2.2 Slope and Rate of Change

In this lesson you will: Find slopes of lines and classify parallel and perpendicular lines. Use slope to solve real-life problems.

The Slope of a Non-vertical line Is the ratio of vertical change (the rise) To the horizontal change (the run). x y “Change in y” “Change in x”

The Slope Formula: m= y x - rise run x y 1 2 x - rise run 1 2 3 4 5 6 7 8 9 10 x y Find the slope for the segments at the right: Segment AB: A(3,2), B(7,5) B x 2 y 2 x 1 y 1 m = - AB 2 5 = -3 -4 = 3 4 3 7 A C D Segment CD: C(8,2), D(10,1) x 2 y 2 x 1 y 1 m = - cd 2 1 = 1 -2 8 10

Using the slope formula below, find the slope for the lines in the coordinate plane: y 1 2 x - Line a: (1,5) (-1,1) x 2 1 y 2 5 x 1 -1 y 1 m = - a 4 = 4 2 4 2 6 -2 -4 -6 8 10 -8 -10 x y = =2 d 1+1 a (-8, 7) Line b: (3,2) (10,2) (1, 5) x 2 3 y 2 x 1 10 y 1 2 m = - b = -7 b (-1, 1) =0 (3, 2) (10, 2) Line c: (6,-8) (2,-5) (2, -5) (-4, -4) y 2 -8 x 2 6 y 1 -5 x 1 2 c m = - c -8+5 = -3 4 (6, -8) = - 3 4 = 4 Line d: (-8,7) (-4,-4) x 2 -8 y 2 7 y 1 -4 x 1 -4 m = - d 7+4 = 11 -4 = - 11 4 = -8+4

Or another way to determine slope is using the “rise” over the “run” method. Find the slope for the lines in the coordinate plane: m= y 1 2 x - + - rise = run + Line a: 4 2 6 -2 -4 -6 8 10 -8 -10 x y - -4 = 4 2 m = a =2 d -2 a 4 + 2 - Line b: - 11 - 4 m = b =0 +7 b 7 + Line c: +3 = - 3 4 m = c c -4 3 + 4 - Line d: -11 = - 11 4 m = d +4

+ + + + FALLING TO THE RIGHT y - - - x - c Slope for c: + m = = - - When we move to the right from one point to the other, we go down the right and the slope is negative!

RISING to the RIGHT x y a Slope for a: + m = =+ + When we move to the left from one point to the other, we go up and the slope is always positive!

Compare the slopes of these lines:

Summary + - y y a b + x x Slope for b: Slope for a: + m = =0 m = =+ + + m = b =0 m = a =+ + + Slope of a horizontal line is always 0. Slope of a line that rises to the right is always POSITIVE! x y x y d c + + - Slope for c: Slope for d: + m = c = - + - m = d = not defined! The slope of a line that falls to the right is always NEGATIVE! The slope of a vertical line IS NEVER DEFINED!

+ + - + PARALLEL and PERPENDICULAR x y PARALLEL PERPENDICULAR x y a c 4 2 6 -2 -4 -6 8 10 -8 -10 x y PARALLEL PERPENDICULAR 4 2 6 -2 -4 -6 8 10 -8 -10 x y a c b 2 + d 2 + 2 + 2 + 2 + 2 + 2 - 2 + +2 +2 Line a: m = a =1 Line c: m = c =-1 m = d m c (-1)(1) =-1 +2 -2 +2 +2 Line b: m = b =1 Line d: m = d =1 +2 +2 The slope product of perpendicular lines is -1 Parallel Lines have the same slope! m = a m b m = d m c -1

Prove that segments AB and BC are perpendicular: Using the slope formula: 1 2 3 4 5 6 7 8 9 10 x y m= y 1 2 x - Segment AB: A(2,1), B(6,4) B x 2 y 2 x 1 y 1 m = - AB 1 4 = -3 -4 = 3 4 2 6 A C Segment BC: B(6,4), C(9,0) x 2 y 2 x 1 y 1 m = - BC 4 = 4 -3 6 9 Is the product of the slopes -1? The product of the slopes is -1. So, They are perpendicular. 3 4 = 4 -3 12 -12 = -1

Slope as a Rate of Change In the Mojave Desert in California, temperatures can drop quickly from day to night. Suppose the temperature drops from 100 degree at 2 p.m. to 68 degrees at 5 a.m. What is the average rate of change in temperature. Use this to estimate the temperature at 10 p.m. Solution: Change in temperature Average rate of change n Temp. = Change in time Because 10 pm is 8 hours after 2pm., the temp. changed 8(-2)=-16 degrees. That means the temperature at 10 pm was about 100-16 or 84 degrees.

Homework 17-47 odd, 49, 52