SLOPE A MEASURE OF STEEPNESS through 7.1.5
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1 SLOPE A MEASURE OF STEEPNESS 7.1. through Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the coefficient of and is the slope of the line. It indicates the direction of the line and its steepness. The constant, b, is the -intercept, written (0, b), and indicates where the line crosses the -ais. See the Math Notes boes on pages 8, 91, and 98. Eample 1 If m is positive, the line goes upward from left to right. If m is negative, the line goes downward from left to right. If m = 0 then the line is horizontal. = =! = 0 + or = Eample When m = 1, as in =, the line goes upward b one unit each time it goes over one unit to the right. Steeper lines have a larger m value, that is, m > 1. Flatter lines have an m value between 0 and 1, usuall in the form of a fraction. = = 1 50 Algebra Connections Parent Guide
2 Eample Slope is written as a ratio. If the line is drawn on a set of aes, a slope triangle can be drawn between an two convenient points (that is, where grid lines cross and coordinates are integers), as shown in the graph at right. Count the vertical distance and the horizontal distance on the dashed sides of the slope triangle. Write the distances in a ratio: m = vertical!!!. Another form used is:. The smbol horizontal!!!! means change. The order in the fraction is important: the numerator (top of the fraction) must be the vertical distance and the denominator (bottom of the fraction) must be the horizontal distance.! =! = Parallel lines have the same steepness and direction, so the have the same slope, as shown in the graph at right. If! = 0, then the line is horizontal and has a slope of zero, that is, m = 0. If! = 0, then the line is vertical and its slope is undefined, so we sa that it has no slope. =!,! = + 1 Eample When the vertical and horizontal distances are not eas to determine, ou can find the slope b drawing a generic slope triangle and using it to find the lengths of the vertical (! ) and horizontal (! ) segments. The figure at right shows how to find the slope of the line that passes through the points (-1, 9) and (19, -15). First graph the points on unscaled aes b approimating where the are located, then draw a slope triangle. Net find the distance along the vertical side b noting that it is 9 units from point B to the -ais then 15 units from the -ais to point C, so! is. Then find the distance from point A to the -ais (1) and the distance from the -ais to point B (19).! is 0. This slope is negative because the line goes downward from left to right, so the slope is! 0 =! 5. A (-1,9)! =0 (19,-15) B C! = Chapter 7: Linear Relationships 51
3 Problems Is the slope of these lines negative, positive or zero? 1... Identif the slope in these equations. State whether the graph of the line is steeper or flatter than = or =!, whether it goes up or down from left to right, or if it is horizontal or vertical.. = + 5. =! = 1! 7.! = 8. =! = = 11. = = 8 Without graphing, find the slope of each line based on the given information. 1.! = 7!! = "8 1.! = 15!! = 15.! = 7!! = Horizontal! = 6 Vertical! = Between (5, 8) and (6, 1) 18. Between (-, ) and (5, -7) Answers 1. zero. negative. positive. Slope =, steeper, up 5. Slope = - 1, flatter, down 6. Slope = 1, flatter, up 7. Slope =, steeper, up 8. Slope = 1, flatter, up 9. Slope =, steeper, up 10. horizontal 11. vertical 1. Slope = -, steeper, down 1.! = undefined ! ! Algebra Connections Parent Guide
4 WRITING AN EQUATION GIVEN THE SLOPE 7..1 and 7.. AND A POINT ON THE LINE In earlier work students used substitution in equations like = + to find and pairs that make the equation true. Students recorded those pairs in a table, then used them as coordinates to graph a line. Ever point (, ) on the line makes the equation true. Later in the course, students used the patterns the saw in the tables and graphs to recognize and write equations in the form of = m + b. The b represents the -intercept of the line, the m represents the slope, while and represent the coordinates of an point on the line. Each line has a unique value for m and a unique value for b, but there are infinite (, ) values for each linear equation. The slope of the line is the same between an two points on that line. We can use this information to write equations without creating tables or graphs. See the Math Notes bo on page 08. Eample 1 What is the equation of the line with a slope of that passes through the point (10, 17)? Substitute the values we know: m,, and. = m + b 17 = (10) + b 17 = 0 + b! = b Write the complete equation using the values of m () and b (-). =! Chapter 7: Linear Relationships 5
5 Eample This algebraic method can help us write equations of parallel and perpendicular lines. Parallel lines never intersect or meet. The have the same slope, m, but different -intercepts, b. What is the equation of the line parallel to =! that goes through the point (, 8)? Substitute the values we know: m,, and. Since the lines are parallel, the slopes are equal. = m + b 8 = () + b 8 = 6 + b = b Write the complete equation. = + Eample Perpendicular lines meet at a right angle, that is, 90. The have opposite, reciprocal slopes and ma have different -intercepts. If one line is steep and goes upward, then the line perpendicular to it must be shallow and go downward. Reciprocals are pairs of numbers that when multiplied equal one. For eample, and 1 are reciprocals as are 5 and. Opposites have different signs: positive or negative. Eamples of 5 opposite reciprocal slopes are! 1 5 and 5;! and ; and! and 1. Write the equation of the line that is perpendicular to = + and goes through the point (6, ). Substitute the values we know: m,, and. Since the line is perpendicular, the new slope is! 1. = m + b =! 1 (6) + b =! + b 6 = b Write the complete equation. =! Algebra Connections Parent Guide
6 Problems Write the equation of the line with the given slope that passes through the given point. 1. slope = 5, (, 1). slope =! 5, (, -1). slope = -, (-, 9). slope =, (6, 8) 5. slope =, (-7, -) 6. slope =, ( 5, -) Write the equation of the line parallel to the given line that goes through the given point. 7. = 5 + (0, 0) 8. =! 1 (-, -6) 9. =! + 5 (-, -) 10. = + 5 (-6, -8) 11. = 1! 1 (6, 9) 1. = + 8 (0, 1 ) Write the equation of the line perpendicular to the given line that goes through the given point. 1. = 5 + (0, 0) 1. =! 1 (-, 6) 15. =! + 5 (-, -) 16. = 1! 1 (6, -) 17. = + 5 (0, -8) 18. = + 1 (0, 1 ) Answers 1. = 5!. =! 5 +. =! + 1. =! 1 5. =! 6. =! 7 7. = 5 8. = + 9. =!! =! 11. = = =! 5 1. =! = =! =! 1! 18. =! Chapter 7: Linear Relationships 55
7 WRITING THE EQUATION OF A LINE GIVEN TWO POINTS 7.. Students now have all the tools the need to find the equation of a line passing through two given points. Recall that the equation of a line requires a slope and a -intercept in = m + b. Students can write the equation of a line from two points b creating a slope triangle and calculating!! Notes bo on page 1. as eplained in sections 7.1. through See the Math Eample 1 Write the equation of the line that passes through the points (1, 9) and (-, -). Draw a generic slope triangle. Calculate the slope using the given two points. (-, -) 1 = m + b (1, 9)!! = 1, m = Substitute m and one of the points for and, in this case (1, 9). 9 = (1) + b 5 = b Write the complete equation. = + 5 Eample Write the equation of the line that passes through the points (8, ) and (, 6). Draw a generic slope triangle. Calculate the slope using the given two points. (, 6) (8, )!! =, m = " = m + b Substitute m and one of the points for and, in this case (8, ). =! (8) + b =!6 + b 9 = b Write the complete equation. =! Algebra Connections Parent Guide
8 Problems Write the equation of the line containing each pair of points. 1. (1, 1) and (0, ). (5, ) and (1, 1). (1, ) and (-5, -15). (-, ) and (, 5) 5. (, -1) and (, -) 6. (, 5) and (-, -) 7. (1, -) and (-, 5) 8. (-, -) and (5, -) 9. (-, 1) and (5, -) Answers 1. =! +. = + 1. =. = =! + 6. =! 1 7. =!! 1 8. =! 9. =! 1! 1 Chapter 7: Linear Relationships 57
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