Traffic signs are used to alert drivers to road conditions. The sign at the right indicates, a hill with a 6% grade . This means that the road will rise or fall 6 feet vertically for every 100 horizontal feet traveled.
HOW is slope used in transportation? Why would a road or train track wind its way up a mountain instead of going directly toward the top? A path going directly toward the top might be too steep for a car or train. To reach the same height, is it easier to push a wheelchair up a long ramp or a short ramp? A long ramp is easier because the climb is less steep, even though you travel father.
Slope of a Line The slope of a line is the ratio of its vertical rise to its horizontal run. Slope = y 2 - y 1 x 2 - x 1 = Vertical Rise Horizontal Run x y Horizontal Run Vertical Rise
Slope The slope m of a line containing two points with coordinates ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = y 2 - y 1 x 2 - x 1 , where x 1 = x 2
Slope The slope of a line indicates whether the line rises to the right, falls to the right, or is horizontal. The slope of a vertical line, where x 1 = x 2 , is undefined.
Find the slope of a line… From (-3, -2) to (-1, 2), Go up four units and right 2 units. Use the slope formula. Let (-4, 0) be ( x 1 , y 1 ) and (0, -1) be ( x 2 , y 2 ). Slope = y 2 - y 1 x 2 - x 1 = -1 - 0 0 - (-4) = -1/4 x y ( -1 , 2 ) ( -3 , -2 ) Use the method. rise run rise run = 4 2 = 2 m = ( -4 , 0 ) ( 0 , -1 )
Study Tip! Lines with positive slope rise as we move from left to right, while lines with negative slope fall as we move from left to right.
Example (-3, 5) (1, 5) What happens with horizontal lines? = 5 - 5 -3 - 1 0 -4 = = 0 So, the slope of every horizontal line is 0! x y m = y 2 - y 1 x 2 - x 1
Example (6, 3) (6, -4) What happens with vertical lines? = 3 - (-4) 6 - 6 = 7 0 , which is undefined. Therefore, all vertical lines are undefined! x y m = y 2 - y 1 x 2 - x 1
WARM UP If we want to get a hint about the positivity or negativity of a slope, what can we look for? HW CHECK: Pg. 142; #1 - 9, 51 - 61 odd
Rate of Change The slope of a line can be used to identify the coordinates of any point on the line. The rate of change describes how a quantity is changing over time.
Use Rate of Change to Solve a Problem Between 1990 and 2000, the annual sales of rollerblade equipment increased by an average rate of $92.4 million per year. In 2000, the total sales were $1074.4 million. If sales increase at the same rate, what will the total be in 2008? Let ( x 1 , y 1 ) = (2000, 1074.4) and m = 92.4 m = y 2 - y 1 x 2 - x 1 Slope formula 92.4 = y 2 - 1074.4 2008 - 2000 Substitution 92.4 = y 2 - 1074.4 8 Simplify 739.2 = y 2 - 1074.4 Multiply by 8 on each side! 1813.6 = y 2 <- Answer!
Therefore, The coordinates of the point representing the sales for 2008 are (2008, 1813.6). Thus, the total sales in 2008 will be about $1813.6 million.
Parallel and Perpendicular Lines Examine the graphs of lines l , m , and n . Lines l and m are parallel, and n is perpendicular to l and m . Let’s investigate! (-3, 5) (2, 2) (1, -3) (0, 4) (4, 2) l m n Slope of line l ? Slope of line m ? Slope of line n ? Because lines l and m are parallel, their slopes are the same. Line n is perpendicular to line l and m , and its slope is the opposite reciprocal of the slopes of l and m . x y m = - 3 5 m = - 3 5 m = 5 3
Slopes of Parallel and Perpendicular Lines Postulate 3.2 - Two nonvertical lines have the same slope if and only if they are parallel. Postulate 3.3 - Two nonvertical lines are perpendicular if and only if the product of their slopes is -1.
My Name is Ned the Ninja! I will be helping you throughout Chapter 3!