2 – 2: Slope and Rate of Change Objective: CA Standard 7.0: Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents.
The Slope of a line The slope of a non-vertical line passing through (x 1, y 1 ) and (x 2, y 2 ) is given by
Example 1 Find the slope of the line passing through (-3, 5) and (2, 1)
Classification of Lines by Slope A line with a positive slope rises from the left to the right m > 0
A line with a negative slope rises from the falls from left to right. m < 0
A line with a slope of zero is horizontal (m = 0)
A line with an undefined slope is a vertical line. (m is undefined)
Classify Lines by their Slopes Tell whether the line through the given points rises, falls, is horizontal or vertical. (3, -4), (1, -6) The slope of the line is positive. The line rises.
(2, -1), (2, 5) The slope of the line is undefined, the line is vertical. Tell whether the line through the given points rises, falls, is horizontal or vertical.
Comparing Steepness of Lines If two lines have positive slopes the line with the greater slope is steepest If two lines have negative slopes the line with the slope of greater absolute value is steeper.
Tell which line is steeper. Line 1: through (2, 3) and (4, 7) Line 2: through (-1, 2) and (4, 5) Line 1 is steeper.
Two lines are parallel if they do not intersect. Two lines are perpendicular if they intersect to form right angles. Slope can be used to determine whether two distinct lines (non- vertical) lines are parallel or perpendicular.
Slopes of Parallel and Perpendicular Lines. Consider two different lines l 1, and l 2 with slopes m 1, m 2 Parallel Lines: have the same slope. m 1 = m 2
Perpendicular Lines: slopes are negative reciprocals of each other.
Classifying Parallel and Perpendicular Lines Tell whether the lines are parallel, perpendicular or neither. Line 1 passes through (-3, 3) and (3, -1) Line 2 passes through (-2, -3) and (2, 3)
Line1 passes through (-3, 1) and (3, 4) Line 2 passes through (-4, -3) and (4, 1)
HOMEWORK Page 79 #17-41 EOO
Geometrical Use of Slope The slope of a road or grade is usually expressed as a percent. For example if a road has a grade of 3%, it rises 3 feet for every 100 feet of horizontal distance.
Find the grade of a road that rises 75 feet over a horizontal distance of 2000 feet
Slope as Rate of Change The number of US cell phones subscribers increased from 16 million in 1993 to 44 million in Find the average rate of change and use it to estimate the number of subscribers in Average rate of change = Change in subscribers Change in time
The number of subscribers will be approximately /3 = 53 1/3 million subscribers.