1. Section 2.2 - Slope and Rate of Change
ALGEBRA TWO
CHAPTER TWO: LINEAR EQUATIONS AND FUNCTIONS
Section Slope and Rate of Change

2. LEARNING GOALS
Goal One - Find slopes of lines and classify parallel and perpendicular lines.
Goal Two - Use slope to solve real-life problems.

3. VOCABULARY
The slope of a nonvertical line is the ratio of vertical change (the rise) to the horizontal change (the run).
The slope of a line is represented by the letter m. 1

4. The Slope of a Line
The slope of the nonvertical line passing through the points (x1, y1) and (x2, y2) is:
slope=m = (y2-y1)
(x2-x1)
Rise: Difference of y values.
Run: Difference of x values. 1

5. Finding the slope of a Line
PROBLEM: Find the slope of a line passing through (-3, 2) and (1, 5).
SOLUTION
Let (x1, y1) = (-3, 2) and (x2, y2) = (1, 5)
slope=m = (y2-y1)
(x2-x1)
Rise: Difference of y values.
Run: Difference of x values.
= (5 - 2)/(1-(-3))
Substitute values.
= (3)/(1+3) = 3/4
Simplify. Slope is positive.
The slope is 3/4. Because the slope is positive, the line rises from left to right.

6. Finding the slope of a Line
PROBLEM: Find the slope of a line passing through (4, 1) and (-6, 3).
SOLUTION
Let (x1, y1) = (4, 1) and (x2, y2) = (-6, 3)
slope =m= (y2-y1)
(x2-x1)
Rise: Difference of y values.
Run: Difference of x values.
= (3 - 1)/(-6-(4))
Substitute values.
= (2)/(-10) = -1/5
Simplify. Slope is negative.
The slope is -1/5. Because the slope is negative, the line falls from left to right.

10. Parallel and Perpendicular Lines
PARALLEL LINES - The lines are parallel if and only if they have the same slope. Parallel lines NEVER intersect.
m1 = m2
These two lines have the same slope, but different y-intercepts. They are parallel. 1

11. Parallel and Perpendicular Lines
PERPENDICULAR LINES - The lines are perpendicular if and only if their slopes are negative reciprocals of each other. Perpendicular lines intersect to form a right angle.
m1 = -1/m2
These two lines intersect at right angles and are perpendicular. Their slopes are m and -1/m . 1

12. Parallel and Perpendicular Lines
PROBLEM: Tell whether the lines are parallel, perpendicular, or neither.
Line 1: through (-1, 3) and (1, -3)
Line 2: through (3, 0) and (0, 1)
SOLUTION
slope = (y2-y1)
(x2-x1)
Rise: Difference of y values.
Run: Difference of x values.
Line 1
Line 2
= (-3 -(-1)/(1-(-1))
= (1 - 0)/(0 - 3)
= (-3 + 1)/(1 + 1)
= (1)/(-3)
= (-2)/(2) = -1
= -1/3 1

13. Line 1: through (-1, 3) and (1, -3) Line 2: through (3, 0) and (0, 1)
Parallel and Perpendicular Lines
PROBLEM: Tell whether the lines are parallel, perpendicular, or neither.
Line 1: through (-1, 3) and (1, -3)
Line 2: through (3, 0) and (0, 1)
SOLUTION
These lines are neither parallel or perpendicular.
Line 1
Line 2
= (-3 -(-1)/(1-(-1))
= (1 - 0)/(0 - 3)
= (-3 + 1)/(1 + 1)
= (1)/(-3)
= (-2)/(2) = -1
= -1/3 1

14. Parallel and Perpendicular Lines
PROBLEM: Tell whether the lines are parallel, perpendicular, or neither.
Line 1: through (-2, 6) and (0, -2)
Line 2: through (-1, 6) and (1, -2)
SOLUTION
slope = (y2-y1)
(x2-x1)
Rise: Difference of y values.
Run: Difference of x values.
Line 1
Line 2
= (-2 - 6)/(0 -(-2))
= (-2 - 6)/(1 - (-1))
= (-8)/(1 + 1)
= (-8)/(2)
= (-8)/(2) = -4
= -4 1

15. Parallel and Perpendicular Lines These lines are parallel.
PROBLEM: Tell whether the lines are parallel, perpendicular, or neither.
Line 1: through (-2, 6) and (0, -2)
Line 2: through (-1, 6) and (1, -2)
SOLUTION
These lines are parallel.
Line 1
Line 2
= (-2 - 6)/(0 -(-2))
= (-2 - 6)/(1 - (-1))
= (-8)/(1 + 1)
= (-8)/(2)
= (-8)/(2) = -4
= -4 1

16. Average rate of change = (change in value) / (change in time)
Slope as Rate of Change
PROBLEM: Six years ago a house was purchased for $89,000. This year it was appraised at $125,000. Find the average rate of change and use it to determine the value after nine years.
SOLUTION
Average rate of change = (change in value) / (change in time)
= ($125,000 - $89,000)/6 years
= $36,000/6 years = $6,000 per year
Over a 9-year period, the value changed 9 • $6, 000/yr. The value after nine years was
$89, $54,000 = $143,000.

17. Homework
Pg 79-80
#17,23,27,29,37,39,41,43,51