1. Section 3.6
Find and Use Slopes of Lines
The _______ of a _____________ line is the ratio of __________ change (_____) to the ____________ change (____) between any two point on a line.
slope
non-vertical
vertical
rise
horizontal
run

2. + + Section 3.6 Find and Use Slopes of Lines
Key Concept: Slopes of Lines in the Coordinate Plane
Slope formula: Given two points
and , the formula to find the slope of a line is:
Example: Find the slope of the line through points (-1, -2) and (-5, 6) + +

3. Section 3.6
Find and Use Slopes of Lines
Negative slope:
Positive slope:
Line falls from left to right.
Line rises from left to right.
Slope results in a negative number.
Slope results in a positive number.

4. Section 3.6
Find and Use Slopes of Lines
Zero slope:
Undefined slope:
Horizontal line with 0 rise.
Vertical line with 0 run.
Can’t divide by 0

5. opposite reciprocal slopes. **All vertical lines are parallel.
Section 3.6
Find and Use Slopes of Lines
Parallel lines:
Perpendicular lines:
Perpendicular lines have
opposite reciprocal slopes.
Parallel lines have same slope, but different y-intercepts.
**All vertical lines are parallel.
+/- and flip

6. Lines have different slopes, but not opposite reciprocals.
Section 3.6
Find and Use Slopes of Lines
Coincident lines:
Intersecting lines not perpendicular:
Coincident lines have same slope and same y-intercepts.
Lines have different slopes, but not opposite reciprocals.
Lines on top of each other

7. undefined (vertical line)
EXAMPLE 1
Find slopes of lines in a coordinate plane
Use rise and run to find the slope of the line a that passes through the points (6, 4) and (8, 2). 2 2
Rise ____ Run ____ Slope _______
Use slope formula to find the slope of line d that passes through the points (6, 4) and (6, 0).
Can’t divide by 0
undefined (vertical line)

8. 4 2 GUIDED PRACTICE for Example 1
Use the graph to find the slopes of the other two lines.
Find the slope of the line b using run and run from points (6, 4) and (4, 0). 4 2
Rise ____ Run ____ Slope _______
Use slope formula to find the slope of the line c that passes through the points (6, 4) and (0, 4).
slope is 0 (horizontal line) 3.
A vertical line has a slope that is ___________.
undefined

9. 4 4 EXAMPLE 2 Find parallel and perpendicular lines
Find the slope of line AB that passes through points
A(3, 1) and B(4, 5)? 4
Slope of is _________
Slopes of parallel line are the __________.
What is the slope of a line parallel to ? _________
same 4
Slopes of perpendicular lines are ____________________.
What is the slope of a line perpendicular to ? _______
opposite reciprocals

10. + GUIDED PRACTICE for Example 2
Line g passes through (–1, 3) and (4, 1). Find the slope of a parallel line and a perpendicular line. Explain how you know.
y2 – y1
x2 – x1
m = =
1 – 3
4 –(–1) = 5
– 2
Slope line g: +
Parallel slope _______
Perpendicular slope_______

11. EXAMPLE 3
Identifying line relationships
Given the graphs of two lines, describe the lines as parallel, perpendicular, coincident, or intersecting, but not perpendicular. b) a)
coincident
parallel

12. EXAMPLE 3
Identifying line relationships
Given the graphs of two lines, describe the lines as parallel, perpendicular, coincident, or intersecting, but not perpendicular. c) d)
intersecting, but not perpendicular
perpendicular

13. EXAMPLE 4
Determining steepness or flatness
Compare slopes. The __________ the slope number, the ___________ the line. The __________ the slope number, the __________ the line. A fraction slope between 0 and 1 is ____________.
bigger
steeper
smaller
flatter
flatter
The slope of line a is –3 and the slope line b is Is line a or line b flatter? __________
line b

14. Find the missing coordinate
EXAMPLE 5
Find the missing coordinate
Determine the value of x so that a line through the points at (5, 3) and (x, 4) has a slope of .
What does x have to be so the denominator is 2
x = 7

15. GUIDED PRACTICE
for Examples 3, 4 and 5
Coincident lines have the ___________ slope and the _____________ y-intercept.
same
same
Compare parallel lines and coincident lines. Explain.
Both have the same slopes, parallel lines have different y-intercepts and coincident lines have same y-intercepts.
The slope of line a is –4, slope line b is , and slope of line c is 3.
Which line is steeper? ______Which line is flatter? _____
line a
line b

16. GUIDED PRACTICE
for Examples 3, 4 and 5
Look at the graph of the two lines.
What do you know about the slopes?
What do you know about the
y-intercepts?
The same, both .
Different y-intercept points, (0, 1) and (0, -3)

17. GUIDED PRACTICE for Examples 3, 4 and 5
Determine the value of x so that a line through the points at (x, 4) and (6, 5) has a slope of .
What does x have to be so the denominator =2
x = 4