The Slope of a Line. units corresponds to a horizontal change of. m y x y 2 y 1. x 1 x 2. Slope is not defined for vertical lines.

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1 0_0P0.qd //0 : PM Page 0 0 CHAPTER P Preparation for Calculus Section P. (, ) = (, ) = change in change in Figure P. Linear Models and Rates of Change Find the slope of a line passing through two points. Write the equation of a line with a given point and slope. Interpret slope as a ratio or as a rate in a real-life application. Sketch the graph of a linear equation in slope-intercept form. Write equations of lines that are parallel or perpendicular to a given line. The Slope of a Line The slope of a nonvertical line is a measure of the number of units the line rises (or falls) verticall for each unit of horizontal change from left to right. Consider the two points, and, on the line in Figure P.. As ou move from left to right along this line, a vertical change of Change in units corresponds to a horizontal change of Change in units. ( is the Greek uppercase letter delta, and the smbols and are read delta and delta. ) Definition of the Slope of a Line The slope m of the nonvertical line passing through, and, is m,. Slope is not defined for vertical lines. NOTE When using the formula for slope, note that. So, it does not matter in which order ou subtract as long as ou are consistent and both subtracted coordinates come from the same point. Figure P. shows four lines: one has a positive slope, one has a slope of zero, one has a negative slope, and one has an undefined slope. In general, the greater the absolute value of the slope of a line, the steeper the line is. For instance, in Figure P., the line with a slope of is steeper than the line with a slope of. (, 0) m = (, ) (, ) m = 0 (, ) (0, ) m = (, ) (, ) m is undefined. (, ) If m is positive, then the line rises from left to right. Figure P. If m is zero, then the line is horizontal. If m is negative, then the line falls from left to right. If m is undefined, then the line is vertical.

2 0_0P0.qd //0 : PM Page SECTION P. Linear Models and Rates of Change EXPLORATION Investigating Equations of Lines Use a graphing utilit to graph each of the linear equations. Which point is common to all seven lines? Which value in the equation determines the slope of each line? a. b. c. d. 0 e. f. g. Use our results to write an equation of a line passing through, with a slope of m. Equations of Lines An two points on a nonvertical line can be used to calculate its slope. This can be verified from the similar triangles shown in Figure P.. (Recall that the ratios of corresponding sides of similar triangles are equal.) ( *, *) (, ) ( *, *) (, ) m = * * = * * An two points on a nonvertical line can be used to determine its slope. Figure P. You can write an equation of a nonvertical line if ou know the slope of the line and the coordinates of one point on the line. Suppose the slope is m and the point is,. If, is an other point on the line, then m. This equation, involving the two variables and, can be rewritten in the form m, which is called the point-slope equation of a line. = -Slope Equation of a Line An equation of the line with slope m passing through the point, is given b m. = = (, ) The line with a slope of passing through the point, Figure P. EXAMPLE Finding an Equation of a Line Find an equation of the line that has a slope of and passes through the point,. Solution m (See Figure P..) -slope form Substitute for, for, and for m. Simplif. Solve for. NOTE Remember that onl nonvertical lines have a slope. Consequentl, vertical lines cannot be written in point-slope form. For instance, the equation of the vertical line passing through the point, is.

3 0_0P0.qd //0 : PM Page CHAPTER P Preparation for Calculus Ratios and Rates of Change The slope of a line can be interpreted as either a ratio or a rate. If the - and -aes have the same unit of measure, the slope has no units and is a ratio. If the - and -aes have different units of measure, the slope is a rate or rate of change. In our stud of calculus, ou will encounter applications involving both interpretations of slope. EXAMPLE Population Growth and Engineering Design Population (in millions) 0, Year Population of Kentuck in census ears Figure P. a. The population of Kentuck was,87,000 in 990 and,0,000 in 000. Over this 0-ear period, the average rate of change of the population was Rate of change If Kentuck s population continues to increase at this same rate for the net 0 ears, it will have a 00 population of,97,000 (see Figure P.). (Source: U.S. Census Bureau) b. In tournament water-ski jumping, the ramp rises to a height of feet on a raft that is feet long, as shown in Figure P.7. The slope of the ski ramp is the ratio of its height (the rise) to the length of its base (the run). Slope of ramp rise run change in population change in ears,0,000,87, ,00 people per ear. feet feet Rise is vertical change, run is horizontal change. 7 In this case, note that the slope is a ratio and has no units. ft ft Dimensions of a water-ski ramp Figure P.7 The rate of change found in Eample (a) is an average rate of change. An average rate of change is alwas calculated over an interval. In this case, the interval is 990, 000. In Chapter ou will stud another tpe of rate of change called an instantaneous rate of change.

4 0_0P0.qd //0 : PM Page SECTION P. Linear Models and Rates of Change Graphing Linear Models Man problems in analtic geometr can be classified in two basic categories: () Given a graph, what is its equation? and () Given an equation, what is its graph? The point-slope equation of a line can be used to solve problems in the first categor. However, this form is not especiall useful for solving problems in the second categor. The form that is better suited to sketching the graph of a line is the slopeintercept form of the equation of a line. The Slope-Intercept Equation of a Line The graph of the linear equation m b is a line having a slope of m and a -intercept at 0, b. EXAMPLE Sketching Lines in the Plane Sketch the graph of each equation. a. b. c. 0 Solution a. Because b, the -intercept is 0,. Because the slope is m, ou know that the line rises two units for each unit it moves to the right, as shown in Figure P.8(a). b. Because b, the -intercept is 0,. Because the slope is m 0, ou know that the line is horizontal, as shown in Figure P.8(b). c. Begin b writing the equation in slope-intercept form. 0 Write original equation. Isolate -term on the left. Slope-intercept form In this form, ou can see that the - intercept is 0, and the slope is m. This means that the line falls one unit for ever three units it moves to the right, as shown in Figure P.8(c). (0, ) = = + = (0, ) = = (0, ) = + = (a) m ; line rises Figure P.8 (b) m 0; line is horizontal (c) m ; line falls

5 0_0P0.qd //0 : PM Page CHAPTER P Preparation for Calculus Because the slope of a vertical line is not defined, its equation cannot be written in the slope-intercept form. However, the equation of an line can be written in the general form A B C 0 General form of the equation of a line where A and B are not both zero. For instance, the vertical line given b a can be represented b the general form a 0. Summar of Equations of Lines. General form: A B C 0,. Vertical line: a. Horizontal line: b. -slope form: m. Slope-intercept form: m b A, B 0 Parallel and Perpendicular Lines The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular, as shown in Figure P.9. Specificall, nonvertical lines with the same slope are parallel and nonvertical lines whose slopes are negative reciprocals are perpendicular. m = m m m m m m = m Parallel lines Figure P.9 Perpendicular lines STUDY TIP In mathematics, the phrase if and onl if is a wa of stating two implications in one statement. For instance, the first statement at the right could be rewritten as the following two implications. a. If two distinct nonvertical lines are parallel, then their slopes are equal. b. If two distinct nonvertical lines have equal slopes, then the are parallel. Parallel and Perpendicular Lines. Two distinct nonvertical lines are parallel if and onl if their slopes are equal that is, if and onl if m m.. Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other that is, if and onl if m m.

6 0_0P0.qd //0 : PM Page SECTION P. Linear Models and Rates of Change EXAMPLE Finding Parallel and Perpendicular Lines + = = (, ) = 7 Lines parallel and perpendicular to Figure P.0 Find the general forms of the equations of the lines that pass through the point, and are a. parallel to the line b. perpendicular to the line. (See Figure P.0.) Solution B writing the linear equation in slope-intercept form,, ou can see that the given line has a slope of m. a. The line through, that is parallel to the given line also has a slope of. m -slope form Substitute. Simplif. 7 0 General form Note the similarit to the original equation. b. Using the negative reciprocal of the slope of the given line, ou can determine that the slope of a line perpendicular to the given line is. So, the line through the point, that is perpendicular to the given line has the following equation. m -slope form Substitute. Simplif. 0 General form TECHNOLOGY PITFALL The slope of a line will appear distorted if ou use different tick-mark spacing on the - and -aes. For instance, the graphing calculator screens in Figures P.(a) and P.(b) both show the lines given b and. Because these lines have slopes that are negative reciprocals, the must be perpendicular. In Figure P.(a), however, the lines don t appear to be perpendicular because the tick-mark spacing on the -ais is not the same as that on the -ais. In Figure P.(b), the lines appear perpendicular because the tick-mark spacing on the -ais is the same as on the -ais. This tpe of viewing window is said to have a square setting (a) Tick-mark spacing on the -ais is not the same as tick-mark spacing on the -ais. Figure P. (b) Tick-mark spacing on the -ais is the same as tick-mark spacing on the -ais.

7 0_0P0.qd //0 : PM Page CHAPTER P Preparation for Calculus Eercises for Section P. In Eercises, estimate the slope of the line from its graph. To print an enlarged cop of the graph, go to the website In Eercises 7 and 8, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate aes. 7., (a) (b) (c) (d) Undefined 8., (a) (b) (c) (d) 0 In Eercises 9, plot the pair of points and find the slope of the line passing through them. 9.,,, 0.,,,.,,,.,,,..,,, In Eercises 8, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Slope Slopes 7 8,,, Slope., m 0., m undefined 7., 7 m 8., m See for worked-out solutions to odd-numbered eercises. 9. Conveor Design A moving conveor is built to rise meter for each meters of horizontal change. (a) Find the slope of the conveor. (b) Suppose the conveor runs between two floors in a factor. Find the length of the conveor if the vertical distance between floors is 0 feet. 0. Rate of Change Each of the following is the slope of a line representing dail revenue in terms of time in das. Use the slope to interpret an change in dail revenue for a one-da increase in time. (a) m 00 (b) m 00 (c) m 0. Modeling Data The table shows the populations (in millions) of the United States for The variable t represents the time in ears, with t corresponding to 99. (Source: U.S. Bureau of the Census) t (a) Plot the data b hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the ear when the population increased least rapidl.. Modeling Data The table shows the rate r (in miles per hour) that a vehicle is traveling after t seconds. t r (a) Plot the data b hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the interval when the vehicle s rate changed most rapidl. How did the rate change? In Eercises, find the slope and the -intercept (if possible) of the line In Eercises 7, find an equation of the line that passes through the point and has the indicated slope. Sketch the line Slope Slope 7. 0, m 8., m undefined 9. 0, 0 m 0. 0, m 0., m., m

8 0_0P0.qd //0 : PM Page 7 SECTION P. Linear Models and Rates of Change 7 In Eercises, find an equation of the line that passes through the points, and sketch the line.. 0, 0,,. 0, 0,,.,, 0,.,,, 7., 8,, 0 8.,,, 9.,,, 8 0.,,,.., 7, 0,. Find an equation of the vertical line with -intercept at.. Show that the line with intercepts a, 0 and 0, b has the following equation. a, b In Eercises 8, use the result of Eercise to write an equation of the line.. -intercept:, 0. -intercept: -intercept: 0, -intercept: 0, 7. on line:, 8. on line:, -intercept: a, 0 -intercept: a, 0 -intercept: 0, a -intercept: 0, a a 0 a 0 a 0, b 0 In Eercises 9, sketch a graph of the equation Square Setting In Eercises 7 and 8, use a graphing utilit to graph both lines in each viewing window. Compare the graphs. Do the lines appear perpendicular? Are the lines perpendicular? Eplain. 7., (a) Xmin = -0 Xma = 0 Xscl = Ymin = -0 Yma = 0 Yscl = 8., (a) Xmin = - Xma = Xscl = Ymin = - Yma = Yscl = (b) (b) 7 8,,, Xmin = - Xma = Xscl = Ymin = -0 Yma = 0 Yscl = Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =, 0 In Eercises 9, write an equation of the line through the point (a) parallel to the given line and (b) perpendicular to the given line. 9., 0.,., ,.,., 0 Rate of Change In Eercises 8, ou are given the dollar value of a product in 00 and the rate at which the value of the product is epected to change during the net ears. Write a linear equation that gives the dollar value V of the product in terms of the ear t. (Let t 0 represent 000.) 00 Value Line Rate. $0 $ increase per ear. $ $.0 increase per ear 7. $0,00 $000 decrease per ear 8. $,000 $00 decrease per ear Line 7 7 In Eercises 9 and 70, use a graphing utilit to graph the parabolas and find their points of intersection. Find an equation of the line through the points of intersection and graph the line in the same viewing window In Eercises 7 and 7, determine whether the points are collinear. (Three points are collinear if the lie on the same line.) 7.,,, 0,, 7. 0,, 7,,, Writing About Concepts In Eercises 7 7, find the coordinates of the point of intersection of the given segments. Eplain our reasoning. 7. (b, c) ( a, 0) (a, 0) Perpendicular bisectors ( a, 0) (a, 0) Altitudes (b, c) ( a, 0) (a, 0) Medians (b, c) 7. Show that the points of intersection in Eercises 7, 7, and 7 are collinear.

9 0_0P0.qd //0 : PM Page 8 8 CHAPTER P Preparation for Calculus 77. Temperature Conversion Find a linear equation that epresses the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. Use the fact that water freezes at 0 C ( F) and boils at 00 C ( F). Use the equation to convert 7 F to degrees Celsius. 78. Reimbursed Epenses A compan reimburses its sales representatives $0 per da for lodging and meals plus per mile driven. Write a linear equation giving the dail cost C to the compan in terms of, the number of miles driven. How much does it cost the compan if a sales representative drives 7 miles on a given da? 79. Career Choice An emploee has two options for positions in a large corporation. One position pas $.0 per hour plus an additional unit rate of $0.7 per unit produced. The other pas $9.0 per hour plus a unit rate of $.0. (a) Find linear equations for the hourl wages W in terms of, the number of units produced per hour, for each option. (b) Use a graphing utilit to graph the linear equations and find the point of intersection. (c) Interpret the meaning of the point of intersection of the graphs in part (b). How would ou use this information to select the correct option if the goal were to obtain the highest hourl wage? 80. Straight-Line Depreciation A small business purchases a piece of equipment for $87. After ears the equipment will be outdated, having no value. (a) Write a linear equation giving the value of the equipment in terms of the time, 0. (b) Find the value of the equipment when. (c) Estimate (to two-decimal-place accurac) the time when the value of the equipment is $ Apartment Rental A real estate office handles an apartment comple with 0 units. When the rent is $80 per month, all 0 units are occupied. However, when the rent is $, the average number of occupied units drops to 7. Assume that the relationship between the monthl rent p and the demand is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving the demand in terms of the rent p. (b) Linear etrapolation Use a graphing utilit to graph the demand equation and use the trace feature to predict the number of units occupied if the rent is raised to $. (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $9. Verif graphicall. 8. Modeling Data An instructor gives regular 0-point quizzes and 00-point eams in a mathematics course. Average scores for si students, given as ordered pairs, where is the average quiz score and is the average test score, are 8, 87, 0,, 9, 9,, 79,, 7, and, 8. (a) Use the regression capabilities of a graphing utilit to find the least squares regression line for the data. (b) Use a graphing utilit to plot the points and graph the regression line in the same viewing window. (c) Use the regression line to predict the average eam score for a student with an average quiz score of 7. (d) Interpret the meaning of the slope of the regression line. (e) The instructor adds points to the average test score of everone in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. 8. Tangent Line Find an equation of the line tangent to the circle 9 at the point,. 8. Tangent Line Find an equation of the line tangent to the circle at the point,. Distance In Eercises 8 90, find the distance between the point and line, or between the lines, using the formula for the distance between the point, and the line A B C 0. Distance A B C A B 8. : 0, 0 8. :, Line: 0 Line: :, 88. :, Line: 0 Line: 89. Line: 90. Line: Line: Line: 0 9. Show that the distance between the point, and the line A B C 0 is Distance A B C. A B 9. Write the distance d between the point, and the line m in terms of m. Use a graphing utilit to graph the equation. When is the distance 0? Eplain the result geometricall. 9. Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.) 9. Prove that the figure formed b connecting consecutive midpoints of the sides of an quadrilateral is a parallelogram. 9. Prove that if the points, and, lie on the same line as, and,, then. Assume and. 9. Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular. True or False? In Eercises 97 and 98, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 97. The lines represented b a b c and b a c are perpendicular. Assume a 0 and b It is possible for two lines with positive slopes to be perpendicular to each other.

1.2 Lines in the Plane

1.2 Lines in the Plane 71_1.qd 1/7/6 1:1 AM Page 88 88 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