1.2 Lines in the Plane

Transcription

1 71_1.qd 1/7/6 1:1 AM Page Chapter 1 Functions and Their Graphs 1. Lines in the Plane The Slope of a Line In this section, ou will stud lines and their equations. The slope of a nonvertical line represents the number of units the line rises or falls verticall for each unit of horizontal change from left to right. For instance, consider the two points and, 1, 1 on the line shown in Figure As ou move from left to right along this line, a change of 1 units in the vertical direction corresponds to a change of 1 units in the horizontal direction. That is, and 1 the change in 1 the change in. The slope of the line is given b the ratio of these two changes. What ou should learn Find the slopes of lines. Write linear equations given points on lines and their slopes. Use slope-intercept forms of linear equations to sketch lines. Use slope to identif parallel and perpendicular lines. Wh ou should learn it The slope of a line can be used to solve real-life problems. For instance, in Eercise 87 on page 99. ou will use a linear equation to model student enrollment at Penn State Universit. 1 (, ) (, ) 1 Sk Bonillo/PhotoEdit 1 Figure 1.16 Definition of the Slope of a Line The slope m of the nonvertical line through and, 1, 1 is m 1 change in 1 change in where 1. When this formula for slope is used, the order of subtraction is important. Given two points on a line, ou are free to label either one of them as 1, 1 and the other as,. However, once ou have done this, ou must form the numerator and denominator using the same order of subtraction. m 1 1 m 1 1 m 1 1 Correct Correct Incorrect Throughout this tet, the term line alwas means a straight line.

2 71_1.qd 1/7/6 1:1 AM Page 89 Section 1. Lines in the Plane 89 Eample 1 Finding the Slope of a Line Find the slope of the line passing through each pair of points. a., and, 1 b. 1, and, c., and 1, 1 Solution a. b. c. Difference in -values m Difference in -values m 1 m The graphs of the three lines are shown in Figure Note that the square setting gives the correct steepness of the lines. Eploration Use a graphing utilit to compare the slopes of the lines.5,,, and. What do ou observe about these lines? Compare the slopes of the lines.5,,, and. What do ou observe about these lines? (Hint: Use a square setting to guarantee a true geometric perspective.) Common Error A common error when finding the slope of a line is combining - and - coordinates in either the numerator or denominator, or both, as in m (, 1) 5 (, ) ( 1, ) (, ) 5 (, ) 8 (1, 1) (a) (b) (c) Figure 1.17 Now tr Eercise 9. The definition of slope does not appl to vertical lines. For instance, consider the points, and, 1 on the vertical line shown in Figure Appling the formula for slope, ou obtain m 1. Undefined Because division b zero is undefined, the slope of a vertical line is undefined. From the slopes of the lines shown in Figures 1.17 and 1.18, ou can make the following generalizations about the slope of a line. 5 (, ) (, 1) Figure 1.18 The Slope of a Line 1. A line with positive slope m > rises from left to right.. A line with negative slope m < falls from left to right.. A line with zero slope m is horizontal.. A line with undefined slope is vertical. Point out to our students that the vertical line shown in Figure 1.18 must be drawn on a graphing utilit with a special command because there is no wa to epress the line s equation in the format.

3 71_1.qd 1/7/6 1:1 AM Page 9 9 Chapter 1 Functions and Their Graphs The Point-Slope Form of the Equation of a Line If ou know the slope of a line and ou also know the coordinates of one point on the line, ou can find an equation for the line. For instance, in Figure 1.19, let 1, 1 be a point on the line whose slope is m. If, is an other point on the line, it follows that (, ) (, ) m. This equation in the variables and can be rewritten in the point-slope form of the equation of a line. Figure 1.19 Point-Slope Form of the Equation of a Line The point-slope form of the equation of the line that passes through the point 1, 1 and has a slope of m is 1 m 1. The point-slope form is most useful for finding the equation of a line if ou know at least one point that the line passes through and the slope of the line. You should remember this form of the equation of a line. Eample The Point-Slope Form of the Equation of a Line Find an equation of the line that passes through the point 1, and has a slope of. Solution 1 m 1 1 The line is shown in Figure 1.. The point-slope form can be used to find an equation of a nonvertical line passing through two points 1, 1 and,. First, find the slope of the line. m 1 1, 5 Now tr Eercise 5. 1 Then use the point-slope form to obtain the equation Point-slope form Substitute for 1, m, and 1. Simplif. Solve for. This is sometimes called the two-point form of the equation of a line Figure 1. = 5 (1, ) STUDY TIP When ou find an equation of the line that passes through two given points, ou need to substitute the coordinates of onl one of the points into the point-slope form. It does not matter which point ou choose because both points will ield the same result.

4 71_1.qd 1/7/6 1:1 AM Page 91 Section 1. Lines in the Plane 91 Eample A Linear Model for Sales Prediction During, Nike s net sales were $1.5 billion, and in 5 net sales were $1.7 billion. Write a linear equation giving the net sales in terms of the ear. Then use the equation to predict the net sales for 6. (Source: Nike, Inc.) Solution Let represent. In Figure 1.1, let, 1.5 and 5, 1.7 be two points on the line representing the net sales. The slope of this line is m Librar of Parent Functions: Linear Function In the net section, ou will be introduced to the precise meaning of the term function. The simplest tpe of function is a linear function of the form f m b. As its name implies, the graph of a linear function is a line that has a slope of m and a -intercept at, b. The basic characteristics of a linear function are summarized below. (Note that some of the terms below will be defined later in the tet.) A review of linear functions can be found in the Stud Capsules. Graph of f m b, m > Graph of f m b, m < Domain:, Domain:, Range:, Range:, -intercept: b m, -intercept: b m, -intercept:, b -intercept:, b Increasing Decreasing f() = m + b, m > ( b, m ( (, b) 1.9. m 1 1 B the point-slope form, the equation of the line is as follows Write in point-slope form Simplif. Now, using this equation, ou can predict the 6 net sales 6 to be $15. billion. Now tr Eercise 5. (, b) When m, the function f b is called a constant function and its graph is a horizontal line. ( f() = m + b, m < b, m ( (6, 15.) (, 1.5) (5, 1.7) = Figure 1.1 STUDY TIP The prediction method illustrated in Eample is called linear etrapolation. Note in the top figure below that an etrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in the bottom figure, the procedure used to predict the point is called linear interpolation. Linear Etrapolation Given points Given points Linear Interpolation 8 Estimated point Estimated point

5 71_1.qd 1/7/6 1:1 AM Page 9 9 Chapter 1 Functions and Their Graphs Sketching Graphs of Lines Man problems in coordinate geometr can be classified as follows. 1. Given a graph (or parts of it), find its equation.. Given an equation, sketch its graph. For lines, the first problem is solved easil b using the point-slope form. This formula, however, is not particularl useful for solving the second tpe of problem. The form that is better suited to graphing linear equations is the slope-intercept form of the equation of a line, m b. Slope-Intercept Form of the Equation of a Line The graph of the equation m b is a line whose slope is m and whose -intercept is, b. Eample Using the Slope-Intercept Form Determine the slope and -intercept of each linear equation. Then describe its graph. a. b. Algebraic Solution a. Begin b writing the equation in slope-intercept form. Write original equation. Subtract from each side. Write in slope-intercept form. From the slope-intercept form of the equation, the slope is 1 and the -intercept is,. Because the slope is negative, ou know that the graph of the equation is a line that falls one unit for ever unit it moves to the right. b. B writing the equation in slope-intercept form ou can see that the slope is and the -intercept is,. A zero slope implies that the line is horizontal. Graphical Solution a. Solve the equation for to obtain. Enter this equation in our graphing utilit. Use a decimal viewing window to graph the equation. To find the -intercept, use the value or trace feature. When,, as shown in Figure 1.(a). So, the -intercept is,. To find the slope, continue to use the trace feature. Move the cursor along the line until 1. At this point, 1. So the graph falls 1 unit for ever unit it moves to the right, and the slope is 1. b. Enter the equation in our graphing utilit and graph the equation. Use the trace feature to verif the -intercept,, as shown in Figure 1.(b), and to see that the value of is the same for all values of. So, the slope of the horizontal line is Now tr Eercise 7. (a) Figure 1..1 (b)

6 71_1.qd 1/7/6 1:1 AM Page 9 Section 1. Lines in the Plane 9 From the slope-intercept form of the equation of a line, ou can see that a horizontal line m has an equation of the form b. This is consistent with the fact that each point on a horizontal line through, b has a -coordinate of b. Similarl, each point on a vertical line through a, has an -coordinate of a. So, a vertical line has an equation of the form a. This equation cannot be written in slope-intercept form because the slope of a vertical line is undefined. However, ever line has an equation that can be written in the general form A B C where A and B are not both zero. General form of the equation of a line Eploration Graph the lines 1 1, 1 1, and 1 in the same viewing window. What do ou observe? Graph the lines 1 1,, and 1 in the same viewing window. What do ou observe? Summar of Equations of Lines 1. General form: A B C. Vertical line: a. Horizontal line: b. Slope-intercept form: m b 5. Point-slope form: 1 m 1 Eample 5 Different Viewing Windows The graphs of the two lines 1 and 1 1 are shown in Figure 1.. Even though the slopes of these lines are quite different ( 1 and 1, respectivel), the graphs seem misleadingl similar because the viewing windows are different. = 1 = = (a) = + 1 Figure 1. 1 Now tr Eercise (b) TECHNOLOGY TIP When a graphing utilit is used to graph a line, it is important to realize that the graph of the line ma not visuall appear to have the slope indicated b its equation. This occurs because of the viewing window used for the graph. For instance, Figure 1. shows graphs of 1 produced on a graphing utilit using three different viewing windows. Notice that the slopes in Figures 1.(a) and (b) do not visuall appear to be equal to. However, if ou use a square setting, as in Figure 1.(c), the slope visuall appears to be (c) Figure 1. 1 = + 1

7 71_1.qd 1/7/6 1:1 AM Page 9 9 Chapter 1 Functions and Their Graphs Parallel and Perpendicular Lines The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular. Parallel Lines Two distinct nonvertical lines are parallel if and onl if their slopes are equal. That is, Eample 6 m 1 m. Equations of Parallel Lines Find the slope-intercept form of the equation of the line that passes through the point, 1 and is parallel to the line 5. Solution Begin b writing the equation of the given line in slope-intercept form Write original equation. Multipl b 1. Add to each side. TECHNOLOGY TIP Be careful when ou graph equations such as 7 with our graphing utilit. A common mistake is to tpe in the equation as Y1 X 7 which ma not be interpreted b our graphing utilit as the original equation. You should use one of the following formulas. Y1 X 7 Y1 X 7 Do ou see wh? 5 Write in slope-intercept form. Therefore, the given line has a slope of m. An line parallel to the given line must also have a slope of. So, the line through, 1 has the following equation. 1 Write in point-slope form. 1 = Simplif. Write in slope-intercept form. 1 5 (, 1) Notice the similarit between the slope-intercept form of the original equation and the slope-intercept form of the parallel equation. The graphs of both equations are shown in Figure 1.5. Now tr Eercise 57(a). Figure 1.5 = 7 Perpendicular Lines Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other. That is, m 1 1 m.

8 71_1.qd 1/7/6 1:1 AM Page 95 Section 1. Lines in the Plane 95 Eample 7 Equations of Perpendicular Lines Find the slope-intercept form of the equation of the line that passes through the point, 1 and is perpendicular to the line 5. Solution From Eample 6, ou know that the equation can be written in the slope-intercept form 5. You can see that the line has a slope of So, an line perpendicular to this line must have a slope of because. is the negative reciprocal of. So, the line through the point, 1 has the following equation. 1 Write in point-slope form. 1 Simplif. Write in slope-intercept form. The graphs of both equations are shown in Figure 1.6. Now tr Eercise 57(b). 7 (, 1) Figure 1.6 = = + 5 Eample 8 Graphs of Perpendicular Lines Use a graphing utilit to graph the lines and 1 in the same viewing window. The lines are supposed to be perpendicular (the have slopes of m 1 1 and m 1). Do the appear to be perpendicular on the displa? Solution If the viewing window is nonsquare, as in Figure 1.7, the two lines will not appear perpendicular. If, however, the viewing window is square, as in Figure 1.8, the lines will appear perpendicular. = + = Figure 1.7 Figure 1.8 Now tr Eercise 67. = + = Activities 1. Write an equation of the line that passes through the points, 1 and,. Answer: 5 7. Find the slope of the line that is perpendicular to the line 7 1. Answer: m 7. Write the equation of the vertical line that passes through the point,. Answer:

9 71_1.qd 1/7/6 1:1 AM Page Chapter 1 Functions and Their Graphs 1. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check 1. Match each equation with its form. (a) A B C (b) a (c) b (d) m b (e) 1 m 1 (i) vertical line (ii) slope-intercept form (iii) general form (iv) point-slope form (v) horizontal line In Eercises 5, fill in the blanks.. For a line, the ratio of the change in to the change in is called the of the line.. Two lines are if and onl if their slopes are equal.. Two lines are if and onl if their slopes are negative reciprocals of each other. 5. The prediction method is the method used to estimate a point on a line that does not lie between the given points. In Eercises 1 and, identif the line that has the indicated slope. 1. (a) m (b) m is undefined. (c) m. (a) m (b) m (c) m 1 Figure for 1 Figure for In Eercises and, sketch the lines through the point with the indicated slopes on the same set of coordinate aes. Point Slopes., (a) (b) 1 (c) (d)., 1 (a) (b) (c) 1 (d) Undefined In Eercises 5 and 6, estimate the slope of the line L 1 L 6 8 L 8 6 L L L In Eercises 7 1, find the slope of the line passing through the pair of points. Then use a graphing utilit to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.) 7., 1,, 8.,,, 9. 6, 1, 6, 1.,, 1, 6 In Eercises 11 18, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are man correct answers.) Point Slope 11., 1 m 1., m 1. 1, 5 m is undefined. 1., 1 m is undefined. 15., 9 m 16. 5, m 17. 7, m , 6 m 1 In Eercises 19, (a) find the slope and -intercept (if possible) of the equation of the line algebraicall, and (b) sketch the line b hand. Use a graphing utilit to verif our answers to parts (a) and (b)

10 71_1.qd 1/7/6 1:1 AM Page 97 Section 1. Lines in the Plane 97 In Eercises 5, find the general form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line b hand. Use a graphing utilit to verif our sketch, if possible. Point Slope ,, 6,, 5 6, 1 m m m 1 m m is undefined.. 1, m is undefined. 1. 1,.., 8.5 m m In Eercises, find the slope-intercept form of the equation of the line that passes through the points. Use a graphing utilit to graph the line.. 5, 1, 5, 5.,,, 5. 8, 1, 8, ,, 6, 7., 1, 1, , 1, 6, , 5, 9 1, 9 5.,,, ,.6,,.6. 8,.6,,. In Eercises and, find the slope-intercept form of the equation of the line shown... (1, ) ( 1, 7) 1, ) 5. Annual Salar A jeweler s salar was $8,5 in and $,9 in 6. The jeweler s salar follows a linear growth pattern. What will the jeweler s salar be in 8? 6. Annual Salar A librarian s salar was $5, in and $7,5 in 6. The librarian s salar follows a linear growth pattern. What will the librarian s salar be in 8? (, 1) ) In Eercises 7 5, determine the slope and -intercept of the linear equation. Then describe its graph In Eercises 51 and 5, use a graphing utilit to graph the equation using each of the suggested viewing windows. Describe the difference between the two graphs Xmin = -5 Xma = 1 Xscl = 1 Ymin = -1 Yma = 1 Yscl = Xmin = -5 Xma = 5 Xscl = 1 Ymin = -1 Yma = 1 Yscl = 1 In Eercises 5 56, determine whether the lines and passing through the pairs of points are parallel, perpendicular, or neither. 5. L 1 :, 1, 5, 9 5. L 1 :, 1, 1, 5 L :,,, 1 Xmin = - Xma = 1 Xscl = 1 Ymin = - Yma = 1 Yscl = 1 Xmin = -5 Xma = 1 Xscl = 1 Ymin = -8 Yma = 8 Yscl = 55. L 1 :, 6, 6, 56. L 1 :, 8,, L :, 1, 5, 7 L :, 5, 1, 1 In Eercises 57 6, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point Line 57., 1 58., 7 59., , , 6., 1 L 1 L : 1,, 5, 5 L

11 71_1.qd 1/7/6 1:1 AM Page Chapter 1 Functions and Their Graphs In Eercises 6 and 6, the lines are parallel. Find the slopeintercept form of the equation of line = ( 1, 1) ( 1, 1) 1 = (b) Find the equation of the line between the ears 1995 and. (c) Interpret the meaning of the slope of the equation from part (b) in the contet of the problem. (d) Use the equation from part (b) to estimate the earnings per share of stock in the ear 1. Do ou think this is an accurate estimation? Eplain. 7. Sales The graph shows the sales (in billions of dollars) for Goodear Tire for the ears 1995 through, where t 5 represents (Source: Goodear Tire) In Eercises 65 and 66, the lines are perpendicular. Find the slope-intercept form of the equation of line (, ) = + 6 (, 5) 6 1 = Sales (in billions of dollars) (7, 1.) (1, 15.1) (1, 1.) (5, 1.) (1, 1.9) (6, 1.1) (9, 1.9) (11, 1.1) (8, 1.6) Year (5 1995) (1, 18.) Graphical Analsis In Eercises 67 7, identif an relationships that eist among the lines, and then use a graphing utilit to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visuall correct. Use the slopes of the lines to verif our results. 67. (a) (b) (c) 68. (a) (b) (c) 69. (a) (b) 1 (c) 7. (a) 8 (b) 1 (c) 71. Earnings per Share The graph shows the earnings per share of stock for Circuit Cit for the ears 1995 through. (Source: Circuit Cit Stores, Inc.) Earnings per share (in dollars) (9, 1.6) (5,.91) (11,.9) (8,.7) (1,.8) (1,.1) (6,.69) (7,.57) (1,.) (1,.) Year (5 1995) (a) Use the slopes to determine the ears in which the earnings per share of stock showed the greatest increase and greatest decrease. 1 (a) Use the slopes to determine the ears in which the sales for Goodear Tire showed the greatest increase and the smallest increase. (b) Find the equation of the line between the ears 1995 and. (c) Interpret the meaning of the slope of the equation from part (b) in the contet of the problem. (d) Use the equation from part (b) to estimate the sales for Goodear Tire in the ear 1. Do ou think this is an accurate estimation? Eplain. 7. Height The rise to run ratio of the roof of a house determines the steepness of the roof. The rise to run ratio of the roof in the figure is to. Determine the maimum height in the attic of the house if the house is feet wide. ft attic height 7. Road Grade When driving down a mountain road, ou notice warning signs indicating that it is a 1% grade. This means that the slope of the road is 1. 1 Approimate the amount of horizontal change in our position if ou note from elevation markers that ou have descended feet verticall.

12 71_1.qd 1/7/6 1:1 AM Page 99 Section 1. Lines in the Plane 99 Rate of Change In Eercises 75 78, ou are given the dollar value of a product in 6 and the rate at which the value of the product is epected to change during the net 5 ears. Write a linear equation that gives the dollar value V of the product in terms of the ear t. (Let t 6 represent 6.) 6 Value Rate 75. $5 $15 increase per ear 76. $156 $.5 increase per ear 77. $, $ decrease per ear 78. $5, $56 decrease per ear Graphical Interpretation In Eercises 79 8, match the description with its graph. Determine the slope of each graph and how it is interpreted in the given contet. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) You are paing $1 per week to repa a $1 loan. 8. An emploee is paid $1.5 per hour plus $1.5 for each unit produced per hour. 81. A sales representative receives $ per da for food plus $.5 for each mile traveled. 8. A computer that was purchased for $6 depreciates $1 per ear. 8. Depreciation A school district purchases a high-volume printer, copier, and scanner for $5,. After 1 ears, the equipment will have to be replaced. Its value at that time is epected to be $. (a) Write a linear equation giving the value V of the equipment during the 1 ears it will be used. (b) Use a graphing utilit to graph the linear equation representing the depreciation of the equipment, and use the value or trace feature to complete the table. (b) (d) 15 6 t V (c) Verif our answers in part (b) algebraicall b using the equation ou found in part (a) Meteorolog Recall that water freezes at C F and boils at 1 C 1 F. (a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. (b) Use the result of part (a) to complete the table. C F Cost, Revenue, and Profit A contractor purchases a bulldozer for $6,5. The bulldozer requires an average ependiture of $5.5 per hour for fuel and maintenance, and the operator is paid $11.5 per hour. (a) Write a linear equation giving the total cost C of operating the bulldozer for t hours. (Include the purchase cost of the bulldozer.) (b) Assuming that customers are charged $7 per hour of bulldozer use, write an equation for the revenue R derived from t hours of use. (c) Use the profit formula P R C to write an equation for the profit derived from t hours of use. (d) Use the result of part (c) to find the break-even point (the number of hours the bulldozer must be used to ield a profit of dollars). 86. Rental Demand A real estate office handles an apartment comple with 5 units. When the rent per unit is $58 per month, all 5 units are occupied. However, when the rent is $65 per month, the average number of occupied units drops to 7. Assume that the relationship between the monthl rent p and the demand is linear. (a) Write the equation of the line giving the demand in terms of the rent p. (b) Use a graphing utilit to graph the demand equation and use the trace feature to estimate the number of units occupied when the rent is $655. Verif our answer algebraicall. (c) Use the demand equation to predict the number of units occupied when the rent is lowered to $595. Verif our answer graphicall. 87. Education In 1991, Penn State Universit had an enrollment of 75,9 students. B 5, the enrollment had increased to 8,1. (Source: Penn State Fact Book) (a) What was the average annual change in enrollment from 1991 to 5? (b) Use the average annual change in enrollment to estimate the enrollments in 198, 1997, and. (c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the contet of the problem.

13 71_1.qd 1/7/6 1:1 AM Page 1 1 Chapter 1 Functions and Their Graphs 88. Writing Using the results of Eercise 87, write a short paragraph discussing the concepts of slope and average rate of change. Snthesis True or False? In Eercises 89 and 9, determine whether the statement is true or false. Justif our answer. 89. The line through 8, and 1, and the line through, and 7, 7 are parallel. 9. If the points 1, and, 9 lie on the same line, then the point 1, 7 also lies on that line. Eploration In Eercises 91 9, use a graphing utilit to graph the equation of the line in the form a 1 1, b a, b. Use the graphs to make a conjecture about what a and b represent. Verif our conjecture In Eercises 95 98, use the results of Eercises 91 9 to write an equation of the line that passes through the points intercept:, 96. -intercept: 5, -intercept:, -intercept:, 97. -intercept: 1 6, 98. -intercept:, -intercept:, -intercept:, 5 Librar of Parent Functions In Eercises 99 and 1, determine which equation(s) ma be represented b the graph shown. (There ma be more than one correct answer.) Librar of Parent Functions In Eercises 11 and 1, determine which pair of equations ma be represented b the graphs shown (a) (b) (c) (d) Think About It Does ever line have both an -intercept and a -intercept? Eplain. 1. Think About It Can ever line be written in slope-intercept form? Eplain. 15. Think About It Does ever line have an infinite number of lines that are parallel to the given line? Eplain. 16. Think About It Does ever line have an infinite number of lines that are perpendicular to the given line? Eplain. Skills Review In Eercises 17 11, determine whether the epression is a polnomial. If it is, write the polnomial in standard form (a) (b) (c) (d) In Eercises , factor the trinomial (a) 1 (b) 1 (c) 1 (d) 1 (a) 5 (b) 5 (c) 5 (d) Make a Decision To work an etended application analzing the numbers of bachelor s degrees earned b women in the United States from 1985 to 5, visit this tetbook s Online Stud Center. (Data Source: U.S. Census Bureau) The Make a Decision eercise indicates a multipart eercise using large data sets. Go to this tetbook s Online Stud Center to view these eercises.