3.4 The Slope of a Line

Transcription

1 CHAPTER Graphs and Functions. The Slope of a Line S Find the Slope of a Line Given Two Points on the Line. Find the Slope of a Line Given the Equation of a Line. Interpret the Slope Intercept Form in an Application. Find the Slopes of Horizontal and Vertical Lines. Compare the Slopes of Parallel and Perpendicular Lines. Finding Slope Given Two Points You ma have noticed b now that different lines often tilt differentl. It is ver important in man fields to be able to measure and compare the tilt, or slope, of lines. For eample, a wheelchair ramp with a slope of means that the ramp rises foot for ever horizontal feet. A road with a slope or grade of % aor b means that the road rises feet for ever 00 horizontal feet. 00 ft ft ft 00 ft We measure the slope of a line as a ratio of vertical change to horizontal change. Slope is usuall designated b the letter m. Suppose that we want to measure the slope of the following line. 6 units units (, 7) (, ) (, ) Vertical change is units 6 7 Horizontal change is units (0, ) The vertical change between both pairs of points on the line is units per horizontal change of units. Then slope m = change in vertical change change in horizontal change = = Notice that slope is a rate of change between points. A slope of or means that between pairs of points on the line, the rate of change is a vertical change of units per horizontal change of unit. Consider the line in the bo on the net page, which passes through the points, and,. (The notation is read -sub-one. ) The vertical change, or rise, between these points is the difference of the -coordinates: -. The horizontal change, or run, between the points is the difference of the -coordinates: -.

2 Section. The Slope of a Line Slope of a Line Given a line passing through points, and,, the slope m of the line is (, ) (, ) horizonatal change, or run vertical change, or rise m = rise run = -, as long as -. CONCEPT CHECK In the definition of slope, we state that. Eplain wh. EXAMPLE Find the slope of the line containing the points (0, ) and (, ). Graph the line. Solution We use the slope formula. It does not matter which point we call, and which point we call,. We ll let, = 0, and, =,. m = - - = = = Upward m (, ) (0, ) Notice in this eample that the slope is positive and that the graph of the line containing (0, ) and (, ) moves upward that is, the -values increase as we go from left to right. Find the slope of the line containing the points (, 0) and -,. Graph the line. Helpful Hint The slope of a line is the same no matter which points of a line ou choose to calculate slope. The line in Eample also contains the point -, 0. Below, we calculate the slope of the line using (0, ) as, and -, 0 as,. Answer to Concept Check: So that the denominator is not 0 m = - = = - - = Same slope as found in Eample.

3 CHAPTER Graphs and Functions EXAMPLE Find the slope of the line containing the points, - and -,. Graph the line. Solution We use the slope formula and let, =, - and, = -,. m = - - = = 7-8 = m (, ) Downward (, ) Notice in this eample that the slope is negative and that the graph of the line through, - and -, moves downward that is, the -values decrease as we go from left to right. Find the slope of the line containing the points -, - and (, ). Graph the line. Helpful Hint When we are tring to find the slope of a line through two given points, it makes no difference which given point is called, and which is called,. Once an -coordinate is called, however, make sure its corresponding -coordinate is called. CONCEPT CHECK Find and correct the error in the following calculation of slope of the line containing the points, and, 7. m = = 8 - = - 8 Finding Slope Given an Equation As we have seen, the slope of a line is defined b two points on the line. Thus, if we know the equation of a line, we can find its slope. EXAMPLE Find the slope of the line whose equation is f = +. Solution Two points are needed on the line defined b f = + or = + to find its slope. We will use intercepts as our two points. If = 0, then If = 0, then = # = + = - = Subtract. Answer to Concept Check: m = = - 8 = = # -6 = Multipl b.

4 Section. The Slope of a Line Use the points (0, ) and -6, 0 to find the slope. Let, be (0, ) and, be -6, 0. Then m = - = = - -6 = Find the slope of the line whose equation is f = Analzing the results of Eample, ou ma notice a striking pattern: The slope of = + is, the same as the coefficient of. Also, the -intercept is (0, ), as epected. When a linear equation is written in the form f = m + b or = m + b, m is the slope of the line and (0, b) is its -intercept. The form = m + b is appropriatel called the slope intercept form. Slope Intercept Form When a linear equation in two variables is written in slope intercept form, slope -intercept is (0, b) T T = m + b then m is the slope of the line and (0, b) is the -intercept of the line. EXAMPLE Find the slope and the -intercept of the line - =. Solution We write the equation in slope intercept form b solving for. - = - = = = - Subtract from both sides. Divide both sides b -. Simplif. The coefficient of,, is the slope, and the -intercept is 0, -. Find the slope and the -intercept of the line - = 9. Interpreting Slope Intercept Form On the following page is the graph of one-da ticket prices at Disne World for the ears shown. Notice that the graph resembles the graph of a line. Recall that businesses often depend on equations that closel fit graphs like this one to model the data and to predict future trends. B the least squares method, the linear function f =. +.0 approimates the data shown, where is the number of ears since 000 and is the ticket price for that ear.

5 6 CHAPTER Graphs and Functions Price of -Da Adult Pass (in dollars) Ticket Prices at Disne World Years (0000) Source: The Walt Disne Compan Helpful Hint The notation means that the number 0 corresponds to the ear 000, corresponds to the ear 00, and so on. EXAMPLE Predicting Future Prices The adult one-da pass price for Disne World is given b =. +.0 where is the number of ears since 000. a. Use this equation to predict the ticket prices for 0. b. What does the slope of this equation mean? c. What does the -intercept of this equation mean? Solution: a. To predict the price of a pass in 0, we need to find when is. (Since the ear 000 corresponds to = 0, the ear 0 corresponds to the ear =.) =. +.0 =. +.0 Let = = 9.90 We predict that in the ear 0, the price of an adult one-da pass to Disne World will be about $9.90. b. The slope of =. +.0 is.. We can think of this number as rise., or run. This means that the ticket price increases on average b $. each ear. c. The -intercept of =. +.0 is.0. Notice that it corresponds to the point (0,.0) on the graph. c c Year price That means that at ear = 0, or 000, the ticket price was about $.0. For the period 980 through 00, the number of people age 8 or older living in the United States is given b the equation = 0,0 +,7,00, where is the number of ears since 980. (Source: Based on data and estimates from the U.S. Bureau of the Census)

6 Section. The Slope of a Line 7 a. Estimate the number of people age 8 or older living in the United States in 00. b. What does the slope of this equation mean? c. What does the -intercept of this equation mean? Finding Slopes of Horizontal and Vertical Lines Net we find the slopes of two special tpes of lines: vertical lines and horizontal lines. EXAMPLE 6 Find the slope of the line = -. Solution Recall that the graph of = - is a vertical line with -intercept -, 0. To find the slope, we find two ordered pair solutions of = -. Of course, solutions of = - must have an -value of -. We will let, = -, 0 and, = -,. Then m = = (, ) = (, 0) 0 6 Since is undefined, we sa that the slope of the vertical line = - is undefined. 0 6 Find the slope of the line =. EXAMPLE 7 Find the slope of the line =. Solution Recall that the graph of = is a horizontal line with -intercept 0,. To find the slope, we find two points on the line, such as 0, and,, and use these points to find the slope. m = = 0 = 0 (, ) (0, ) The slope of the horizontal line = is 0. 7 Find the slope of the line = -.

7 8 CHAPTER Graphs and Functions From the previous two eamples, we have the following generalization. The slope of an vertical line is undefined. The slope of an horizontal line is 0. Helpful Hint Slope of 0 and undefined slope are not the same. Vertical lines have undefined slope, whereas horizontal lines have slope of 0. The following four graphs summarize the overall appearance of lines with positive, negative, zero, or undefined slopes. Appearance of Lines with Given Slopes Increasing line, positive slope Decreasing line, negative slope Horizontal line, zero slope Vertical line, undefined slope The appearance of a line can give us further information about its slope. The graphs of = + and = + are shown to the right. Recall that the graph of = + has a slope of and that the graph of = + has a slope of. m q m q Notice that the line with the slope of is steeper than the line with the slope of. This is true in general for positive slopes. For a line with positive slope m, as m increases, the line becomes steeper. To see wh this is so, compare the slopes from above. means a vertical change of unit per horizontal change of units; or 0 means a vertical change of 0 units per horizontal change of units. For larger positive slopes, the vertical change is greater for the same horizontal change. Thus, larger positive slopes mean steeper lines. Comparing Slopes of Parallel and Perpendicular Lines Slopes of lines can help us determine whether lines are parallel. Parallel lines are distinct lines with the same steepness, so it follows that the have the same slope.

8 Section. The Slope of a Line 9 Parallel Lines Two nonvertical lines are parallel if the have the same slope and different -intercepts. Different -intercepts Same slope How do the slopes of perpendicular lines compare? (Two lines intersecting at right angles are called perpendicular lines.) Suppose that a line has a slope of a. If the line b is rotated 90, the rise and run are now switched, ecept that the run is now negative. This means that the new slope is - b. Notice that a a a b b # a - b a b = - b a 90 b a This is how we tell whether two lines are perpendicular. Perpendicular Lines Two nonvertical lines are perpendicular if the product of their slopes is -. b m a a m b In other words, two nonvertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. EXAMPLE 8 Are the following pairs of lines parallel, perpendicular, or neither? a. + 7 = b. - + = 6 + = = Solution Find the slope of each line b solving each equation for. a. + 7 = 6 + = 7 7 = - + = = = = = c a c a a0, 7 b a0, b The slopes of both lines are - 7. The -intercepts are different, so the lines are not the same. Therefore, the lines are parallel. (Their graphs are shown in the margin.)

9 60 CHAPTER Graphs and Functions b. - + = + 6 = = + 6 = - + = = = + = c a c a a0, b a0, 6 b The slopes are not the same and their product is not -. ca b # a - b = - 9 d Therefore, the lines are neither parallel nor perpendicular. (Their graphs are shown in the margin.) 8 Are the following pairs of lines parallel, perpendicular, or neither? a. - = b. - = + = = -6 CONCEPT CHECK What is different about the equations of two parallel lines? Graphing Calculator Eplorations Man graphing calculators have a TRACE feature. This feature allows ou to trace along a graph and see the corresponding - and -coordinates appear on the screen. Use this feature for the following eercises. Graph each function and then use the TRACE feature to complete each ordered pair solution. (Man times, the tracer will not show an eact - or -value asked for. In each case, trace as closel as ou can to the given - or -coordinate and approimate the other, unknown coordinate to one decimal place.). = = =., =? = -.8, =?. = = =?, = 7. =?, = -.. = =., =? =?, = 6 (There will be two answers here.) 6. = =., =? =?, = (There will be two answers here.) Answer to Concept Check: -intercepts are different

10 Section. The Slope of a Line 6 Vocabular, Readiness & Video Check Use the choices below to fill in each blank. Some choices ma be used more than once and some not at all. horizontal the same - -intercepts 0, b slope vertical different m -intercepts b, 0 slope intercept. The measure of the steepness or tilt of a line is called.. The slope of a line through two points is measured b the ratio of change to change.. If a linear equation is in the form = m + b, or f = m + b, the slope of the line is and the -intercept is.. The form = m + b or f = m + b is the form.. The slope of a line is The slope of a line is undefined. 7. Two non-vertical perpendicular lines have slopes whose product is. 8. Two non-vertical lines are parallel if the have slope and different. Martin-Ga Interactive Videos See Video. Watch the section lecture video and answer the following questions. 9. Based on Eamples and, complete the following statements. A positive slope means the line from left to right. A negative slope means the line from left to right. 0. From Eample, how do ou write an equation in slope intercept form? Once the equation is in slope intercept form, how do ou determine the slope?. Eample gives a linear equation that models a real-life application. The equation is rewritten in slope intercept form. How does this help us better understand how the equation relates to the application? What specific information is discovered?. In the lecture after Eample 6, different slopes are summarized. What s the difference between zero slope and undefined slope? What does no slope mean?. From the lecture before Eample 7, wh do the slope rules for parallel or perpendicular lines indicate nonvertical lines onl?. Eercise Set Find the slope of the line that goes through the given points. See Eamples and..,, 8,., 6, 7,.,,, 8., 9, 6,. -, 8,, 6., 7, -, 7. -, -6,, , -, -, , -, -, 0., -, -6,. -,,,.,,, 0. -,, -, -. -, -,, -. 0, 6, -, 0 6.,, 0, 7. -,, -, 8., -, -, -6 Decide whether a line with the given slope slants upward or downward from left to right or is horizontal or vertical. 9. m = m = -. m = 0. m is undefined

11 6 CHAPTER Graphs and Functions Two lines are graphed on each set of aes. Decide whether l or l has the greater slope. See the boed material on page 8... l l l l C D f = + 8. f = - 9. f = f = - - l Find the slope of each line. See Eamples 6 and 7. l l l. =. = -. = -. =. + = = l l Find the slope and the -intercept of each line. See Eamples and. 9. f = - 0. f = = = 0. - = = 6. f = 6. f = - Match each graph with its equation. See Eamples and. l l MIXED Find the slope and the -intercept of each line. See Eamples through f = f = = = =. - 7 =. =. = 7. f = 7 6. f = = = = = -7 Decide whether the lines are parallel, perpendicular, or neither. See Eample = + 6 = - 6. = + 8 = = = - = - 7 = - 6 A B 6. f = f = 7-6 g = = - = = + = g = = 7 - = = -0 + =

12 Section. The Slope of a Line 6 Use the points shown on the graphs to determine the slope of each line. See Eamples and Solve. See Eample. 79. The life epectanc for females born in the United States is given b the equation = , where is the number of ears after 90. (Source: U.S. National Center for Health Statistics) a. Find the life epectanc of an American female born in 980. b. Find and interpret the slope of the equation. c. Find and interpret the -intercept of the equation. 80. The average annual income of an American woman with a bachelor s degree is given b the equation = ,97, where is the number of ears after 00. (Source: Based on data from U.S. Bureau of the Census, ) a. Find the average income of an American woman with a bachelor s degree in 008. b. Find and interpret the slope of the equation. c. Find and interpret the -intercept of the equation. Find each slope. See Eamples and. 7. Find the pitch, or slope, of the roof shown. ft 8 ft 76. Upon takeoff, a Delta Airlines jet climbs to miles as it passes over miles of land below it. Find the slope of its climb. 8. One of the top 0 occupations in terms of job growth in the net few ears is epected to be phsician assistants. The number of people,, in thousands, emploed as phsician assistants in the United States can be estimated b the linear equation 9-0 = -70, where is the number of ears after 008. (Source: Based on projections from the U.S. Bureau of Labor Statistics, ) a. Find the slope and the -intercept of the linear equation. mi mi b. What does the slope mean in this contet? c. What does the -intercept mean in this contet? 77. Driving down Bald Mountain in Woming, Bob Dean finds that he descends 600 feet in elevation b the time he is. miles (horizontall) awa from the high point on the mountain road. Find the slope of his descent rounded to two decimal places mile = 80 feet. 78. Find the grade, or slope, of the road shown. 00 ft ft 8. One of the fastest growing occupations over the net few ears is epected to be network sstem and data communications analsts. The number of people,, in thousands, emploed as network sstem and data communications analsts in the United States can be estimated b the linear equation 78 - = -60, where is the number of ears after 008. (Source: Based on projections from the U.S. Bureau of Labor Statistics ) a. Find the slope and the -intercept of the linear equation. b. What does the slope mean in this contet? c. What does the -intercept mean in this contet?

13 6 CHAPTER Graphs and Functions 8. The number of U.S. admissions (in billions) to movie theaters can be estimated b the linear equation = , where is the number of ears after 00. (Source: Motion Picture Association of America) a. Use this equation to estimate the number of movie admissions in the United States in 007. b. Use this equation to predict in what ear the number of movie admissions in the United States will be below billion. (Hint: Let = and solve for.) c. Use this equation to estimate the number of movie admissions in the present ear. Do ou go to the movies? Do our friends? 8. The amount of restaurant sales (in billions of dollars) in the United States can be estimated b the linear equation =. +.6, where is the number of ears after 970. (Source: Based on data from the National Restaurant Association) a. Use this equation to approimate the amount of restaurant sales in 00. b. Use this equation to approimate the ear in which the amount of restaurant sales will eceed $700 billion. c. Use this equation to approimate the amount of restaurant sales in the current ear. Do ou go out to eat often? Do our friends? REVIEW AND PREVIEW Simplif and solve for. See Section = = -[ - -0] = = -8[ - -] CONCEPT EXTENSIONS Each slope calculation is incorrect. Find the error and correct the calculation. See the second Concept Check in this section , 6 and 7, - m = Find the slope of a line parallel to the line f = Find the slope of a line parallel to the line f =. 9. Find the slope of a line perpendicular to the line f = = -0 = , and -, 9 m = = - or , -0 and -, - m = = - -9 = , - and -6, -6 m = = 6 = 96. Find the slope of a line perpendicular to the line f =. 97. Find the slope of a line parallel to the line - = Find the slope of a line parallel to the line - + = Find the slope of a line perpendicular to the line - = Find the slope of a line perpendicular to the line - + = 0. Each line below has negative slope. l ( 8, 6) l 7 6 (, ) l (, ) ( 6, 0) (, ) (0, ) 0. Find the slope of each line. 0. Use the result of Eercise 0 to fill in the blank. For lines with negative slopes, the steeper line has the (greater/lesser) slope. The following graph shows the altitude of a seagull in flight over a time period of 0 seconds. Altitude (ards) 6 A 0 B 8 6 C 0 G 8 6 D E F Time (seconds) 0. Find the coordinates of point B. 0. Find the coordinates of point C. 0. Find the rate of change of altitude between points B and C. (Recall that the rate of change between points is the slope between points. This rate of change will be in ards per second.) 06. Find the rate of change of altitude (in ards per second) between points F and G. 07. Eplain how merel looking at a line can tell us whether its slope is negative, positive, undefined, or zero. 08. Eplain wh the graph of = b is a horizontal line. 09. Eplain whether two lines, both with positive slopes, can be perpendicular. 0. Eplain wh it is reasonable that nonvertical parallel lines have the same slope.

14 Section. Equations of Lines 6. Professional plumbers suggest that a sewer pipe should be sloped 0. inch for ever foot. Find the recommended slope for a sewer pipe. (Source: Rules of Thumb b Tom Parker, Houghton Mifflin Compan). a. On a single screen, graph = +, = +, and = +. Notice the change in slope for each graph. b. On a single screen, graph = - +, = - +, and = - +. Notice the change in slope for each graph. c. Determine whether the following statement is true or false for slope m of a given line. As 0 m 0 becomes greater, the line becomes steeper.. Support the result of Eercise 67 b graphing the pair of equations on a graphing calculator.. Support the result of Eercise 70 b graphing the pair of equations on a graphing calculator. (Hint: Use the window showing [-, ] on the -ais and [-0, 0] on the -ais.). Equations of Lines S Graph a Line Using Its Slope and -Intercept. Use the Slope Intercept Form to Write the Equation of a Line. Use the Point Slope Form to Write the Equation of a Line. Write Equations of Vertical and Horizontal Lines. Find Equations of Parallel and Perpendicular Lines. Graphing a Line Using Its Slope and -Intercept In the last section, we learned that the slope intercept form of a linear equation is = m + b. Recall that when an equation is written in this form, the slope of the line is the same as the coefficient m of. Also, the -intercept of the line is 0, b. For eample, the slope of the line defined b = + is and its -intercept is 0,. We ma also use the slope intercept form to graph a linear equation. EXAMPLE Graph = -. Solution Recall that the slope of the graph of = - is and the -intercept is 0, -. To graph the line, we first plot the -intercept 0, -. To find another point on the line, we recall that slope is rise run =. Another point ma then be plotted b starting at 0, -, rising unit up, and then running units to the right. We are now at the point, -. The graph is the line through these two points. m ~ 6 Run Rise (, ) (0, ) 6 Notice that the line does have a -intercept of 0, - and a slope of. Graph = +. EXAMPLE Graph + =. Solution First, we solve the equation for to write it in slope intercept form. In slope intercept form, the equation is = - +. Net we plot the -intercept 0,. To find another point on the line, we use the slope -, which can be written as rise run = -. We start at 0, and move down units since the numerator of the slope