1.2: Slope

-slope formula

M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope.

GSE

Understanding Slope

• If a line rises as you move from left to right, then the slope is positive.

2

-2

F: (1, 2)

E: (-2, -2)

Riding a bike uphill

Understanding Slope

• If a line drops as you move from left to right, then the slope is negative.

4

2

H: (1, 2)G: (-2, 3)

Skiing Downhill

Understanding Slope

• A horizontal line has zero slope: m = 0 2

K: (2, 1)J: (-2, 1)

Running on a flat surface like a track Or any athletic fieldRunning on a flat surface like a track Or any athletic fieldRunning on a flat surface like a track Or any athletic field

Understanding Slope

• A vertical line has no slope: m is undefined.

4

2

N: (2, 1)

M: (2, 3)

Running into a wall, youcant get past itRunning into a wall, youcant get past it

Slope Formula

The slope of a line through the points (x1, y1) and (x2, y2) is as follows:

yy22 –– yy11 xx22 –– xx11

m =

Ex.

Find the slope of the line that passes through (–2, –3) and (4, 6).

Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6).

6 – (–3)4 – (–2)

Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.

=y2 – y1

x2 – x1

96= 3

2=

The slope of the line that passes through (–

2, –3) and (4, 6) is . 32

*** Always reduce your fractions****

Understanding Slope

• Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel.)

4

2

-2

-4

-6

5

D: (4, -1)

C: (-2, -4)

B: (3, 3)

A: (-1, 1)

Understanding Slope

• The slope of AB is:

• The slope of CD is:

• Since m1=m2, AB || CD

4

2

-2

-4

-6

5

D: (4, -1)

C: (-2, -4)

B: (3, 3)

A: (-1, 1) 1

3 1 2 1

3 1 4 2m

2

1 4 3 1

4 2 6 2m

Perpendicular Lines

• (┴)Perpendicular Lines- 2 lines that intersect forming 4 right angles

Right angle

Slopes of Lines

• In a coordinate plane, 2 non vertical lines are iff the product of their slopes is -1.

• This means, if 2 lines are their slopes are opposite reciprocals of each other; such as ½ and -2.

• Vertical and horizontal lines are to each other.

Example• Line l passes through (0,3) and (3,1).

• Line m passes through (0,3) and (-4,-3).

Are they ?

Slope of line l =

Slope of line m =

l m

30

13

3

2-or

3

2

40

33

2

3or

4

6

Opposite Opposite Reciprocals!Reciprocals!

Equation of a line in slope intercept form (y = mx+b)

Now that we know how to find slope given any two points, we cangenerate an equation of the line connecting the two points.

Example : points (3,2) and (6,9)

2nd example

Slope-Intercept Form (y = mx+b)

• Find the equation of a line passing through the points P(0, 2) and Q(3, –2).

2

-2

Q: (3, -2)

P: (0, 2)

•Is this line parallel to a line with the equation

43?

3y x

a) Find the equation of a line that passes through the points G ( -4, 5) and H (-8, 3)

b) Write the equation of a line that passes through point P (1, -2) and is perpendicular to the one from part a

Slope Graphically

You can always count ! (not suggested as you advance In your math courses)

Homework

Another application of Slope

run

rise

Slope is rise

run

The steepness of the ramp matters to people who need to walk on it.

or Rise:Run

No minimum pitch in the code.

Required slope would be determined by roofing materials.

Most shingle type roofs require minimum 4 in 12 pitch.

You can go as low as 2.5 in 12 with special underlayment.

Some local jurisdictions with heavy snowfall require a steep pitched roof for obvious reasons.

Rhode Island Code

• The Maximum slope for wheel chair ramps is 1:8 .

The house has a platform 6 ft off the ground. If RI code says the maximum slope is 1:8What could the lengths of the run be for a ramp?

Assignments