Algebra 1- Slope

Things that involve slope!

Slope is the ratio between vertical rise and horizontal run between any 2 points on the same line.

Slope= vertical rise ÷ horizontal run

The Slope Ratio

Find the slope of a hill that has a vertical rise of 40ft and a horizontal run of 200 ft.

(Let m represent slope)

Real Life Examples

Vertical Rise= 40 ft

Horizontal Run= 200 ft

m= vertical/horizontal

m= 40/200 = 1/5

The slope m of a line that passes through points (x₁, y₁) and (x₂, y₂) is

The Slope of a Line

m=riserun=

change in ychange in x

y₂-y₁=x₂-x₁

Where (x₁, y₁) and (x₂, y₂) are any 2 points on the same line and x₁≠x₂.

Let (x₁, y₁)= (1,0) and (x₂,y₂)=(3,4)Find m.

Positive Slope

y₂-y₁x₂-x₁

m= = 4-03-1

=24 =2

There is a positive slope of 2 therefore the line rises from left to right.

Negative Slope

Let (x₁, y₁)= (0,3) and (x₂,y₂)=(4,1)

Find m.y₂-y₁x₂-x₁

m= = 1-34-0

=4-2

= -1

There is a negative slope of -1/2 therefore the line falls from left to right.

2

Zero Slope

Let (x₁, y₁)= (1,2) and (x₂,y₂)=(4,2)Find m.

y₂-y₁x₂-x₁

m= = 2-24-1

=30 =0

There is a zero slope therefore the line is horizontal. We can compare this to walking across the floor.

Undefined Slope

Let (x₁, y₁)= (3,-1) and (x₂,y₂)=(3,3)Find m.

y₂-y₁x₂-x₁

m= = 3-⁻15-5

=04 =Undefin

ed!The slope is undefined because division by 0 is undefined and the expression 4/0 has no meaning. This line is vertical and we can compare it to walking up a wall, impossible.

Summary: Slope of Lines

Positive- rises left to right

Negative- falls left to right

Zero- horizontal

Undefined- vertical

y=mx+b where m and b are constants, then the graph of that equation is a line with slope m and y-intercept b.

The bigger the slope, the steeper the line.

Even of you choose completely different points on the same line, the slope will remain constant.

Slope Intercept Form of a Line

Example

Find the slope and y-intercept of the line and then graph the line.

3x + 5y −15 = 0

y= x+3

-35

Solve for y by subtracting 5y from both side and then divinding both sides by -5.

slope

y-intercept

The top of the ladder is 12ft from the ground. The base of the ladder is 5ft from the wall. What is the slope of the ladder?

Ladder

rise 12run5

=12ft

5ft

Making Sense of Slope!

A

D

CBE

Choose different pairs of points on the line and find their slope.What do you notice? What conclusion can you draw?

Making Sense of Slope! Anwsers

mAB=

mBC=

mCD=

mDE=

12 12 12 12

Now, based on the conclusion that the slope is ½ between each set of points, complete the sentence:No matter what pair of points you choose on a line, the ______ remains constant.

*slope

The Slope of the Road

The slope of the road is the vertical rise divided by the horizontal run. If the vertical rise is 24 feet for a horizontal run of 1 mile, determine the slope of the road. *Hint- 1 mile=5280 feet 24ft 24ft

1 1mi 5280ft 220

==

*This means that for every 1ft that you move up, you also move 220ft forward.

CREDITS

Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 1: Concepts and Skills. Teacher's ed. Evanston, IL: McDougal Littell, 2001. 229-233. Print.

Gustafson, R. David, and Peter D. Frisk. Beginning Algebra. Instructor's 3rd ed. Pacific Grove, CA: Brooks/Cole, 1992. 285-289. Print.

Håkan Dahlström, Six Flags Holland, 2/28/11 via Flickr, Creative Commons Attribution

Chad K., Hills Outside Silgo, 2/28/11 via Flickr, Creative Commons Attribution

Tony the Misfit, Snowboarding Along the East River, 2/28/11 via Flickr, Creative Commons Attribution

Pocius, Rooftops in Reykjavik, Iceland, 2/28/11 via Flickr, Creative Commons Attribution

Robert Couse-Baker, Up and Down the Ladder, 2/28/11 via Flickr, Creative Commons Attribution

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