The slope m of a nonvertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. F INDING THE S LOPE OF A L INE

x y Two points on a line are all that is needed to find its slope. slope = = = F INDING THE S LOPE OF A L INE The slanted line at the right rises 3 units for each 2 units of horizontal change from left to right. So, the slope m of the line is The slope m of a nonvertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. nonvertical line (2, 5) rise = = 3 units rise run run = = 2 units (0, 2)

x y run change in x The slope m of the nonvertical line passing through the points (x 1, y 1 ) and (x 2, y 2 ) is Read y 1 as “y sub one” Read x 1 as “x sub one” m = = (x1, y1)(x1, y1) (x 1, y 1 ) (x 2, y 2 ) (x2, y2)(x2, y2) y2 - y1y2 - y1 (y 2 - y 1 ) x2 - x1x2 - x1 (x 2 - x 1 ) F INDING THE S LOPE OF A L INE (x2, y2)(x2, y2) (x1, y1)(x1, y1) rise change in y =

m = run change in x rise change in y = = y2 - y1y2 - y1 x2 - x1x2 - x1 y2 - y1y2 - y1 m = x2 - x1x2 - x1 The order of subtraction is important. You can label either point as (x 1, y 1 ) and the other point as (x 2, y 2 ). However, both the numerator and denominator must use the same order. numerator y2 - y1y2 - y1 denominator x2 - x1x2 - x1 F INDING THE S LOPE OF A L INE When you use the formula for the slope, Subtraction order is the same the numerator and denominator must use the same subtraction order. CORRECT x1 - x2x1 - x2 y2 - y1y2 - y1 Subtraction order is different INCORRECT

Let (x 1, y 1 ) = (-1, 2) and (x 2, y 2 ) = (3, 2) (3, 2)(3, 2)(x2, y2)(x2, y2) (-1, 2) (x1, y1)(x1, y1) Find the slope of a line passing through (-1, 2) and (3, 2). Slope is zero. Line is horizontal. x y line S OLUTION m =m = run = 3 - ( -1) = 4 units 3 - (-1) rise = = 0 units = Substitute values. = 0 A Line with a Zero Slope is Horizontal x 2 - x 1 Run: difference of x-values Rise: difference of y-values y 2 - y 1 Simplify. = 0404 (-1, 2) (3, 2).(3, 2). (3, 2)(3, 2) (3, 2).(3, 2). (3, 2)(3, 2) (x1, y1) (x1, y1) (x2, y2)(x2, y2)

x y Height (feet) Time (seconds) You are parachuting. At time t = 0 seconds, you open your parachute at h = 2500 feet above the ground. At t = 35 seconds, you are at h = 2115 feet. a. What is your rate of change in height? b. About when will you reach the ground? Slope as a Rate of Change I NTERPRETING S LOPE AS A R ATE OF C HANGE (0, 2500) t = 0 seconds, h = 2500 feet t = 35 seconds,h = 2115 feet. t = 0 seconds, h = 2500 feet t = 35 seconds,h = 2115 feet. (35, 2115)

S OLUTION a. Use the formula for slope to find the rate of change. The change in time is = 35 seconds. Subtract in the same order. The change in height is = feet. Rate of Change = rate of change. Change in Time change in time Rate of Change = m (ft/sec) Change in Height = (ft) m m = Change in Time = 35 (sec) 35 = - 11 Slope as a Rate of Change V ERBAL M ODEL Your rate of change is - 11 ft/sec. The negative value indicates you are falling. Change in Height change in height L ABELS A LGEBRAIC M ODEL

b. Falling at a rate of -ll ft/sec, find the time it will take you to fall 2500 ft. Time = Distance Rate - 11 ft/sec You will reach the ground about 227 seconds after opening your parachute. Time Slope as a Rate of Change Distance ft = 227 sec