1 . Slope of a Line (-) SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail and stud the concept of slope of a line. Slope Using Coordinates to Find Slope Parallel Lines Perpendicular Lines Applications of Slope Slope If a highwa has a 6% grade, then in 00 feet (measured horizontall) the road rises 6 feet (measured verticall). See Fig..0. The ratio of 6 to 00 is 6%. If a roof rises 9 feet in a horizontal distance (or run) of feet, then the roof has a 9 pitch. A roof with a 9 pitch is steeper than a roof with a 6 pitch. The grade of a road and the pitch of a roof are measurements of steepness. In each case the measurement is a ratio of rise (vertical change) to run (horizontal change). 6% GRADE 6 00 SLOW VEHICLES KEEP RIGHT ft run 9 ft rise helpful hint Since the amount of run is arbitrar, we can choose the run to be. In this case slope ri se rise. So the slope is the amount of change in for a change of in the -coordinate.this is wh rates like 0 miles per hour (mph), 8 hours per da, and two people per car are all slopes. FIGURE.0 9 pitch We measure the steepness of a line in the same wa that we measure steepness of a road or a roof. The slope of a line is the ratio of the change in -coordinate, or the rise, to the change in -coordinate, or the run, between two points on the line. Slope Slope change in -coordinate change in -coordinate Consider the line in Fig..(a) on the net page. In going from (0, ) to (, ), there is a change of in the -coordinate and a change of in the -coordinate, rise run
2 (-) Chapter Graphs and Functions in the Cartesian Coordinate Sstem (, ) + (0, ) + (, ) (0, ) (a) FIGURE. (b) or a run of and a rise of. So the slope is or. If we move from (, ) to (0, ) as in Fig..(b) the rise is and the run is. So the slope is or. If we start at either point and move to the other point, we get the same slope. E X A M P L E Finding the slope from a graph Find the slope of each line b going from point A to point B. a) b) c) A B A B 6 6 B A a) A is located at (0, ) and B at (, 0). In going from A to B, the change in is and the change in is. So slope. b) In going from A(, ) to B(6, ), we must rise and run. So slope. c) In going from A(0, 0) to B(6, ), we find that the rise is and the run is 6. So slope 6.
3 . Slope of a Line (-) Note that in Eample (c) we found the slope of the line of Eample (b) b using two different points. The slope is the ratio of the lengths of the two legs of a right triangle whose hpotenuse is on the line. See Fig... As long as one leg is vertical and the other leg is horizontal, all such triangles for a given line have the same shape: The are similar triangles. Because ratios of corresponding sides in similar triangles are equal, the slope has the same value no matter which two points of the line are used to find it. Hpotenuse Run Hpotenuse Rise Run Rise (, ) (, ) Run Rise (, ) FIGURE. FIGURE. Using Coordinates to Find Slope We can obtain the rise and run from a graph, or we can get them without a graph b subtracting the -coordinates to get the rise and the -coordinates to get the run for two points on the line. See Fig... Slope Using Coordinates The slope m of the line containing the points (, ) and (, ) is given b m, provided that 0. E X A M P L E stud tip Don t epect to understand a new topic the first time that ou see it. Learning mathematics takes time, patience, and repetition. Keep reading the tet, asking questions, and working problems. Someone once said, All mathematics is eas once ou understand it. Finding slope from coordinates Find the slope of each line. a) The line through (, ) and (6, ) b) The line through (, ) and (, ) c) The line through (6, ) and the origin a) Let (, ) (, ) and (, ) (6, ). The assignment of (, ) and (, )is arbitrar. m 6 b) Let (, ) (, ) and (, ) (, ): () m ()
4 (-) Chapter Graphs and Functions in the Cartesian Coordinate Sstem c) Let (, ) (0, 0) and (, ) (6, ): 0 m CAUTION Do not reverse the order of subtraction from numerator to denominator when finding the slope. If ou divide b, ou will get the wrong sign for the slope. E X A M P L E Slope for horizontal and vertical lines Find the slope of each line. a) b) helpful hint Think about what slope means to skiers. No one skis on cliffs or even refers to them as slopes. Zero slope (, ) (, ) a) Using (, ) and (, ) to find the slope of the horizontal line, we get m (, ) (, ) Small slope Larger slope b) Using (, ) and (, ) to find the slope of the vertical line, we get 0. Because the definition of slope using coordinates sas that must be nonzero, the slope is undefined for this line. Since the -coordinates are equal for an two points on a horizontal line, 0 and the slope is 0. Since the -coordinates are equal for an two points on a vertical line, 0 and the slope is undefined. Horizontal and Vertical Lines The slope of an horizontal line is 0. Slope is undefined for an vertical line. CAUTION Do not sa that a vertical line has no slope because no slope could be confused with 0 slope, the slope of a horizontal line. Undefined slope As ou move the tip of our pencil from left to right along a line with positive slope, the -coordinates are increasing. As ou move the tip of our pencil from
5 . Slope of a Line (-) left to right along a line with negative slope, the -coordinates are decreasing. See Fig... Increasing -coordinates Positive slope Negative slope Decreasing -coordinates Slope Slope FIGURE. FIGURE. Parallel Lines Consider the two lines shown in Fig... Each of these lines has a slope of, and these lines are parallel. In general, we have the following fact. Parallel Lines Nonvertical parallel lines have equal slopes. Of course, an two vertical lines are parallel, but we cannot sa that the have equal slopes because slope is not defined for vertical lines. E X A M P L E (, ) Slope Slope FIGURE.6 Parallel lines Line l goes through the origin and is parallel to the line through (, ) and (, ). Find the slope of line l. The line through (, ) and (, ) has slope m 8 () 6. Because line l is parallel to a line with slope, the slope of line l is also. Perpendicular Lines The lines shown in Fig..6 have slopes and. These two lines appear to be perpendicular to each other. It can be shown that a line is perpendicular to another line if its slope is the negative of the reciprocal of the slope of the other.
6 6 (-6) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Perpendicular Lines Two lines with slopes m and m are perpendicular if and onl if m. m Of course, an vertical line and an horizontal line are perpendicular, but we cannot give a relationship between their slopes because slope is undefined for vertical lines. E X A M P L E Perpendicular lines Line l contains the point (, 6) and is perpendicular to the line through (, ) and (, ). Find the slope of line l. The line through (, ) and (, ) has slope () m 7. 7 Because line l is perpendicular to a line with slope 7, the slope of line l is 7. E X A M P L E 6 D (, ) A (, ) C (, ) B (, ) FIGURE.7 Applications of Slope When a geometric figure is located in a coordinate sstem, we can use slope to determine whether it has an parallel or perpendicular sides. Using slope with geometric figures Determine whether (, ), (, ), (, ), and (, ) are the vertices of a rectangle. Figure.7 shows the quadrilateral determined b these points. If a parallelogram has at least one right angle, then it is a rectangle. Calculate the slope of each side. () m AB () m CD m BC 6 m AD 6 Because the opposite sides have the same slope, the are parallel, and the figure is a parallelogram. Because is the opposite of the reciprocal of, the intersecting sides are perpendicular. Therefore the figure is a rectangle. The slope of a line is a rate. The slope tells us how much the dependent variable changes for a change of in the independent variable. For eample, if the horizontal ais is hours and the vertical ais is miles, then the slope is miles per hour (mph).
7 . Slope of a Line (-7) 7 If the horizontal ais is das and the vertical ais is dollars, then the slope is dollars per da. E X A M P L E 7 Slope as a rate Worldwide carbon dioide (CO ) emissions have increased from billion tons in 970 to billion tons in 99 (World Resources Institute, CO emission (in billions of tons) Year FIGURE FOR EXAMPLE 7 stud tip Finding out what happened in class and attending class are not the same. Attend ever class and be attentive. Don t just take notes and let our mind wander. Use class time as a learning time. a) Find and interpret the slope of the line in the accompaning figure. b) Predict the amount of worldwide CO emissions in 00. a) Find the slope of the line through (970, ) and (99, ): m The slope of the line is 0. billion tons per ear. b) If the (CO ) emissions keep increasing at 0. billion tons per ear, then in 0 ears the level will go up 0(0.) or billion tons. So in 00 CO emissions will be 8 billion tons. WARM-UPS True or false? Eplain our answer.. Slope is a measurement of the steepness of a line. True. Slope is run divided b rise. False. The line through (, ) and (, ) has undefined slope. False. The line through (, 6) and (, ) has undefined slope. True. Slope cannot be negative. False 6. The slope of the line through (0, ) and (, 0) is. False 7. The line through (, ) and (, ) has slope. False 8. If a line contains points in quadrants I and III, then its slope is positive. True 9. Lines with slope and are perpendicular to each other. False 0. An two parallel lines have equal slopes. False
8 8 (-8) Chapter Graphs and Functions in the Cartesian Coordinate Sstem. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What does slope measure? Slope measures the steepness of a line.. What is the rise and what is the run? The rise is the change in -coordinates and run is the change in -coordinates.. Wh does a horizontal line have zero slope? A horizontal line has zero slope because it has no rise. Determine the slope of each line. See Eample Wh is slope undefined for vertical lines? Slope is undefined for vertical lines because the run is zero and division b zero is undefined.. What is the relationship between the slopes of perpendicular lines? If m and m are the slopes of perpendicular lines, then m. m 6. What is the relationship between the slopes of parallel lines? If m and m are the slopes of parallel lines, then m m. Undefined
9 . Slope of a Line (-9) 9 6. Find the slope of the line that contains each of the following pairs of points. See Eamples and. 7. (, 6), (, ) 8. (, ), (6, 0) 9. (, ), (, ) 0. (, ), (, ) 7. (, ), (, 7). (, ), (, 6). (, ), (0, 0). (0, 0), (, ). (0, ), (, 0) 6. (, 0), (0, 0) 0 7.,,, 8.,,, 6 9. (6, ), (7, 09) 0. (988, 06), (990, ) 9. (, 7), (, 7) 0. (, ), (9, ) 0. (, 6), (, 6) Undefined. (, ), (, 0) Undefined. (.,.9), (.7, 8.) (.7, 9.), (.6,.8) ,,, ,, 6, 0.90 In each case, make a sketch and find the slope of line l. See Eamples and. 9. Line l contains the point (, ) and is perpendicular to the line through (, ) and (, ) Line l goes through (, ) and is perpendicular to the line through (, 6) and (, ). 7. Line l goes through (, ) and is parallel to the line through (, ) and (, ). 7. Line l goes through the origin and is parallel to the line through (, ) and (, ). 7. Line l is perpendicular to a line with slope. Both lines contain the origin.. Line l is perpendicular to a line with slope. Both lines contain the origin. Solve each geometric figure problem. See Eample 6.. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Use slope to determine whether the points (6, ), (, ), (0, ), and (, ) are the vertices of a parallelogram. Yes 6. Use slope to determine whether the points (7, 0), (, 6), (, ), and (6, ) are the vertices of a parallelogram. See Eercise. No 7. A trapezoid is a quadrilateral with one pair of parallel sides. Use slope to determine whether the points (, ), (, ), (, 6), and (6, ) are the vertices of a trapezoid. No 8. A parallelogram with at least one right angle is a rectangle. Determine whether the points (, ), (, ), (0, 6), and (, 0) are the vertices of a rectangle. Yes 9. If a triangle has one right angle, then it is a right triangle. Use slope to determine whether the points (, ), (, 6), and (0, 0) are the vertices of a right triangle. No 0. Use slope to determine whether the points (0, ), (, ), and (, ) are the vertices of a right triangle. See Eercise 9. Yes Solve each problem. See Eample 7.. Pricing the Crown Victoria. The list price of a new Ford Crown Victoria four-door sedan was $0, in 99 and $, in 998 (Edmund s New Car Prices, a) Find the slope of the line shown in the figure. 0 b) Use the graph to predict the price in 00. $,00 c) Use the slope to predict the price of a new Crown Victoria in 00. $,6 List price (in thousands of dollars) 0 (99, 0,) (998,,) Year FIGURE FOR EXERCISE. Depreciating Monte Carlo. In 998 the average retail price of a one-ear-old Chevrolet Monte Carlo was $,9, whereas the average retail price of a -ear-old Monte Carlo was $,09 (Edmund s Used Car Prices).
10 0 (-0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Selling price (in thousands of dollars) 0 a) Use the graph on the net page to estimate the average retail price of a -ear-old car in 998. $,000 b) Find the slope of the line shown in the figure. 0 c) Use the slope to predict the price of a -ear-old car. $, (,,9) (,,09) Age (in ears) FIGURE FOR EXERCISE MISCELLANEOUS. The points (, ) and (,7) are on the line that passes through (, ) and has slope. Find the missing coordinates of the points. (, ), (0, 7). If a line passes through (, ) and has slope, then what is the value of on this line when 8,, and?, 6, 6. Find k so that the line through (, k) and (, ) has slope. 6. Find k so that the line through (k,)and(, 0) has slope. or 7. What is the slope of a line that is perpendicular to a line with slope 0.7? What is the slope of a line that is perpendicular to the line through (.7,.6) and (.8,.6)?.76 GETTING MORE INVOLVED 9. Writing. What is the difference between zero slope and undefined slope? A horizontal line has a zero slope and a vertical line has undefined slope. 60. Writing. Is it possible for a line to be in onl one quadrant? Two quadrants? Write a rule for determining whether a line has positive, negative, zero, or undefined slope from knowing in which quadrants the line is found. Ever line goes through at least two quadrants. A nonhorizontal, nonvertical line that misses quadrant II or IV or both has a positive slope. A nonhorizontal, nonvertical line that misses quadrant I or III or both has a negative slope. 6. Eploration. A rhombus is a quadrilateral with four equal sides. Draw a rhombus with vertices (, ), (0, ), (, ), and (, ). Find the slopes of the diagonals of the rhombus. What can ou conclude about the diagonals of this rhombus?,, perpendicular 6. Eploration. Draw a square with vertices (, ), (, ), (, ), and (, ). Find the slopes of the diagonals of the square. What can ou conclude about the diagonals of this square?,, perpendicular GRAPHING CALCULATOR EXERCISES 6. Graph,,, and together in the standard viewing window. These equations are all of the form m. What effect does increasing m have on the graph of the equation? What are the slopes of these four lines? Increasing m makes the graph increase faster. The slopes of these lines are,,, and. 6. Graph,,, and together in the standard viewing window. These equations are all of the form m. What effect does decreasing m have on the graph of the equation? What are the slopes of these four lines? Decreasing m makes the graph decrease faster. The slopes of these lines are,,, and. In this section Point-Slope Form Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Linear Functions. THREE FORMS FOR THE EQUATION OF A LINE In Section. ou learned how to graph a straight line corresponding to a linear equation. The line contains all of the points that satisf the equation. In this section we start with a line or a description of a line and write an equation corresponding to the line. Point-Slope Form Figure.8 shows the line that has slope and contains the point (, ). In Section. ou learned that the slope is the same no matter which two points of the line
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