Section 4.3 Features of a Line

Transcription

1 Section.3 Features of a Line Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif the - and -intercepts of a line. Plotting points in the --plane (.1) Calculate the slope of a line b counting Drawing a line in the --plane (.1) rise and run. Choosing values of and to find points Using the slope ratio to find points on a line. on a line (.) Identif the slope and -intercept of a line from an equation of the form = m + b. Graph a line of the form = m + b. INTRODUCTION In Section. we learned the technique for graphing a line from an equation. This is done b selecting either an -value and solving for the corresponding -value, or b first selecting a -value and solving for. In this section, we learn some of the features of a line and use them to graph lines using a non-algebraic method. - AND -INTERCEPTS Ever line that is neither horizontal nor vertical has two aial points called intercepts. The value at which a line crosses the -ais is called the -intercept. Likewise, the value at which a line crosses the -ais is called the -intercept. For eample, in the graph of 3 = -1, the - intercept is - and the -intercept is. When labeled as an ordered pair, the -intercept point is (-, 0) and the -intercept point is (0, ). Because the -intercept point and the -intercept point are aial points, it is required that at least one of the coordinates is 0. -intercept point (-, 0) intercept point = -1 (0, ) When graphing lines in Section., we often found the - or -intercept point. Eample of Section. has an equation in the form a + b = c. In the equation 3 + = -, we found both - and - intercept points: when we chose = 0, we found the -intercept, -, and the -intercept point, (0, -); when we chose = 0, we found the -intercept, - 3, and the -intercept point, (- 3, 0). Features of a Line Robert Prior, 010 page.3-1

2 Eample 3 of Section. has an equation in the form = m + b. In the equation = + 1, we chose = 0 and found the -intercept, 1, and the -intercept point, (0, 1). Notice that the -intercept, 1, is the same as the value of b (the constant) in the equation. Will this alwas be the case? Yes. To see wh this is so, we can replace with 0 in the general equation and find the corresponding -value: = m + b (, ) 0 = m + b = m(0) + b = 0 + b (0, b) = b For ever linear equation written in the form = m + b, b is the -intercept and (0, b) is the -intercept point. This is true regardless of the value of m. ~Instructor Insight~ This section focuses on the -intercept. The -intercept will be discussed in more detail in Section.. Eample 1: For each linear equation, identif the -intercept point. a) = + 5 b) = 3 5 c) = d) = Procedure: The -intercept is the constant term in the equation, and the -coordinate is 0. For part d), there is no visible constant term, so the constant is 0, as in = + 0. Answer: a) (0, 5) b) (0, -) c) (0, 1 ) d) (0, 0) Features of a Line Robert Prior, 010 page.3 -

3 YTI 1 For each linear equation, identif the -intercept point. Use Eample 1 as a guide. a) = + 3 b) = c) = 5 9 d) = -3 e) = 3 f) = Think about it 1 Does a vertical line have an -intercept. a -intercept, or neither. Eplain our answer. THE SLOPE RATIO Just like a wheelchair ramp, a mountain, or a staircase, ever non-vertical line has a slope to it. This slope can also be described as a slant or steepness. A wheelchair ramp generall has a ver shallow slant so that the wheelchair driver can easil navigate the ramp. Man wheelchair ramps are like a line in that the are straight and the steepness does not change. For eample, Mark built a wheelchair ramp for his grandfather. Mark made the ramp so that it had a 1 vertical foot slope ratio of 1 to 8, or 8 horizontal feet = 1 8. This means that the slant of the ramp should have a vertical rise of 1 foot for ever 8 horizontal feet of length. The ramp Mark made has a horizontal length of1 feet, and the vertical rise is feet: feet 1 feet = 1 8, the same slope ratio. 1 feet feet A mountain has steepness to it, as well, but it is unlike a line because the mountain will be steeper in some parts than in others. Often, a geographer will consider the average steepness, or slope, of a mountain, possibl from a point somewhere on the mountain to the top of the mountain (the summit). Features of a Line Robert Prior, 010 page.3-3

4 In the eample at right, Joann wants to know the average slope of the mountain from her camp at the 3,000 foot elevation level. Using her watch with GPS (global positioning sstem), she is able to calculate that the summit (the top most point on the mountain) is at 3,500 feet and the horizontal distance from her camp is 300 feet. This line represents the average slope. Camp elevation is 3,000 feet. Summit elevation is 3,500 feet. Note: The elevations mentioned in the eample above indicate elevations above sea level. So, the camp is at 3,000 feet above sea level and the summit is at 3,500 feet above sea level. Joann determines the slope ratio is 500 vertical feet 300 horizontal feet. Because can simplif b a factor of 100 to just 5 3, we can sa that the slope ratio for that part of the mountain is 5 3. For the sake of consistenc, the slope ratio of phsical objects is alwas defined as This is true in the eamples of the wheelchair ramp and of the mountain. vertical distance horizontal distance. YTI Identif the slope ratio of the diagram. Simplif the ratio to lowest terms. Use the discussion above as a guide. a) A ramp at a warehouse loading dock ends 8 feet above the drivewa. The start of the ramp is 30 feet from the loading dock, as shown. What is the slope ratio of the ramp? 30 feet 8 feet b) The elevation at the base of a mountain is,300 feet, and the summit is at,700 feet. The horizontal distance from the base to the summit is 900 feet. What is the (average) slope ratio of the mountain? Summit elevation is,700 feet 900 feet across Base elevation is,300 feet Features of a Line Robert Prior, 010 page.3 -

5 A staircase has a consistent slant to it as well. B itself it isn t a line, but we might imagine a line that supports the steps from underneath. A staircase step has a vertical riser and a horizontal tread. In the world of stair making, the height of the riser is called the rise and the length of the tread is called the run. Source: riser tread rise run For eample, in the staircase at right, the riser (rise) is inches and the tread (run) is 1 inches. The slope ratio of the line underneath is rise run = 1 = 3 7. in. 1 in. We can use the analog of the staircase to help us understand the slope of a line. The notion of rise and run is similar to the eercise of locating a new point, as we did in Section.1. In that eercise, we used directions of 1. up or down, and then. left or right to locate a new point in the --plane. Doing this eercise repeatedl is like creating stair steps from point to point, and the points that are created are collinear. Eample : Directions: Procedure: From the point (-8, -5), locate and label four new points in the --plane according to the directions. Then draw the unique line that passes through these five points. 1. Count up spaces, and. Count to the right spaces. 3. Plot and label the new point.. Repeat. Starting at (-8, -5), use the directions to locate a second point (the first new point). From this point, repeat the directions to locate a third point, and so on. Features of a Line Robert Prior, 010 page.3-5

6 right (8, 3) (, 1) 8 - (0, -1) up (-8, -5) (-, -3) - - (8, 3) (, 1) (0, -1) (-, -3) - (-8, -5) - YTI 3 From the given point, locate and label three new points in the --plane according to the directions. Then draw the unique line that passes through these four points. Use Eample as a guide. a) The given point is (, -). b) The given point is (7, 5). 1. Count up 3 spaces, and 1. Count down 1 spaces, and. Count to the left spaces.. Count to the left spaces. 3. Plot and label the new point. 3. Plot and label the new point.. Repeat.. Repeat ~Instructor Insight~ A somewhat rare instance of a phsical object with a negative slope is a backward cliff face. The slope is still calculated using distances, but it is assigned a negative value. THE SLOPE OF A LINE In the discussion on the slope ratio, it is mentioned that the slope ratio of phsical objects is defined as vertical distance horizontal distance. Because distance is alwas a positive measure, the slope ratio for phsical objects Features of a Line Robert Prior, 010 page.3 -

7 is alwas positive. This is due to the fact that a phsical object, such as a ramp, a staircase, or a mountain, can be looked at from different points of view, different orientations. vertical change (rise) rise For a line, the slope ratio is horizontal change (run), or simpl run, and either change can be positive or negative. This is because in the --plane, there is onl one point of view, one orientation. The -plane is centered at the origin and the direction of up is positive and down is negative. Similarl, the direction right is positive and left is negative. In general, we call the slope ratio of a line just the slope, and we give it the value m: The slope of line is defined as m = rise run. We can use the rise and run directions of the slope to locate points on the line. That is, given one point on the line, the slope indicates what up/down and left/right directions we are to count to locate other points. In the numerator of the slope, a positive rise indicates counting upward and a negative rise indicates counting downward. In the denominator a positive run indicates counting to the right and a negative run indicates counting to the left. To prepare for graphing lines using the slope, let s first eplore how to interpret a given rise run. Eample 3: Given the slope, m, describe the directions of the slope. a) m = 7 b) m = 5 - c) m = -3 d) m = -1 - Procedure: The numerator is the rise (up/down direction) and the denominator is the run (left/right direction). In part c), we can write it as a fraction, m = Slope interpretation: a) m = 7 = up right 7 b) m = 5 - = up 5 left c) m = -3 = -3 1 = down 3 right 1 d) m = -1 - = down 1 left Answer: Up and right 7 Up 5 and left Down 3 and right 1 Down 1 and left Features of a Line Robert Prior, 010 page.3-7

8 YTI Given the slope, m, describe the directions of the slope. Use Eample 3 as a guide. a) m = -9 b) m = 8 3 c) m = d) m = - -5 Let s put this understanding to use in drawing the graph of a line based on knowing one point on the line and the slope of the line. Eample : Draw the line that passes through the point (-, 5) and has slope m = -3. Procedure: m = -3 = down 3 right and has directions down 3 and right, starting from the point (-, 5). We should be able to locate three other points on the line within the --grid. Preparation: (-, -3) down 3 (-, ) (-, -3) (-, ) (0, -1) 8 right - - (, -) (0, -1) (, -) - - Note: Because it is difficult to show the step-b-step drawing of a graph on a page, the preparation graph is shown in Eample. However, onl the answer graph is epected to be shown in You Tr It 5. Features of a Line Robert Prior, 010 page.3-8

9 YTI 5 Draw the line that passes through the given point and has given slope. Use Eample as a guide. a) Given point: (-, 1); given slope m = 5. b) Given point: (-, ); given slope m = MORE ABOUT THE SLOPE OF A LINE Ever positive and negative slope can be interpreted in two was. To illustrate this, let s consider the slopes m = 3 5 and m = - 1. In a positive fraction, such as m = 3 5 have the same sign, either both positive or both negative. So,, the numerator and denominator m = 3 5 can be thought of as either 1. m = m = -3-5, up 3 and right 5; or, down 3 and left 5. In a negative fraction, such as m = - 1, the numerator and denominator have different signs, one is positive and the other is negative. So, m = - 1 can be thought of as either 1. m = -1 +, down 1 and right, or. m = +1 -., up 1 and left. Features of a Line Robert Prior, 010 page.3-9

10 Eample 5: Given the slope, m, describe two possible directions of the slope. a) m = b) m = Procedure: Consider the different was the numerator and denominator can be positive or negative without changing the value of the m. Slope interpretations: a) m = can be written as either m = -1 3 = down 1 right 3 or as m = 1-3 = up 1 left 3 Answer: Down 1 and right 3 Up 1 and left 3 b) m = can be written as either m = 1 = up right 1 or as m = - -1 = down left 1 Up and right 1 Down and left 1 YTI Given the slope, m, describe two possible directions of the slope. Use Eample 5 as a guide. a) m = 5 b) m = - 7 Consider a line that has slope m = 5 and passes through the points (-8, -), (-3, -), (, ), and (7, ). We can verif that these points are on the line b starting at (-8, -) and using the slope directions up and right 5 to locate the other points on the line. Or, we can verif that these points are on the line b starting at (7, ) and using the slope directions down and left 5 to locate the other points on the line. Features of a Line Robert Prior, 010 page.3-10

11 Start here (7, ) right 5 (, ) (-3, -) up - (-8, -) - Start here or (7, ) down (, ) (-3, -) - left 5 - (-8, -) - What this illustration shows is that we can actuall use both sets of directions together: Because m = 5 = up right 5 be written as m = - -5 = down left 5, we get the directions up and right 5. In addition, this slope can, giving us the directions down and left 5. This idea is especiall useful if the onl point we are given is in the middle region of the --grid, near to the origin. For eample, we can start at (, ) and use the two sets of directions, up and right 5 and down and left 5, to find more points on the line: right 5 (7, ) up (, ) down (-3, -) - Start here (-8, -) - left 5-1 up and right 5 to find (7, ). down and left 5 to find (-3, -) and (-8, -). Features of a Line Robert Prior, 010 page.3-11

12 Eample : Draw the graph of the line that has slope m = - 3 and passes through the point (-, 1). Procedure: We can think of the slope in two was: as m = -3 = down 3 right and as m = 3 - = up 3 left. Using m = down 3 right we get both (, -) and (, -5); using m = up 3 left we get (-, ). The Preparation: Start here left (-, ) up 3 (-, 1) down (, -) - (, -5) -right (-, ) (-, 1) (, -) - (, -5) - YTI 7 Draw the graph of the line that has the given slope and passes through the given point. Use Eample as a guide. a) Given slope m = 1 ; given point: (-, -3). b) Given slope m = -3; given point: (1, ) Features of a Line Robert Prior, 010 page.3-1

13 Think about it Match up the line (A, B, C, or D) with each given slope. 1. m = C D. 3.. m = - 1 m = 1 m = - 7 A B C D B A THE SLOPE-INTERCEPT FORM OF A LINE In the beginning of this section we were introduced to the -intercept and how it is easil identified in the equation = m + b: the -intercept is b, and the -intercept point is (0, b). The other constant value in the equation = m + b is m, the slope of the line. Wh is the coefficient of the slope of the line? Recall, from Section., that to find points on the line of, sa, = 3 + 1, we are encouraged to choose values of that are multiples of the denominator, 3, such as -, -3, 0, 3, and : = + 1 (, ) - = (-) + 1 (-, -3) = -3-3 = (-3) + 1 (-3, -1) = -1 = (0) + 1 = 1 = (3) + 1 = 3 = () + 1 = 5 (0, 1) (3, 3) (, 5) (-3, -1) (-, -3) - - (0, 1) (3, 3) (, 5) Because we have chosen -values that are multiples of 3, there is a horizontal change of 3 from point to point. Notice also that there is a vertical change of from point to point. This suggests a slope is up m = =, the coefficent of. right 3 3 Features of a Line Robert Prior, 010 page.3-13

14 This means that the equation = m + b contains both the slope, m, and the -intercept, b. For this reason, = m + b is called the slope-intercept form of a line. The Slope-Intercept Form of a Line In an linear equation in the form = m + b, the slope of the line is m and the -intercept point is (0, b). Eample 7: Given the equation of the line, identif its slope and -intercept point. a) = 1 3 b) = c) = - 1 d) = + 5 Procedure: The slope, m, is the coefficient of. The -intercept point is (0, b). a) m = 1 ; -intercept point is (0, -3) b) m = ; -intercept point is (0, ) c) m = - or - 1 ; -intercept point is (0, -1) d) m = 1 or 1 1 ; -intercept point is (0, 5) YTI 8 Given the equation of the line, identif its slope and -intercept point. Use Eample 7 as a guide. a) = b) = 3 c) = 5 1 d) = - 3 GRAPHING FROM THE SLOPE-INTERCEPT FORM If a linear equation is in slope-intercept form, then we can graph the line b first identifing the - intercept point and the slope. We can then plot the -intercept point and use the slope to find two or three more points on the line, as demonstrated in this net eample. Eample 8: Identif the slope and the -intercept point of the line. Then use them to graph the line. a) = 3 b) = Features of a Line Robert Prior, 010 page.3-1

15 Procedure: The slope, m, is the coefficient of, and the -intercept is the point (0, b). Plot the point (0, b) and use the slope to locate two more points on the line. a) The -intercept point is (0, -), b) The -intercept point is (0, -3), and the slope is 3. We can use both and the slope is - = - 1. We can use both 3 = up - right 3 and -3 = down left 3 to locate - 1 = down right 1 and -1 = up left 1 to locate other points on the line. other points on the line. (3, 0) (, ) (0, -) (-3, -) - - (-1, 5) (0, 3) (1, 1) (, -1) - - YTI 9 Identif the slope and the -intercept of the line. Then use them to graph the line. Use Eample 8 as a guide. a) = b) = Features of a Line Robert Prior, 010 page.3-15

16 Answers: You Tr It and Think About It YTI 1: a) (0, 3) b) (0, 1) c) (0, -9) d) (0, 0) e) (0, - 3 ) f) (0, 1.5) YTI : a) The slope ratio of the ramp is 15. b) The (average) slope ratio of the mountain is 8 3. YTI 3: a) (, -1) b) (3, ) YTI : a) down 9 and right b) up 8 and right 3 c) up and right 1 d) down and left 5 YTI 5: Some points shown ma be different from ours. a) b) (-1, ) (, 5) (-1, 3) (-, 1) (-1, ) (0, -) YTI : a) m = 5 1 = -5-1 b) m = -7 = 7 - Features of a Line Robert Prior, 010 page.3-1

17 YTI 7: Some points shown ma be different from ours. a) b) (-1, 5) (1, ) (-, -3) - (, -) - (-, -) (, -1) - - YTI 8: a) m = - 3 ; -intercept point is (0, ) b) m = 3; -intercept point is (0, 0) c) m = 5 ; -intercept point is (0, -1) d) m = -1; -intercept point is (0, -3) YTI 9: You ma have other points than what are shown here. (-3, 5) (0, ) (3, 3) (1, 3) (0, ) - - (-1, -5) - - Think About It: 1. Answers ma var. One possibilit is, A vertical line crosses onl the -ais, so it has an -intercept onl. (Also, a vertical line is parallel to the -ais, so it cannot have a -intercept.) Features of a Line Robert Prior, 010 page.3-17

18 Section.3 Eercises Think Again. 1. Does a vertical line have an -intercept. a -intercept, or neither. Eplain our answer. (Refer to Think About It 1). For an -intercept point, which coordinate must be 0? Eplain our answer. 3. Is it possible for a slope to be 0? Eplain our answer.. If a line crosses through the origin, what are the -intercept point and the -intercept point? Focus Eercises. Identif the slope ratio of the diagram. Simplif the ratio to lowest terms. 5. A ramp at a courthouse has a vertical rise of feet and a horizontal run of feet before it makes a turn. What is the slope ratio of this ramp?. To get from the third level to the fourth level at a baseball stadium, a ramp has a vertical rise of 15 feet and a horizontal run of 10 feet. What is the slope ratio of the ramp? 7. A mountain has its base at sea level (0 feet), and the summit is at 1,800 feet. The horizontal distance from the base to the summit is 800 feet. What is the (average) slope ratio of the mountain? 8. The elevation at a camp on a mountain is,00 feet, and the summit is at 5,00 feet. The horizontal distance from the base to the summit is 00 feet. What is the (average) slope ratio of the mountain? Features of a Line Robert Prior, 010 page.3-18

19 From the given point, locate and label three new points in the --plane according to the directions. Then draw the unique line that passes through these four points. 9. The given point is (-, ). 10. The given point is (, ). 1. Count down 3 spaces, and 1. Count down spaces, and. Count to the right 1 spaces.. Count to the left 3 spaces. 3. Plot and label the new point. 3. Plot and label the new point.. Repeat.. Repeat. 11. The given point is (-, -5). 1. The given point is (3, -). 1. Count up 3 spaces, and 1. Count up spaces, and. Count to the right spaces.. Count to the left 3 spaces. 3. Plot and label the new point. 3. Plot and label the new point.. Repeat.. Repeat. Draw the line that passes through the given point and has given slope. Find and label two other points on the line. 13. (0, -) and slope m = 1 1. (0, 5) and slope m = (, 0) and slope m = (-3, 0) and slope m = (3, -5) and slope m = (-, -) and slope m = (-, -1) and slope m = (-5, ) and slope m = (0, 1) and slope m = 5. (0, ) and slope m = (0, -) and slope m = 5. (0, -1) and slope m = (-1, ) and slope m = - 1. (, -3) and slope m = (1, ) and slope m = 7 8. (, -1) and slope m = - 5 Features of a Line Robert Prior, 010 page.3-19

20 Identif the slope and the -intercept of the line, and use them to graph the line. 9. = = = 3 3. = = = = = = = = = = - +. = = = = 3. = + 7. = = - THINK OUTSIDE THE BOX: For each: a) Plot the given point in the --plane. b) Use the given slope to locate the -intercept. c) Graph the line that passes through these points d) Write the equation of the line. 9. (, ); m = (8, 5); m = (-, -5); m = 5. (-3, 5); m = -3 Features of a Line Robert Prior, 010 page.3-0