. = + 0. = - +. = - +. =. = Year If talk about a federal budget surplus sounded too good to be true, that s because it probabl was. The Congressional Budget Office s estimates for 00 range from a $. trillion budget surplus to a $86 billion deficit. Use the information provided b the Congressional Budget Office graphs to solve Eercises Budget Surplus (billions) Number of Downhill Skiers (millions) $ Federal Budget Projections Optimistic: The boom goes on Year Source: Congressional Budget Office CBO base line Pessimistic: Hard times ahead

16 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs 70. Turn back a page and look at the line that indicates hard times ahead. Find the slope of this line using (00, 0) and (00, 86). Use a calculator and round to the nearest whole number. Describe what the slope represents. 7. Turn back a page and look at the line that indicates the boom goes on. Find the slope of this line using (00, 00) and (00, 00). Use a calculator and round to the nearest whole number. Describe what the slope represents. 7. Horrified at the cost the last time ou needed a prescription drug? The graph shows that the cost of the average retail prescription has been rising steadil since 99. Source: Newsweek a. According to the graph, what is the -intercept? Describe what this represents in this situation. b. Use the coordinates of the two points shown to compute the slope. What does this mean about the cost of the average retail prescription? c. Write a linear equation in slope-intercept form that models the cost of the average retail prescription,, ears after 99. d. Use our model from part (c) to predict the cost of the average retail prescription in For 6 ears, Social Securit has been a huge success. It is the primar source of income for 66% of Americans over 6 and the onl thing that keeps % of the elderl from povert. However, the number of workers per Social Securit beneficiar has been declining steadil since 90. Workers per Beneficiar Cost (dollars) 0 0 $ Number of Workers per Social Securit Beneficiar (0, 6) Average Cost of a Retail Prescription (0, ) (7, 8) Years after Year after 90 Source: Social Securit Administration (0, ) a. According to the graph, what is the -intercept? Describe what this represents in this situation. b. Use the coordinates of the two points shown to compute the slope. What does this mean about the number of workers per beneficiar? c. Write a linear equation in slope-intercept form that models the number of workers per beneficiar,, ears after 90. d. Use our model from part (c) to predict number of workers per beneficiar in 00. For ever 8 workers, how man beneficiaries will there be? 7. We seem to be fed up with being lectured at about our waistlines. The points in the graph show the average weight of American adults from 990 through 000. Also shown is a line that passes through or near the points. U.S. Average Weight (pounds) Source: Diabetes Care a. Use the two points whose coordinates are shown b the voice balloons to find the point-slope equation of the line that models average weight of Americans,, in pounds, ears after 990. b. Write the equation in part (a) in slope-intercept form. c. Use the slope-intercept equation to predict the average weight of Americans in Films ma not be getting an better, but in this era of moviegoing, the number of screens available for new films and the classics has eploded. The points in the graph show the number of screens in the United States from 99 through 000. Also shown is a line that passes through or near the points. Number of Screens (thousands) Average Weight of Americans (, 6) Years after 990 Number of Movie Screens in the U.S. (, 0) (8, 68) (, 8) Years after 99 Source: Motion Picture Association of America

17 BLITMC0A.099_7- /0/0 :07 AM Page 9 Eercise Set. 9 a. Use the two points whose coordinates are shown b the voice balloons to find the point-slope equation of the line that models the number of screens,, in thousands, ears after 99. b. Write the equation in part (a) in slope-intercept form. c. Use the slope-intercept equation to predict the number of screens, in thousands, in The scatter plot shows the relationship between the percentage of married women of child-bearing age using contraceptives and the births per woman in selected countries. Also shown is the regression line. Use two points on this line to write both its point-slope and slope-intercept equations. Then find the number of births per woman if 90% of married women of child-bearing age use contraceptives. Births per Woman Source: Population Reference Bureau 77. Shown, again, is the scatter plot that indicates a relationship between the percentage of adult females in a countr who are literate and the mortalit of children under five. Also shown is a line that passes through or near the points. Find a linear equation that models the data b finding the slope-intercept equation of the line. Use the model to make a prediction about child mortalit based on the percentage of adult females in a countr who are literate. Under-Five Mortalit (per thousand) Contraceptive Prevalence and Births per Woman, Selected Countries 0 9 Kena 8 Sria Jordan Botswana 7 Zimbabwe 6 Lesotho Honduras Vietnam Costa Rica Fiji Chile Spain Denmark Percentage of Married Women of Child-Bearing Age Using Contraceptives Source: United Nations Literac and Child Mortalit Percentage of Adult Females Who Are Literate In Eercises 78 80, find a linear equation in slope-intercept form that models the given description. Describe what each variable in our model represents. Then use the model to make a prediction. 78. In 99, the average temperature of Earth was 7.7ºF and has increased at a rate of 0.0ºF per ear since then. 79. In 99, 60% of U.S. adults read a newspaper and this percentage has decreased at a rate of 0.7% per ear since then. 80. A computer that was purchased for $000 is depreciating at a rate of $90 per ear. 8. A business discovers a linear relationship between the number of shirts it can sell and the price per shirt. In particular, 0,000 shirts can be sold at $9 each, and 000 of the same shirts can be sold at $ each. Write the slope-intercept equation of the demand line that models the number of shirts that can be sold,, at a price of dollars. Then determine the number of shirts that can be sold at $0 each. Writing in Mathematics 8. What is the slope of a line and how is it found? 8. Describe how to write the equation of a line if two points along the line are known. 8. Eplain how to derive the slope-intercept form of a line s equation, = m + b, from the point-slope form - = m( - ). 8. Eplain how to graph the equation =. Can this equation be epressed in slope-intercept form? Eplain. 86. Eplain how to use the general form of a line s equation to find the line s slope and -intercept. 87. If two lines are parallel, describe the relationship between their slopes. 88. If two lines are perpendicular, describe the relationship between their slopes. 88. If ou know a point on a line and ou know the equation of a line perpendicular to this line, eplain how to write the line s equation. 90. A formula in the form = m + b models the cost,,of a four-ear college ears after 00.Would ou epect m to be positive, negative, or zero? Eplain our answer. 9. We saw that the percentage of people satisfied with their lives remains relativel constant for all age groups. Eercise 69 showed that the number of skiers in the United States has remained relativel constant over time. Give another eample of a real-world phenomenon that has remained relativel constant. Tr writing an equation that models this phenomenon.

18 BLITMC0A.099_7- /0/0 :07 AM Page 9 9 Chapter Functions and Graphs Technolog Eercises Use a graphing utilit to graph each equation in Eercises 9 9. Then use the TRACE feature to trace along the line and find the coordinates of two points. Use these points to compute the line s slope. our result b using the coefficient of in the line s equation. 9. = + 9. = = = Is there a relationship between alcohol from moderate wine consumption and heart disease death rate? The table gives data from 9 developed countries. France Countr A B C D E F G Liters of alcohol from drinking wine, per person, per ear () Deaths from heart disease, per 00, people per ear () U.S. Critical Thinking Eercises 97. Which one of the following is true? a. A linear equation with nonnegative slope has a graph that rises from left to right. b. The equations = and = - have graphs that are perpendicular lines. c. The line whose equation is = 0 passes through the point (6, 0) and has slope - 6. d. The graph of = 7 in the rectangular coordinate sstem is the single point (7, 0). 98. Prove that the equation of a line passing through (a, 0) and (0, b) (a Z 0, b Z 0) can be written in the form Wh is this called the intercept form of a line? a + b =. 99. Use the figure shown to make the following lists. a. List the slopes m, m, m, and m in order of decreasing size. b. List the -intercepts b, b, b, and b in order of decreasing size. = m + b Countr H I J K L M N O P Q R S () () Source: New York Times, December 8, 99 a. Use the statistical menu of our graphing utilit to enter the 9 ordered pairs of data items shown in the table. b. Use the DRAW menu and the scatter plot capabilit to draw a scatter plot of the data. c. Select the linear regression option. Use our utilit to obtain values for a and b for the equation of the regression line, = a + b. You ma also be given a correlation coefficient, r. Values of r close to indicate that the points can be described b a linear relationship and the regression line has a positive slope. Values of r close to - indicate that the points can be described b a linear relationship and the regression line has a negative slope.values of r close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accuratel describe the data. d. Use the appropriate sequence (consult our manual) to graph the regression equation on top of the points in the scatter plot. 00. Ecited about the success of celebrit stamps, post office officials were rumored to have put forth a plan to institute two new tpes of thermometers. On these new scales, E represents degrees Elvis and M represents degrees Madonna. If it is known that 0 E = M, 80 E = M, and degrees Elvis is linearl related to degrees Madonna, write an equation epressing E in terms of M. Group Eercise = m + b = m + b = m + b 0. Group members should consult an almanac, newspaper, magazine, or the Internet to find data that lie approimatel on or near a straight line.working b hand or using a graphing utilit, construct a scatter plot for the data. If working b hand, draw a line that approimatel fits the data and then write its equation. If using a graphing utilit, obtain the equation of the regression line. Then use the equation of the line to make a prediction about what might happen in the future. Are there circumstances that might affect the accurac of this prediction? List some of these circumstances.

19 BLITMC0A.099_7- /0/0 :07 AM Page 9 Section. Distance and Midpoint Formulas; Circles 9 SECTION. Distance and Midpoint Formulas; Circles Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation is in standard form.. Convert the general form of a circle s equation to standard form. It s a good idea to know our wa around a circle. Clocks, angles, maps, and compasses are based on circles. Circles occur everwhere in nature: in ripples on water, patterns on a butterfl s wings, and cross sections of trees. Some consider the circle to be the most pleasing of all shapes. The rectangular coordinate sstem gives us a unique wa of knowing a circle. It enables us to translate a circle s geometric definition into an algebraic equation. To do this, we must first develop a formula for the distance between an two points in rectangular coordinates. Find the distance between two points. The Distance Formula Using the Pthagorean Theorem, we can find the distance between the two points P(, ) and P(, ) in the rectangular coordinate sstem. The two points are illustrated in Figure.. P(, ) d P(, ) (, ) Figure. The distance that we need to find is represented b d and shown in blue. Notice that the distance between two points on the dashed horizontal line is the absolute value of the difference between the -coordinates of the two points. This distance, -, is shown in pink. Similarl, the distance between two points on the dashed vertical line is the absolute value of the difference between the -coordinates of the two points. This distance, -, is also shown in pink.

20 BLITMC0A.099_7- /0/0 :07 AM Page 9 9 Chapter Functions and Graphs P (, ) Because the dashed lines are horizontal and vertical, a right triangle is formed. Thus, we can use the Pthagorean Theorem to find distance d. B the Pthagorean Theorem, P (, ) d (, ) d = d = d = ( - ) + ( - ). Figure., repeated This result is called the distance formula. The Distance Formula The distance, d, between the points (, ) and (, ) in the rectangular coordinate sstem is d = ( - ) + ( - ). When using the distance formula, it does not matter which point ou call (, ) and which ou call (, ). EXAMPLE Using the Distance Formula Find the distance between (-, -) and (, ). Solution Letting (, ) = (-, -) and (, ) = (, ), we obtain d = ( - ) + ( - ) Use the distance formula. (, ) = [ - (-)] + [ - (-)] = ( + ) + ( + ) = + 6 = Substitute the given values. Appl the definition of subtraction within the grouping smbols. Perform the resulting additions. Square and 6. (, ) Distance is units. = = L 6.7. Add. = 9 = 9 = Figure. Finding the distance between two points The distance between the given points is units, or approimatel 6.7 units. The situation is illustrated in Figure.. Find the distance between (, ) and (, ). Find the midpoint of a line segment. The Midpoint Formula The distance formula can be used to derive a formula for finding the midpoint of a line segment between two given points. The formula is given as follows:

21 BLITMC0A.099_7- /0/0 :07 AM Page 9 Section. Distance and Midpoint Formulas; Circles 9 The Midpoint Formula Consider a line segment whose endpoints are (, ) and (, ). The coordinates of the segment s midpoint are +, +. To find the midpoint, take the average of the two -coordinates and the average of the two -coordinates ( 8, ) 7 (, ) Midpoint (, 6) EXAMPLE Using the Midpoint Formula Find the midpoint of the line segment with endpoints (, -6) and (-8, -). Solution To find the coordinates of the midpoint, we average the coordinates of the endpoints. Midpoint = + (-8) Average the -coordinates (-), Average the -coordinates. = -7, -0 = - 7, - Figure.6 Finding a line segment s midpoint Figure.6 illustrates that the point (, -6) and (-8, -). (- 7, -) is midwa between the points Find the midpoint of the line segment with endpoints (, ) and (7, -). Circles Our goal is to translate a circle s geometric definition into an equation. We begin with this geometric definition. Center (h, k) Radius: r (, ) An point on the circle Figure.7 A circle centered at (h, k) with radius r Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fied point, called the center. The fied distance from the circle s center to an point on the circle is called the radius. Figure.7 is our starting point for obtaining a circle s equation. We ve placed the circle into a rectangular coordinate sstem. The circle s center is (h, k) and its radius is r.we let (, ) represent the coordinates of an point on the circle. What does the geometric definition of a circle tell us about point (, ) in Figure.7? The point is on the circle if and onl if its distance from the center is r.we can use the distance formula to epress this idea algebraicall: The distance between (, ) and (h, k) ( - h) + ( - k) is alwas r. = r

22 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs Squaring both sides of ( - h) + ( - k) = r ields the standard form of the equation of a circle. The Standard Form of the Equation of a Circle The standard form of the equation of a circle with center (h, k) and radius r is ( - h) + ( - k) = r. Write the standard form of a circle s equation. (0, 0) Figure.8 The graph of + = (, ) + = EXAMPLE Finding the Standard Form of a Circle s Equation Write the standard form of the equation of the circle with center (0, 0) and radius. Graph the circle. Solution The center is (0, 0). Because the center is represented as (h, k) in the standard form of the equation, h = 0 and k = 0. The radius is, so we will let r = in the equation. ( - h) + ( - k) = r ( - 0) + ( - 0) = + = This is the standard form of a circle s equation. Substitute 0 for h, 0 for k, and for r. Simplif. The standard form of the equation of the circle is + =. Figure.8 shows the graph. Technolog = Write the standard form of the equation of the circle with center (0, 0) and radius. To graph a circle with a graphing utilit, first solve the equation for. + = = - Graph the two equations =; - = - and =- - = in the same viewing rectangle. The graph of = - is the top semicircle because is alwas positive.the graph of =- - is the bottom semicircle because is alwas negative. Use a ZOOM SQUARE setting so that the circle looks like a circle. (Man graphing utilities have problems connecting the two semicircles because the segments directl across horizontall from the center become nearl vertical.) Eample and involved circles centered at the origin. The standard form of the equation of all such circles is + = r, where r is the circle s radius. Now, let s consider a circle whose center is not at the origin.

23 BLITMC0A.099_7- /0/0 :07 AM Page 97 EXAMPLE Section. Distance and Midpoint Formulas; Circles 97 Finding the Standard Form of a Circle s Equation Write the standard form of the equation of the circle with center (-, ) and radius. Solution The center is (-, ). Because the center is represented as (h, k) in the standard form of the equation, h =-and k =. The radius is, so we will let r = in the equation. ( - h) + ( - k) = r This is the standard form of a circle s equation. [ - (-)] + ( - ) = Substitute for h, for k, and for r. ( + ) + ( - ) = 6 Simplif. The standard form of the equation of the circle is ( + ) + ( - ) = 6. Write the standard form of the equation of the circle with center (, -6) and radius 0. Give the center and radius of a circle whose equation is in standard form. EXAMPLE Using the Standard Form of a Circle s Equation to Graph the Circle Find the center and radius of the circle whose equation is and graph the equation. ( - ) + ( + ) = 9 Solution In order to graph the circle, we need to know its center, (h, k), and its radius, r. We can find the values for h, k, and r b comparing the given equation to the standard form of the equation of a circle. (, ) 6 7 (, ) 6 7 (, ) (, 7) Figure.9 The graph of ( - ) + ( + ) = 9 (, ) ( - ) + [ - (-)] = This is ( h), with h =. ( - ) + ( + ) = 9 This is ( k), with k =. This is r, with r =. We see that h =, k =-, and r =. Thus, the circle has center (h, k) = (, -) and a radius of units. To graph this circle, first plot the center (, -). Because the radius is, ou can locate at least four points on the circle b going out three units to the right, to the left, up, and down from the center. The points three units to the right and to the left of (, -) are (, -) and (-, -), respectivel. The points three units up and down from (, -) are (, -) and (, -7), respectivel. Using these points, we obtain the graph in Figure.9. Find the center and radius of the circle whose equation is and graph the equation. ( + ) + ( - ) =

24 BLITMC0A.099_7- /0/0 :07 AM Page Chapter Functions and Graphs If we square - and + in the standard form of the equation from Eample, we obtain another form for the circle s equation. ( - ) + ( + ) = 9 This is the standard form of the equation from Eample = 9 Square - and = 9 Combine numerical terms and rearrange terms = 0 Subtract 9 from both sides. This result suggests that an equation in the form + + D + E + F = 0 can represent a circle. This is called the general form of the equation of a circle. The General Form of the Equation of a Circle The general form of the equation of a circle is + + D + E + F = 0. Convert the general form of a circle s equation to standard form. Stud Tip To review completing the square, see Section., pages (, ) Figure.0 The graph of ( + ) + ( - ) = We can convert the general form of the equation of a circle to the standard form ( - h) + ( - k) = r. We do so b completing the square on and. Let s see how this is done. EXAMPLE 6 Write in standard form and graph: Converting the General Form of a Circle s Equation to Standard Form and Graphing the Circle Solution Because we plan to complete the square on both and, let s rearrange terms so that -terms are arranged in descending order, -terms are arranged in descending order, and the constant term appears on the right = 0 ( + ) + ( - 6 ) = ( + + ) + ( ) = Remember that numbers added on the left side must also be added on the right side. ( + ) + ( - ) = 6 This is the given equation. Rewrite in anticipation of completing the square. Complete the square on : = and =, so add to both sides. Complete the square on : and (-) (-6) =- = 9, so add 9 to both sides. Factor on the left and add on the right. This last equation is in standard form. We can identif the circle s center and radius b comparing this equation to the standard form of the equation of a circle, ( - h) + ( - k) = r. [ - (-)] + ( - ) = 6 This is ( h), with h = = 0. ( + ) + ( - ) = 6 This is ( k), with k =. This is r, with r = 6. We use the center, (h, k) = (-, ), and the radius, r = 6, to graph the circle. The graph is shown in Figure.0.