6-1 Slope Objectives 1. find the slope of a line 2.use rate of change to solve problems.
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Transcript of 6-1 Slope Objectives 1. find the slope of a line 2.use rate of change to solve problems.
![Page 1: 6-1 Slope Objectives 1. find the slope of a line 2.use rate of change to solve problems.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed15503460f94be01e5/html5/thumbnails/1.jpg)
6-1 Slope
Objectives
1. find the slope of a line2. use rate of change to solve problems
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What is the meaning of this sign?
1. Icy Road Ahead
2. Steep Road Ahead
3. Curvy Road Ahead
4. Trucks Entering Highway Ahead
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What does the 7% mean?
7% is the slope of the road. It means the road drops 7 feet vertically for every 100 feet
horizontally.
7%
So, what is slope???Slope is the steepness of a line.
7 feet
100 feet
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Slope can be expressed different ways:
A line has a positive slope if it is going uphill from left to right.
A line has a negative slope if it isgoing downhill from left to right.
2 1
2 1
( ) vertical change
( ) horizontal change
y y risem
x x run
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When given the graph, it is easier to apply “rise over run”.
1) Determine the slope of the line.
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Start with the lower point and count how much you rise and run to get to the other
point!
Determine the slope of the line.
6
3
run
3
6= =
1
2
rise
Notice the slope is positive AND the line increases!
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2) Find the slope of the line that passes through the points (-2, -2) and (4, 1).
(1 ( 2))
(4 ( 2))m
2 1
2 1
( )
( )
y ym
x x
(1 2)
(4 2)
When given points, it is easier to use the formula!
y2 is the y coordinate of the 2nd ordered pair (y2 = 1)
y1 is the y coordinate of the 1st ordered pair (y1 = -2)
13
6 2
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Did you notice that Example #1 and Example #2 were the same problem
written differently?
(-2, -2) and (4, 1)6
31
2slope
You can do the problems either way!Which one do you think is easiest?
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Find the slope of the line that passes through (3, 5) and (-1, 4).
1. 4
2. -4
3. ¼
4. - ¼
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3) Find the slope of the line that goes through the points (-5, 3)
and (2, 1).
)5(2
31
m
52
31
mm y2 y1
x2 x1
7
2m
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Determine the slope of the line shown.
1. -2
2. -½
3. ½
4. 2
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Determine the slope of the line.
The line is decreasing (slope is negative).
2
-1
rise
run
2
1 2
Find points on the graph. Use two of them and apply rise over run.
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What is the slope of a horizontal line?
The line doesn’t rise!
m 0
number0
All horizontal lines have a slope of 0.
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What is the slope of a vertical line?
The line doesn’t run!
All vertical lines have an undefined slope.
m number
0undefined
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Remember the word “VUXHOY”
Vertical lines
Undefined slope
X = number; This is the equation of the line.
Horizontal lines
O - zero is the slope
Y = number; This is the equation of the line.
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1
Draw a line through the point (2,0) that has a slope of 3.
1. Graph the ordered pair (2, 0).
2. From (2, 0), apply rise over run (write 3 as a fraction).
3. Plot a point at this location.
4. Draw a straight line through the points.
3
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The slope of a line that goes through the points (r, 6) and (4, 2) is 4. Find r.
To solve this, plug the given information into the formula
m
( y2 y
1)
(x2 x
1)
.
2 64
4 r
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4 4
1 4 r
To solve for r, simplify and write as a proportion.
4 4
1 4 r
2 6
44 r
Cross multiply.
1(-4)4(4 – r) =
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Simplify and solve the equation.4(4 – r) = 1(-4)16 – 4r = -4
-16 -16
-4r = -20-4 -4
r = 5
The ordered pairs are (5, 6) and (4, 2)
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Slope can be used to describe a rate of change. Rate of change tells, on average, how a quantity is changing over time.
change in quantityriserun change in t ime
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Travel The graph to the right shows the number of U.S. passports issued in 1991, 1995, and 1999.
Find the rates of change for 1991-1995 and 1995-1999.
Use the formula for slope.
millions of passportsyears
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1991-1995:
Substitute.
Answer: The number of passports issued increased by 1.9 million in a 4-year period for a rate of change of 475,000 per year.
Simplify.
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1995-1999:
Substitute.
Simplify.
Answer: Over this 4-year period, the number of U.S. passports issued increased by 1.4 million for a rate of change of 350,000 per year.
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Explain the meaning of slope in each case.
Answer: For 1991-1995, on average, 475,000 more passports were issued each year than the last. For 1995-1999, on average, 350,000 more passports were issued each year than the last.
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How are the different rates of change shown on the graph?
Answer: There is a greater rate of change from 1991-1995 than from 1995-1999. Therefore, the section of the graph for 1991-1995 has a steeper slope.
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Airlines The graph shows the number of airplane departures in the United States in recent years.
a. Find the rates of changefor 1990-1995 and 1995-2000.
Answer: 240,000 per year; 180,000 per year
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b. Explain the meaning of the slope in each case.
Answer: For 1990-1995, the number of airplane departures increased by about 240,000 flights each year. For 1995-2000, the number of airplane departures increased by about 180,000 flights each year.
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c. How are the differentrates of change shownon the graph?
Answer: There is a greater vertical change for 1990-1995 than for 1995-2000. Therefore, the section of the graph for 1990-1995 has a steeper slope.