5-1 Slope October 25, 2010 What do you think of when you hear the word slope? Are there different types of slopes?

FINDING THE SLOPE OF A L INE Math Definition 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. Slope is the ratio of change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move from one point to the other. The number describes how steep the line is.

rise slope = = = run FINDING THE SLOPE OF A L INE x y 1 3 5 7 9 -1 - 1 9 7 5 3 1 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 . 3 2 (2, 5) rise = 5 - 2 = 3 units rise (0, 2) run run = 2 - 0 = 2 units Two points on a line are all that is needed to find its slope. 5 - 2 3 = slope = = 2 - 0 2

m = y2 - y1 = = x2 - x1 FINDING THE SLOPE OF A L INE and (x2, y2) is The slope m of the nonvertical line passing through the points (x1, y1) and (x2, y2) is x y (x2, y2) (x1, y1) (x1, y1) (x2, y2) rise change in y = y2 - y1 (y2 - y1 ) Read y1 as “y sub one” Read x1 as “x sub one” run change in x x2 - x1 (x2 - x1 ) m = =

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

Find the slope of the line that passes through (–3, –4) and (–2, –8). Let and Example 1-2a

Example 1-2b

Rise: difference of y-values Run: difference of x-values A Line with a Zero Slope is Horizontal Find the slope of a line passing through (-1, 2) and (3, 2). line (-1, 2) (3, 2). (-1, 2) (3, 2). (3, 2) x y 1 3 5 7 9 -1 9 7 5 3 1 SOLUTION (3, 2) (x1, y1) (-1, 2) (x2, y2) Let (x1, y1) = (-1, 2) and (x2, y2) = (3, 2) (-1, 2) (x1, y1) (3, 2) (x2, y2) Rise: difference of y-values y2 - y1 m = rise = 2 - 2 = 0 units 2 - 2 = x2 - x1 Run: difference of x-values Substitute values. run = 3 - ( -1) = 4 units 3 - (-1) Simplify. = 4 Slope is zero. Line is horizontal. =

Find the slope of the line that passes through (–3, 4) and (4, 4). Let and Example 1-3a

Example 1-3b

Find the slope of the line that passes through (–2, –4) and (–2, 3). Let and Answer: Since division by zero is undefined, the slope is undefined. These two points are on what kind of a line? Example 1-4a

INTERPRETING SLOPE AS A RATE OF CHANGE Slope as a Rate of Change – tells, on average, how a quantity is changing over time 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. t = 0 seconds, h = 2500 feet t = 35 seconds, h = 2115 feet. t = 0 seconds, h = 2500 feet t = 35 seconds, h = 2115 feet. x y 5 15 25 35 45 2700 2500 2300 2100 1900 Height (feet) Time (seconds) a. What is your rate of change in height? (0, 2500) b. About when will you reach the ground? (35, 2115)

Slope as a Rate of Change SOLUTION a. Use the formula for slope to find the rate of change. The change in time is 35 - 0 = 35 seconds. Subtract in the same order. The change in height is 2115 - 2500 = -385 feet. Rate of Change = rate of change. Change in Time change in time Change in Height change in height VERBAL MODEL LABELS Rate of Change = m (ft/sec) m = ALGEBRAIC MODEL Change in Height = - 385 (ft) - 385 Change in Time = 35 (sec) 35 = -11 Your rate of change is -11 ft/sec. The negative value indicates you are falling.

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

Concept Summary Slope: Rate of change SLOPE of a line Four types of slopes: Positive Negative Zero Undefined (no slope)

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