10.1 Slope of a Line

CORD Math

Mrs. Spitz

Fall 2006

Objective

• Find the slope of a line, given the coordinates of two points on the line.

• pgs. 403-404 #1-39 all

Assignment

Application

• Do you ever recall seeing a sign along the highway like the one above? These signs are designed to inform the driver that there is a steep hill ahead. If a hill has a grade of 6%, this means that for every 100 feet of horizontal change, there is a vertical change of 6 feet.

Ex. 1: How many feet does a road with a 6% grade drop in 3 miles?

Using .06 for 6%, the grade can be expressed as follows:

v)3(06.

h

v06.0

hh

vh )06(.

v18.0

vh06.

v represents the vertical change and h represents the horizontal change.

Multiply each side by h to isolate v.

Simplify

Substitute value h with 3

Multiply

Since 1 mile = 5,280 ft., 0.18(5280) or 950.4 feet. Thus, a road with a 6% grade drops 950.4 feet in 3 miles.

What is slope?

• Sometimes the vertical distance is referred to as the rise, and the horizontal distance is referred to as the run. The ratio of rise to run is called slope. The slope of a line describes its steepness of rate of change.

• On the graph below, the line passes through the origin, (0, 0), and (4, 3). The change in y or rise is 3, while the change in x or run is 4. Therefore the slope of this line is .

4

3

rise

runslope =

change in y

change in x=

Definition of Slope

The slope m of a line is the ratio of the change in y to the corresponding change in x.

slope = or m =

Change can also be expressed with the Greek letter, (delta).

m =

change in y

change in x

change in y

change in x

y

x

Ex. 2: Determine the slope of each line.

change in y

change in x=

2

3

change in y

change in x=

2

-1= -2

change in y

change in x=

0

1= 0

POSITIVE SLOPE NEGATIVE

SLOPE

SLOPE = 0Vertical lines have no slope.

Ex. 3: Determine the slope of the line containing the points with the coordinates listed in the table.

• Notice that y increases 3 units for every 2 units that x increases.

These examples suggest that the slope of a non-vertical line can be determined from the coordinates of any two points on the line.

Determining Slope Given Two Points

• Given the coordinates of two points on a line, (x1, y1) and (x2, y2), the slope, m, can be found as follows:

where x2 ≠ x1

,12

12

xx

yym

NOTE: y2 is read “y sub 2.” The 2 is called a subscript.

Ex. 4: Determine the slope of a line passing through (3, -9) and (4, -12).

12

12

xx

yym

34

)9(12

m

1

3m

3m

Formula for slope of a line.

Substitute values into formula

Simplify

Simplify

What happens if you mess up and use the other point first?

• The difference of the y-coordinates was expressed as -12 – (-9). Suppose -9 – (-12) had been used as the change in y-coordinates and 3 – 4 had been used as the change in x-coordinates. Since -9 – (-12)/3-4 is also equal to -3, it does not matter which point is chosen to be (x1, y1). However, the coordinates of both points must be used in the same order.

Ex. 5: Determine the value of r so the line through (r, 4) and (9, -2) has a slope of

12

12

xx

yym

r

9

42

2

3

Formula for slope of a line.

Substitute values into formula

Simplify

Add 27 to both sides

2

3

r

9

6

2

3

)2(6)9(3 r

12327 r153 r5r

Means-extremes property

Distributive property

Simplify