Characteristics of Linear Relations
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1 HW Mark: RE-Submit Characteristics of Linear Relations This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher has important instructions for you to write down below. P a g e 1 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
2 Characteristics of Linear Relations Relations SPECIFIC OUTCOMES Demonstrate an understanding of slope with respect to: rise and run line segments and lines rate of change parallel lines perpendicular lines. TOPICS 3.1 Determine the slope of a line segment by measuring or calculating the rise and run. 3.2 Classify lines in a given set as having positive or negative slopes. 3.3 Explain the meaning of the slope of a horizontal or vertical line. 3.4 Explain why the slope of a line can be determined by using any two points on that line. 3.5 Explain, using examples, slope as a rate of change. 3.6 Draw a line, given its slope and a point on the line. 3.7 Determine another point on a line, given the slope and a point on the line. 3.8 Generalize and apply a rule for determining whether two lines are parallel or perpendicular. 3.9 Solve a contextual problem involving slope. REVIEW Note or Example Determine the characteristics of the graphs of linear relations, including the: intercepts slope domain range. 5.1 Determine the intercepts of the graph of a linear relation, and state the intercepts as values or ordered pairs. 5.2 Determine the slope of the graph of a linear relation. [C] Communication [PS] Problem Solving, [CN] Connections [R] Reasoning, [ME] Mental Mathematics [T] Technology, and Estimation, [V] Visualization P a g e 2 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
3 Characteristics of Linear Relations Key Terms Term Definition Example Line Line segment Linear relation Slope Positive slope Negative slope Zero slope Undefined slope Intercepts Parallel lines Parallel slopes Perpendicular lines Perpendicular slopes Midpoint formula Distance formula Parallelogram P a g e 3 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
4 Linear Relations: A relationship between two quantities that when graphed will produce a straight line. One quantity increases or decreases at a constant rate with respect to another. Eg. LINE SEGMENT: A part of a line that has two endpoints and includes all the points between the endpoints. 1. Using a dashed or coloured line, graph the relation represented by the equation. 2. Using a solid or different coloured line graph the same relation if the domain is. The solid section you just plotted is a line segment, a section of the dashed line. 3. What are the endpoints of the line segment? 4. What are the endpoints of the dashed line? 5. What are 5 properties you could use to describe the line segment above? 6. Which of these properties are also true for the dashed line above? P a g e 4 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
5 7. Challenge Question: Find the slope (rate of change) of the line below. 8. Challenge Question: Find the slope of the line segment with end points at A(-4,0) and B(0,3). P a g e 5 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
6 Slope of a Line (or Line Segment): (Rate of Change) Consider the line segment below. 9. What is the vertical change (rise) between the endpoints? 10. What is the horizontal change between the two endpoints? 11. What is the ratio of rise to run as a fraction? 12. How fast does the relationship change in the vertical direction when compared to the horizontal direction? Your notes here P a g e 6 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
7 Slope of a Line (or Line Segment) Slope is the measure of the steepness of a line. It is represented with the symbol (m). Slope also describes the direction of the line. The slope is found by dividing the vertical change (the rise or fall) by the horizontal change (the run). Be Careful with Negatives! Positive Slope Negative Slope Zero Slope Undefined Slope Eg. Eg. Eg. Eg. Rises from left to right. Falls from left to right. Rise is 0. 0 divided by any run will still = 0. Think the run is 0. Division by 0 is undefined. 13. Describe, in your own words, how you find the slope of a line segment. 14. How does a line segment differ from a line? P a g e 7 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
8 Find the rise, the run and the slope for the following lines by counting units. In most cases, you will need to pick two points on the line to use. 15. rise= 16. rise= 17. rise= 18. rise= run= run= run= run= slope= slope= slope= slope= 19. rise= 20. rise= 21. rise= 22. rise= run= run= run= run= slope= slope= slope= slope= Use the formula to find the slopes of line segments with the following endpoints. 23. and 24. and 25. and P a g e 8 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
9 Use the formula to find the slopes of line segments with the following endpoints. 26. and 28. and 30. and 27. Find the coordinates of another point on this line. 29. Find the coordinates of another point on this line. 31. Find the coordinates of another point on this line. 32. The slope of a line is -2. If the line passes through (t, -1) and (-4,9), find the value of t. 33. The slope of a line is. If the line passes through (5,2) and (b,-4), find the value of b. Given a point on the line and the slope, sketch the graph of the line P a g e 9 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
10 Given a point on the line and the slope, sketch the graph of the line A fallen tree leans against a vertical cliff. The tree was 15 m from the cliff and now rests against the cliff 25 m from the ground. Find the positive slope of the fallen tree. 41. A section of roller coaster falls 52 m in a horizontal distance of 4 m. Find the slope of this section of track? Slope is a measure of Rate of Change for a relation. That is, how fast one quantity increases or decreases in respect to another. 42. The cost for 8 students to go to the movies is $80. What is the cost per student, or rate? 44. To fill my gas tank that holds 70 litres, I paid $ What is the rate for gasoline per litre (in cents to the nearest tenth)? 45. Tspray drove 735 kilometres in 7 hours. Find his rate of travel per hour. 43. Write two ordered pairs for this relation. 46. What name is given to this quantity? P a g e 10 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
11 Slope is a measure of Rate of Change for a relation. That is, how fast one quantity increases or decreases in respect to another. 47. A round of golf for a group of hackers consists of the green fee and the club rental fee. Clubs are rented on a fee per club basis. Jack pays $72.25 for his green fee and 3 clubs, and Jill pays $95 for her green fee and 10 clubs. What is the rate to rent one club? 48. Pro-lectric charges their customers a fixed cost plus an hourly rate. To work in my basement, they charged me $210 for 5 hours work. To complete my upstairs renovations they charged me $720 for 22 hours work. What is the hourly rate? What is the green fee? What is the fixed cost? 49. Plot the relation above. 50. Plot the relation above. COST COST Number of clubs Hours worked P a g e 11 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
12 51. Below is a scale drawing of a bridge support. Perform the necessary measurements to determine the slope of the indicated beam. 52. Below is a scale diagram of a section of road between Sidney and Victoria. Measure and calculate the slope of the road. 53. A fishing boat moving at 12 knots is shown below. Calculate the slope of the line in the water behind the boat. 54. Terraced landscapes are used by farmers to create useable space from seemingly unusable geography. Calculate the slope of the hill that has been terraced to support crops. 55. The pitch of a roof is a measure of its steepness. Calculate the height of the roof truss below if its total span is 20 feet and the pitch (slope)is 6/ Mr. J is building a hide-away cabin with a roof that has a pitch of 9/12. T-spray is also building a hut but his roof is one-third as steep. If both roofs have the same total height, how many times wider is T-spray s roof? span P a g e 12 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
13 Since slope compares two quantities, it is a measure of rate of change. For each of the following scenarios, what rate does the slope represent? Rate: Rate: What are the units of the slope? What are the units of the slope? 59. Rate: What are the units of the slope? P a g e 13 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
14 60. Challenge # 5 Determine if AB is parallel to CD given the following points: A(1,2), B(5,4), C(0,-2), D(6,1). 61. What can you say about the slopes of parallel line segments? P a g e 14 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
15 Slopes of Parallel Lines (or segments) Recall two lines are parallel if they do not ever intersect. Parallel lines have equal slopes. Any two horizontal lines are parallel. Any two vertical lines are parallel. To determine if line segments are parallel, calculate their slopes. Eg.1. Determine if AB is parallel to CD. A(1,2), B(5,4), C(0,-2), D(6,1). Slope of AB: Slope of CD: SAME SLOPES PARALLEL Eg.2. The following are slopes of two lines. Find the value of k so that the two lines are parallel. and Since the lines are parallel, slopes must be equal. Cross Multiply: P a g e 15 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
16 Determine if the following pairs of line segments are parallel. 62. A(-2,-1), B(1,5) and 63. E(-3, 0), F(1, 5) and C(2, -1), D(4,3) G(0, -6), H(2, -1) 64. I(-4,0), J(8, 2) and K(2, 8), L(-2, 4) The following are slopes of two lines. Find the value of k so that the two lines are parallel. 65. and 66. and 67. and 68. The points T(2, -5), U(-2, 1), and V(3, -1) are given. Determine the coordinates of point W so that TUVW is a parallelogram. 69. The points A(6,3), B(2,9), and C(2,3) are given. Determine the coordinates of point D so that CD is parallel to AB and D is on the y-axis. P a g e 16 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
17 Slopes of Perpendicular Line Segments. The slopes of perpendicular lines are negative reciprocals. The product of perpendicular slopes is Plot the right triangle with vertices: A(2,2), B(5,7), and C(10,4). 71. Find the slope of AB. m = 72. Find the slope of BC. m = These segments form the right angle in the triangle. 73. What do you notice about the slopes of the two segments. 74. Multiply the two slopes. What is the result? 75. Is the triangle with vertices X(-9,-1), Y(-7,7), Z(3,-4) a right triangle? P a g e 17 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
18 Perpendicular Lines will have slopes that are NEGATIVE RECIPROCALS. Examples of perpendicular slopes are:. Examples of perpendicular slopes are:. Perpendicular slopes will have a product of. Look at the example above Determine the slope of a line segment perpendicular to a segment with each given slope The following are slopes of two lines. Find the value of k so that the two lines are perpendicular. 79. and 80. and 81. and Graph each pair of line segments. Determine if they are perpendicular or not. 82. A(0,0), B(6,4) and 83. G(2,10), H(-7,-2) and C(7,3), D(-11,1) J(7,0), K(-5,9) P a g e 18 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
19 Intercepts Non-vertical and non-horizontal lines are called oblique lines. Oblique lines will cross both the x-axis and the axis. These points are called the x-intercept and the y-intercept. Line crosses x-axis here. Line crosses y-axis here. 84. Challenge Question: Find the intercepts for the line. 85. Challenge Question: Find the intercepts for the line. P a g e 19 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
20 Finding the Intercepts from a graph. The location where a line passes through the x-axis is called the x-intercept. This point will have the coordinates The location where a line passes through the y-axis is called the y-intercept. This point will have the coordinates Consider: This line has an x-intercept at (8, 0). And a y-intercept at (0, 4). You may see this written as: x-intercept is 8. y-intercept is 4. Find the x- and y-intercepts from the graph below x-intercept: y-intercept: x-intercept: y-intercept: P a g e 20 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
21 Finding the Intercepts from an equation. The x-intercept will have coordinates (x, 0). This means we can substitute 0 in for y and solve to find the x- intercept. The y-intercept will have coordinates (0, y). Eg. Find the x-intercept for Find the y-intercept: Calculate the x- and y-intercepts P a g e 21 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
22 Using and Interpreting Intercepts 94. Find the intercepts and graph the line. 95. Find the intercepts and graph the line. 96. Based on the equation for the linear relation, when do you think it is most appropriate to graph the relation using intercepts? 97. The cost of a new pair of shoes at ShoeInc is reduced at a constant rate. The graph below shows the profit ShoeInc makes on each sale. In what month does ShoeInc break even on these shoes? 98. Use the graph below to plot the fuel consumed on Sandy s last road trip. She started out with 72 litres of fuel and drove for 2 hours. At that point she had 54 litres left. After driving another 1.5 hours she had 40.5 litres remaining. At this rate, when will she run out of fuel? P a g e 22 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
23 Mixed Practice: 99. A triangle has vertices A(-2,3), B(8,-2), and C(4,6). Determine whether it is a right triangle P (5,4) and Q(1,-2) are points on a line. Find the coordinates of a point, R, so that PR is perpendicular to PQ Find the value of k so that the two slopes are perpendicular. and 102. Two vertices of an isosceles triangle are A(-5,4) and B(3,8).The third vertex is on the x-axis. What are the possible coordinates of the third vertex, C? P a g e 23 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
24 Additional Material Distance between points (length of line segments) 103. Challenge Question: How can you find the EXACT distance between and? EXACT length means find the answer as a radical or fraction if it cannot be expressed as a whole number Challenge Question: How can you find the EXACT distance between and? P a g e 24 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
25 Finding the Distance Between Points The distance between two points can be found using The Pythagorean Theorem Plot points A and B. Connect the endpoints of segment AB Draw a vertical line down from B. Draw a horizontal line right from A. Notice that a right triangle is formed. The distance between A and B is the hypotenuse of the right triangle. The distance between A and B is 5 units. From this method we develop the formula for the distance between two points: Eg.1. Consider points A and B. from above. To use the distance formula, we substitute values from the ordered pairs into the formula. Put both x-coordinates into the first bracket. Put both y-coordinates into the second bracket (same order). Eg.2. Use the distance formula to find the lenth of the line segment connecting point C and D. Put both x-coordinates into the first bracket. Put both y-coordinates into the second bracket (same order as above) Can you see how the formula above is derived from the Pythagorean Theorem? (some of you will often find it easier to draw a right triangle and use the Pythagorean Th.) Explain P a g e 25 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
26 Find the distance between the pairs of points and 109. and 110. and 111. and 112. and 113. and 114. Find the radius of a circle with the centre at (-3, 4) and a point on the circumference at (-4, -6) Find the length of AB Find the length of CD On the grid draw the rectangle with vertices at A, B, C, D A Coast Guard patrol boat is located 5 km east and 8 km north of the entrance to St. John s harbour. A tanker is 9 km east and 6 km south of the entrance. Find the distance between the two ships to the nearest tenth of a km. Find the side lengths and the perimeter. Leave answers in simplest radical form if necessary. P a g e 26 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
27 119. Challenge Question: Draw a line connecting the two points and. What are the coordinates of the midpoint of that line segment? 120. Challenge Question: Given that the midpoint of a line segment is M(-1,3) and one endpoint is E(-4,5). Find the coordinates of the other endpoint P a g e 27 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
28 Finding the Midpoint of A Line Segment The midpoint of a line segment can be found by plotting the points on a grid. STEP 1: Plot points A and B. STEP 2: Draw a vertical line down from B. STEP 3: Draw a horizontal line right from A. STEP 4: Find the middle of each line you just drew. STEP 5: Find the intersection of the two middles. This will be the midpoint. Midpoint is at Using the Midpoint Formula: Notice that you are finding the average x value and average y value. Eg.1. Find the midpoint of the line connecting points A and B. Midpoint is at (3,2) Eg.2. The midpoint of a line segment is M(-1,3) and one endpoint is E(-4,5). Find the coordinates of the other endpoint. Using the formula. The other endpoint is at (2,1) Graphically. Plot the endpoint and the midpoint. Count grid lines to locate the other endpoint. P a g e 28 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
29 Find the midpoint of each line segment from the given endpoints and 122. and 123. and 124. and 125. and 126. and 127. A diameter of a circle has endpoints at (-7,-4) and (-1,10). What are the coordinates of the centre of the circle? 128. One endpoint is at (6,10) and the midpoint is at (0,0). Find the other endpoint One endpoint is at (4,0) and the midpoint is at (7,2). Find the other endpoint The centre of a circle has coordinates (-1,-3). One endpoint of a diameter is at (-3,0). What are the coordinates of the other endpoint? P a g e 29 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
30 The endpoints of a diameter of a circle are (-3,4) and (6,0) Find the centre of the circle What is the length of the radius? Answer to two decimal places. Line segment JK has endpoints at and What is the slope of a line segment perpendicular to JK Calculate the exact length of segment JK Find the value of q if is the midpoint of JK Give the coordinates of a point such that JL has an undefined slope. P a g e 30 Linear Characteristics Copyright Mathbeacon.com. Use with permission. Do not use after June 2011
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