Upload
torn
View
39
Download
0
Embed Size (px)
DESCRIPTION
Section 1.3. Slope of a Line. Section 1.3. Slide 2. Introduction. Comparing the Steepness of Two Objects. Two ladders leaning against a building. Which is steeper? - PowerPoint PPT Presentation
Citation preview
Slope of a Line
Section 1.3
Lehmann, Intermediate Algebra, 3edSection 1.3 Slide 2
Introduction
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.
Comparing the Steepness of Two Objects
Lehmann, Intermediate Algebra, 3edSection 1.3 Slide 3
Introduction
Ratio of vertical distance to the horizontal distance:
Latter A:
Latter B:
So, Latter B is steeper.
8 feet 42 feet 1
8 feet 24 feet 1
Comparing the Steepness of Two Objects
Lehmann, Intermediate Algebra, 3edSection 1.3
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.
Slide 4
Property of Comparing the Steepness of Two Objects
vertical distancehorizontal distance
Property
Comparing the Steepness of Two Objects
Lehmann, Intermediate Algebra, 3edSection 1.3
Road A climbs steadily for 135 feet over a horizontal distance of 3900 feet. Road B climbs steadily for 120 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
Slide 5
Comparing the Steepness of Two Roads
Example
Solution
Comparing the Steepness of Two Objects
Lehmann, Intermediate Algebra, 3edSection 1.3
A: = = ≈
B: = = ≈
Slide 6
Comparing the Steepness of Two Roads
vertical distancehorizontal distance
135 feet3900 feet
0.0351
vertical distancehorizontal distance
120 feet3175 feet
0.0381
• Road B is a little steeper than road A
Solution Continued
Comparing the Steepness of Two Objects
Lehmann, Intermediate Algebra, 3edSection 1.3
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 = 0.038(100%) = 3.8%.
Slide 7
Comparing the Steepness of Two Roads
Definition
Solution
Example
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
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.Slide 8
Slope of a Non-vertical Line
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
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
Let (x1, y1) and (x2, y2) be two distinct point of a non-vertical line. The slope of the line is
Slide 9
Slope of a Non-vertical Line
vertical changehorizontal change
riserun
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.
Definition
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
A formula is an equation that contains two or more variables. We will refer to the equation a
Slide 10
Slope of a Non-vertical Line
2 1
2 1
y ym
x x
as the slope formula.
Sign of rise or run
run is positive run is negativerise is positiverise is negative
Direction (verbal)
goes to the rightgoes to the left
goes upgoes down
(graphical)
Definition
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slope of the line that contains the points (1, 2) and (5, 4).
(x1, y1) = (1, 2)
(x2, y2) = (5, 4).
Slide 11
Finding the Slope of a Line
4 2 2 15 1 4 2
m
Example
Solution
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
A common error is to substitute the slope formula incorrectly:
Slide 12
Finding the Slope of a Line
Correct Incorrect Incorrect
2 1 2 1 2 1
2 1 1 2 2 1
y y y y x xm m m
x x x x y y
Warning
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slope of the line that contains the points (2, 3) and (5, 1).
Slide 13
Finding the Slope of a Line
rise 2 2run 3 3
m
By plotting points, the run is 3 and the rise is –2.
Example
Solution
Finding a Line’s Slope
Lehmann, Intermediate Algebra, 3edSection 1.3
Increasing: Positive Slope Decreasing: Negative Slope
Slide 14
Definition
Positive risePositive run
m =
= Positive slope
negative risepositive run
m =
= negative slope
Increasing and Decreasing Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slope of the line that contains the points (– 9 , –4) and (12, –8).
Slide 15
Finding the Slope of a Line
8 4 8 4 4 412 9 12 9 21 21
m
•The slope is negative
•The line is decreasing
–
Example
Solution
Increasing and Decreasing Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slope of the two lines sketched on the right.
Slide 16
Comparing the Slopes of Two Lines
For line l1 the run is 1 and the rise is 2.
rise 12
run 2m
Example
Solution
Increasing and Decreasing Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
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.
Slide 17
Comparing the Slopes of Two Lines
rise 44
run 1m
For line l2 the run is 1 and the rise is 4.
Solution Continued
Increasing and Decreasing Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slope of the line that contains the points (2, 3) and (6, 3).
Slide 18
Investigating Slope of a Horizontal Line
Plotting the points (above) and calculating the slope we get 3 3 0
06 2 4
m
The slope of the horizontal line is zero, no steepness.
Example
Solution
Horizontal and Vertical Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slope of the line that contains the points (4, 2) and (4, 5).
Slide 19
Investigating the slope of a Vertical Line
Plotting the points (above) and calculating the slope we get
5 2 3, division by zero is undefined.
4 4 0m
The slope of the vertical line is undefined.
Example
Solution
Horizontal and Vertical Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
• A horizontal line has slope of zero (left figure).
• A vertical line has undefined slope (right figure).
Slide 20
Property
Property
Horizontal and Vertical Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
Two lines are called parallel if they do not intersect.
Slide 21
Finding Slopes of Parallel Lines
Find the slopes of the lines l1 and l2 sketch to the right.
Definition
Example
Parallel and Perpendicular Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
• Both lines the run is 3, the rise is 1
• The slope is,
Slide 22
Finding Slopes of Parallel Lines
• It makes sense that the nonvertical parallel lines have equal slope
• Since they have the same steepness
rise 1run 3
m
Solution
Parallel and Perpendicular Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
If lines l1 and l2 are nonvertical parallel lines on the same coordinate system, then the slopes of the lines are equal: m1 = m2Also, if two distinct lines have equal slope, then the lines are parallel.
Slide 23
Property
Two lines are perpendicular is they intercepts at a 90o angle.
Property
Definition
Parallel and Perpendicular Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
Find the slopes of the perpendicular lines l1 and l2.
Slide 24
Finding Slopes of Perpendicular Lines
• The slope of line l1 is m1 = 2/3 and l2 is m2 = –3/2
Example
Solution
Parallel and Perpendicular Lines
Lehmann, Intermediate Algebra, 3edSection 1.3
If lines l1 and l2 are nonvertical perpendicular lines, then the slope of one line is the opposite of the reciprocal of the slope of the other line:
Also, if the slope of one line is the opposite of the reciprocal of another line’s slope, then the lines are perpendicular.
Slide 25
Property
21
1m
m
Property
Parallel and Perpendicular Lines
Lehmann, Intermediate Algebra, 3edSection 1.3 Slide 26
Finding Slopes of Parallel Lines
Line l1 has a slope of
1.If l2 has is parallel to l1, find the slope of l2.
2.If l3 is perpendicular to l1, find the slope of l3.
1. The slopes of l1 and l2 are equal, so l2 has a slope
2. The slope of l3 is the opposite of the reciprocal of . or
3.
7
3.
7
3,
73
.7
Example
Solution
Parallel and Perpendicular Lines