1. Section 1.3
Slope of a Line

2. Comparing the Steepness of Two Objects
Introduction
Comparing the Steepness of Two Objects
Two ladders leaning against a building. Which is steeper?
We compare the vertical distance from the base of the building to the ladder’s top with the horizontal distance from the ladder’s foot to the building.
Section 1.3
Slide 2

3. Comparing the Steepness of Two Objects
Introduction
Comparing the Steepness of Two Objects
Ratio of vertical distance to the horizontal distance:
Latter A:
Latter B:
So, Latter B is steeper.
Section 1.3
Slide 3

4. Property Property of Comparing the Steepness of Two Objects
To compare the steepness of two objects such as two ramps, two roofs, or two ski slopes, compute the ratio
for each object. The object with the larger ratio is the steeper object.
Section 1.3
Slide 4

5. Comparing the Steepness of Two Roads
Comparing the Steepness of Two Objects
Example
Road A climbs steadily for 135 feet over a horizontal distance of 3900 feet. Road B climbs steadily for feet over a horizontal distance of 3175 feet. Which road is steeper? Explain.
These figures are of the two roads, however they are not to scale
Solution
Section 1.3
Slide 5

6. Comparing the Steepness of Two Roads
Comparing the Steepness of Two Objects
Solution Continued
A: = = ≈
B: = = ≈
vertical distance
horizontal distance
135 feet
3900 feet
0.035 1
vertical distance
horizontal distance
120 feet
3175 feet
0.038 1
Road B is a little steeper than road A
Section 1.3
Slide 6

7. Comparing the Steepness of Two Roads
Finding a Line’s Slope
Definition
The grade of a road is the ratio of the vertical to the horizontal distance written as a percent.
What is the grade of roads A?
Ratio of vertical distance to horizontal distance is for road A is = 0.038(100%) = 3.8%.
Example
Solution
Section 1.3
Slide 7

8. Slope of a Non-vertical Line
Finding a Line’s Slope
We will now calculate the steepness of a non-vertical line given two points on the line.
Pronounced x sub
1 and y sub 1
Pronounced x sub 1 and y sub 1
Let’s use subscript 1 to label x1 and y1 as the coordinates of the first point, (x1, y1). And x2 and y2 for the second point, (x2, y2).
Run: Horizontal Change = x2 – x1
Rise: Vertical Change = y2 – y1
The slope is the ratio of the rise to the run.
Section 1.3
Slide 8

9. Slope of a Non-vertical Line
Finding a Line’s Slope
Definition
Let (x1, y1) and (x2, y2) be two distinct point of a non-vertical line. The slope of the line is
vertical change
horizontal change
rise
run
y2 – y1
x2 – x1
m = = =
In words: The slope of a non-vertical line is equal to the ratio of the rise to the run in going from one point on the line to another point on the line.
Section 1.3
Slide 9

10. Slope of a Non-vertical Line
Finding a Line’s Slope
Definition
A formula is an equation that contains two or more variables. We will refer to the equation a
as the slope formula.
(graphical)
Sign of rise or run
run is positive
run is negative
rise is positive
rise is negative
Direction (verbal)
goes to the right
goes to the left
goes up
goes down
Section 1.3
Slide 10

11. Finding the Slope of a Line
Finding a Line’s Slope
Example
Find the slope of the line that contains the points (1, 2) and (5, 4).
(x1, y1) = (1, 2)
(x2, y2) = (5, 4).
Solution
Section 1.3
Slide 11

12. A common error is to substitute the slope formula incorrectly:
Finding the Slope of a Line
Finding a Line’s Slope
Warning
A common error is to substitute the slope formula incorrectly:
Correct Incorrect Incorrect
Section 1.3
Slide 12

13. Find the slope of the line that contains the points (2, 3) and (5, 1).
Finding the Slope of a Line
Finding a Line’s Slope
Example
Find the slope of the line that contains the points (2, 3) and (5, 1).
Solution
By plotting points, the run is 3 and the rise is –2.
Section 1.3
Slide 13

14. Increasing: Positive Slope Decreasing: Negative Slope
Definition
Increasing and Decreasing Lines
Increasing: Positive Slope Decreasing: Negative Slope
Positive rise
Positive run
negative rise
positive run
m =
m =
= Positive slope
= negative slope
Section 1.3
Slide 14

15. Finding the Slope of a Line
Increasing and Decreasing Lines
Example
Find the slope of the line that contains the points (– 9 , –4) and (12, –8).
Solution –
The slope is negative
The line is decreasing
Section 1.3
Slide 15

16. Find the slope of the two lines sketched on the right.
Comparing the Slopes of Two Lines
Increasing and Decreasing Lines
Example
Find the slope of the two lines sketched on the right.
Solution
For line l1 the run is 1 and the rise is 2.
Section 1.3
Slide 16

17. Comparing the Slopes of Two Lines
Increasing and Decreasing Lines
Solution Continued
For line l2 the run is 1 and the rise is 4.
Note that the slope of l2 is greater than the slope of l1, which is what we expected because line l2 looks steeper than line l1.
Section 1.3
Slide 17

18. Find the slope of the line that contains the points (2, 3) and (6, 3).
Investigating Slope of a Horizontal Line
Horizontal and Vertical Lines
Example
Find the slope of the line that contains the points (2, 3) and (6, 3).
Solution
Plotting the points (above) and calculating the slope we get
The slope of the horizontal line is zero, no steepness.
Section 1.3
Slide 18

19. Find the slope of the line that contains the points (4, 2) and (4, 5).
Investigating the slope of a Vertical Line
Horizontal and Vertical Lines
Example
Find the slope of the line that contains the points (4, 2) and (4, 5).
Solution
Plotting the points (above) and calculating the slope we get
The slope of the vertical line is undefined.
Section 1.3
Slide 19

20. Horizontal and Vertical Lines
Property
Horizontal and Vertical Lines
Property
A horizontal line has slope of zero (left figure).
A vertical line has undefined slope (right figure).
Section 1.3
Slide 20

21. Two lines are called parallel if they do not intersect.
Finding Slopes of Parallel Lines
Parallel and Perpendicular Lines
Definition
Two lines are called parallel if they do not intersect.
Example
Find the slopes of the lines l1 and l2 sketch to the right.
Section 1.3
Slide 21

22. Both lines the run is 3, the rise is 1 The slope is,
Finding Slopes of Parallel Lines
Parallel and Perpendicular Lines
Solution
Both lines the run is 3, the rise is 1
The slope is,
It makes sense that the nonvertical parallel lines have equal slope
Since they have the same steepness
Section 1.3
Slide 22