4.5 Finding The Slope of a

Line

What is the meaning of this sign?

1. Icy Road Ahead

2. Steep Road Ahead

3. Curvy Road Ahead

4. Trucks Entering Highway Ahead

Slopes are commonly associated with mountains.

The slope we are studying is associated with the graph of a

line.

Definitions of Slope

• The tilt or inclination of a line

• The ratio of vertical change to horizontal change.

• The change in y over the change in x

ΔyΔx

Tilt or Inclination

FROM LEFT TO RIGHT!!!!

POSITIVE SLOPE

NEGATIVESLOPE

SLOPE IS ZERO

SLOPE ISUNDEFINED

y = #

x = #

Tell me slope of each line

POSITIVE NEGATIVE

ZEROPOSITIVE

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

UP or DOWN RIGHT

Graph (3,2) and (-1,-1)

Draw a line through the points.

Rise

3

4

Run

THE SLOPE IS

34

Find the slope of the following line.

The slope is…

1

2

Find the slope of the line.

The slope is….

-3

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

1) Determine the slope of the line.

€

3

6=

1

2

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

Find the slope of these lines.

• Black line 3

• Red Line 1

• Blue Line -1/2

Find the slope of these lines

• Orange line = 0

• Green Line = Undefined

Determining Slope

Rise=1

Slope= 12

Slope=3

Rise=6Run =2

Run =2Rise=12Run =4

Let’s find the slope of a line with two points, WITHOUT LOOKING AT A GRAPH.

REMEMBER THE SLOPE FORMULA???!!!!!

€

m =y2 − y1

x2 − x1

Determining Slope without graphing

(-4, -6)

(2, 5)

•Pick 2 points on the line

•Find the change in the Y-coordinates by subtracting(rise)

•Find the change in the X-coordinates by subtracting(run)

5-(-6)2-(-4)

= 11 6

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

3) Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 3

2 ( 5)m

1 3

2 5m

2

7m

x1 y1 x2 y2

Sample Problems:• Find the slope of the line thru the points given:

(-3,-1) and (-2,4)

(-3,4) and (2,-2)

m 4 1 2 3

5

15

m 2 4

2 3 6

5

x1 y1 x2 y2

x1 y1 x2 y2

Independent Practice: Calculate the slope for each pair of points. Classify each slope as positive, negative, zero, or undefined.

1. (2, 2) and (3, 5) 2. (0, 0) and (3, 0)

3. (-2, -1) and (-1, -4)

4. (2, 3) and (2, 7)

5. (-1, -1) and (5, 5)

6. (8, 4) and (6, 4)

Slope: 3 Classification: Positive

Slope: -3

Classification: Negative

Slope: 1 Classification: Positive

Slope: 0 Classification: Zero

Slope: Undefined Classification: Vertical

Slope: 0

Classification: Horizontal

EVERYONE GET A COMMUNICATOR!!!

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

If you know the slope and one point on the line you can draw the line.

Example: slope is 1/3 and the point (-2,-1)

Now you draw a line.

• Given: slope is 2/3 and the point is (1,0)

• Given: slope is –3 and the point is (–2,4)

• Given: slope is 1 and the point is (2,3)

Graph the line y = 3x +1,then tell me the slope of that line

after you graphed it.The slope is 3

x y

-2

-1

0

1

2

-5

-2

1

4

7

Graph the line 2x + y = 4,then tell me the slope of that line

after you graphed it.

x y

-2

-1

0

1

2

8

6

4

2

0

The slope is -2y = -2x + 4

Graph the line y = -5,then tell me the slope of that line

after you graphed it.

x y

-2

-1

0

1

2

-5

-5

-5

-5

-5

The slope is 0

Graph the line y = 2x,then tell me the slope of that line

after you graphed it.

x y

-2

-1

0

1

2

-4

-2

0

2

4

The slope is 2

Graph the line y = 3x - 2,then tell me the slope of that line

after you graphed it.

x y

-2

-1

0

1

2

-8

-5

-2

1

4

The slope is 3

Graph the line 2x + 3y = 6,then tell me the slope of that line

after you graphed it.

x y

-6

-3

0

3

6

6

4

2

0

-2

The slope is -2/3

€

y =−2

3x + 2

Graph the line y = -x - 2,then tell me the slope of that line

after you graphed it.

x y

-2

-1

0

1

2

0

-1

-2

-3

-4

The slope is -1

Graph the line 3x – 4y = 12,then tell me the slope of that line

after you graphed it.

x y

-8

-4

0

4

8

-9

-6

-3

0

3

The slope is 3/4

€

€

y =3

4x − 3

Graph the line x – y = 5,then tell me the slope of that line

after you graphed it.

x y

-2

-1

0

1

2

-7

-6

-5

-4

-3

The slope is 1y = x - 5

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.1(-4) = 4(4 – r)

-4 = 16 – 4r-16 -16

-20 = -4r-4 -45 = r

The ordered pairs are (5, 6) and (4, 2)