copyright © 2015, 2008, 2011 pearson education, inc. section 1.3, slide 1 chapter 1 linear...
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Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 3 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.TRANSCRIPT
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 1
Chapter 1Linear Equations and Linear Functions
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 2
1.3 Slope of a Line
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 3
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 4
Comparing the Steepness of Two Objects
• Ratio of vertical distance to the horizontal distance:
• Ladder A:
• Ladder B:
• So, Ladder B is steeper.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 5
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.
vertical distancehorizontal distance
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 6
Example: Comparing the Steepness of Two Roads
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 7
Solution
Sketches of the two roads are shown below. Note that the distances are not drawn to scale.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 8
Solution
Calculate the approximate ratio of the vertical distance to the horizontal distance for each road:
Road B is a little steeper because Road B’s ratio is greater than Road A’s.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 9
Grade of a Road
The grade of a road is the ratio of the vertical distance to the horizontal distance, written as a percentage.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 10
Slope of a nonvertical line
Definition Let (x1, y1) and (x2, y2) be two distinct points of a
nonvertical line. The slope of the line is
In words, the slope of a nonvertical 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).
2
2 1
1
horizontal changevertical change rise
runy yx x
m
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 11
SlopeA formula is an equation that contains two or more variables. We refer to the equation as the slope formula. Here we list the directions associated with the signs of rises and runs:
2 1
2 1
y ymx x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 12
Example: Finding the Slope of a Line
Find the slope of the line that contains the points (1, 2) and (5, 4).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 13
Solution
Using the slope formula, where (x1, y1) = (1, 2) and (x2, y2) = (5, 4), we have
4 2 2 15 1 4 2
m
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 14
Solution
By plotting the points, we find that if the run is 4, then the rise is 2. So, the slope is
which is our result from using the slope formula.
run1r se ,2
i 24
m
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 15
Slope
Warning
It is a common error to substitute into the slope formula incorrectly.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 16
Example: Finding the Slope of a Line
Find the slope of the line that contains the points (2, 3) and (5, 1).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 17
Solution
1 3 2 25 2 3 3
m
By plotting the points, we find that if the run is 3, then the rise is –2 . So, the slope is
which is our result from using the slope formula.
223
,3
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 18
Increasing and Decreasing Lines
An increasing line has positive slope
A decreasing line has negative slope
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 19
Example: Finding the Slope of a Line
Find the slope of the line that contains the points (–9, –4) and (12, –8).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 20
Solution
8 ( 4) 8 4 412 ( 9) 12 9 21
m
Since the slope is negative, the line is decreasing.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 21
Example: Comparing the Slopes of Two Lines
Find the slopes of the two lines sketched at the right. Which line has the greater slope? Explain why this makes sense in terms of the steepness of a line.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 22
SolutionFor line l1, if the run is 1, the rise is 2. So,
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 23
SolutionFor line l2, if the run is 1, the rise is 4. So,
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 24
Solution
Note that the slope of line l2 is greater than the slope of line l1, which is what we would expect because line l2 looks steeper than line l1.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 25
Example: Investigating the Slops of a Horizontal Line
Find the slope of the line that contains the points (2, 3) and (6, 3).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 26
Solution
We plot the points (2, 3) and (6, 3) and sketch the line that contains the points.
3 3 0 06 2 4
m
So, the slope of the horizontal line is zero, because such a line has “no steepness.”
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 27
Example: Investigating the Slops of a Vertical Line
Find the slope of the line that contains the points (4, 2) and (4, 5).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 28
Solution
We plot the points (4, 2) and (4, 5) and sketch the line that contains the points.
5 2 34 4 0
m
Since division by zero is undefined, the slope of the vertical line is undefined.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 29
Slopes of Horizontal and Vertical Lines
• A horizontal line has slope equal to zero.• A vertical line has undefined slope.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 30
Parallel Lines
Two lines are called parallel if they do not intersect.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 31
Example: Finding Slopes of Parallel Lines
Find the slopes of the parallel lines l1 and l2.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 32
SolutionFor both lines, if the run is 3, the rise is 1. So, the slope of both lines is
rise 1run 3
m
It makes sense that nonvertical parallel lines have equal slope, since parallel lines have the same steepness.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 33
Slopes of Parallel Lines
If lines l1 and l2 are nonvertical parallel lines on the same coordinate system, then the slopes of the lines are equal:
m1 = m2
Also, if two distinct lines have equal slope, then the lines are parallel.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 34
Perpendicular Lines
Two lines are called perpendicular if they intersect at a 90° angle.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 35
Example: Finding Slopes of Perpendicular Lines
Find the slopes of the perpendicular lines l1 and l2.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 36
SolutionWe see that the slope of line l1 is
and the slope of line l2 is
123
m
23 3
2.
2m
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 37
Slopes of Perpendicular Lines
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.
21
1mm
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 38
Example: Finding Slopes of Parallel and Perpendicular Lines
A line l1 has slope 3.7
1. If line l2 is parallel to line l1, find the slope of line l2.
2. If line l3 is perpendicular to line l1, find the slope of line l3.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 39
Solution
1. The slopes of lines l2 and l1 are equal, so line l2
3.7
2. The slope of line l3 is the opposite of the3,7
has slope
reciprocal of or 7.3