6-1 Slope

Objectives

1. find the slope of a line2. use rate of change to solve problems

What is the meaning of this sign?

1. Icy Road Ahead

2. Steep Road Ahead

3. Curvy Road Ahead

4. Trucks Entering Highway Ahead

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

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

When given the graph, it is easier to apply “rise over run”.

1) Determine the slope of the line.

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!

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

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?

Find the slope of the line that passes through (3, 5) and (-1, 4).

1. 4

2. -4

3. ¼

4. - ¼

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

Determine the slope of the line shown.

1. -2

2. -½

3. ½

4. 2

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.

What is the slope of a horizontal line?

The line doesn’t rise!

m 0

number0

All horizontal lines have a slope of 0.

What is the slope of a vertical line?

The line doesn’t run!

All vertical lines have an undefined slope.

m number

0undefined

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.

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

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

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) =

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)

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

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

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.

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.

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.

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.

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

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.

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.