Slopes of Equations and Lines

Honors GeometryHonors GeometryChapter 2Chapter 2

Nancy Powell, 2007Nancy Powell, 2007

Objectives:•Calculate and interpret the slope of a line

•Graph lines given a point and the slope

•Use the point-slope form of a line

•Find the equation of a line given two points

•Write the equation of a line in slope-intercept form

Let and be two distinct points with . The slope slope mm of the non-vertical line is defined by the formula

11, yxP 22 , yxQ

21 xx

2 11 2

2 1

y y

m x xx x

If , then is a vertical line and the slope m is undefinedundefined (since this results in division by 0).

21 xx

PQ�������������� �

PQ�������������� �NOTE:NOTE:

x x2 1

y y2 1

P = ( , )x y1 1

Q = ( , )x y2 2

y

x

Slope can be thought of as the ratio of the vertical change ( ) to the horizontal change ( ), often termed

y y2 1

x x2 1

RISERISE

RUNRUN

RISEover RUN

x x2 1

y y2 1

P = ( , )x y1 1

Q = ( , )x y2 2

y

x

+RISE+RISE

+RUN+RUN

2 11 2

2 1

y y

m x xx

RISE

RUN x

Because this line rises from left to right with a +RISE+RISE and a +RUN +RUN , we’ll call this a RISINGRISING line.

x

x x2 1

y y2 1P = ( , )x y1 1

Q = ( , )x y2 2

y

x

-RISE-RISE

+RUN+RUN

2 11 2

2 1

y y

m x xx

RISE

RUN x

Because this line falls from left to right or has a

– RISE – RISE and a +RUN +RUN, we’ll call this a FALLINGFALLING line.

1P=( , )a y

2Q=( , )a y

y

x

If , then is zero and the slope is UNDEFINEDUNDEFINED. (+ or - RISE + or - RISE and Zero RUN) Zero RUN)

21 xx x x2 1

x aPlotting the two points results in the graph of a VERTICAL line with the equation .

x1

2P=( , )x b1Q=( , )x b

y

xIf , then is zero and the slope is ZEROZERO.

( Zero RISE Zero RISE and + or - RUN) + or - RUN)

1 2y y 2 1y y

y bPlotting the two points results in the graph of a HoRIZ0ntal line with the equation .

Example: Find the slope of the line joining the points (3,8) and (-1,2).

x y x y1 1 2 23 8 1 2, , , ,

my yx x

2 1

2 1

m 2 81 3

64

32

Example: Draw the graph of the line passing through (1,4) with a slope of -3/2.

Step 1: Plot the given point.

Step 2: Use the slope to find another point on the line (vertical change = , horizontal change = ).

y

x

(1,4)

2

-3

(3,1)

Find another point on this line.

( ____ , ____ )

- 3 2

Example: Draw the graph of the equation x = 2.

y

x

x = 2

What do you know about the slope of this line?

The slope is undefined.

Theorem: Point-Slope Form of an Equation of a Line

An equation of a non-vertical line of slope m that passes through the point (x1, y1) is:

y y m x x 1 1

Example: Find an equation of a line with slope -2 passing through (-1,5).

m x y 2 1 51 1 and , ,

y y m x x 1 1

y x 5 2 1

y x 5 2 2

y x 2 3

1 1y y m x x

A HoRIZ0ntal line is given by an equation of the form y = b, where (0,b) is the y-intercept. Slope =

Example: Graph the line y=4.

y

x

y = 4

0

The equation of a line L is in general form with it is written as

Ax By C 0

where A, B, and C are three integers and A and B are not both 0.

The equation of a line in slope-intercept form is written as

y = mx + b

where m is the slope of the line and (0,b) is the y-intercept.

Example: Find the slope m and y-intercept (0,b) of the graph of the line 3x - 2y + 6 = 0.

3x - 2y + 6 = 0

-2y = -3x - 6

y x 32

3

3

2m slope 3b

Solve for y in terms of x and find the slope and the y-intercept!

So, (0,3) is the y-intercept

Objectives (part 2):•Define parallel, perpendicular, and oblique lines

• Find equations of parallel Lines

• Find equations of perpendicular Line

• Determine whether lines are parallel, perpendicular or oblique

Definition: Parallel Linesl

m

n

Two distinct non-vertical lines are parallelparallel if and only if they

1. are in the same planesame plane,

2. have the same slopesame slope and

3. have different y-interceptsdifferent y-intercepts.

Find the equation of the line parallel to y = -3x + 5 passing through (1,7).

Since parallel lines have the same slope, the slope of the parallel line must also be equal to -3.

y y m x x 1 1

7 3 1y x

7 3 3y x

3 10y x

Isn’t thisSlope-

intercept form?

x1 = 1 and y1 = 7

Definition: Perpendicular LinesTwo lines are said to be perpendicularperpendicular if they

intersect at a right angle.

Two non-vertical lines are perpendicular if

•Their slopes are opposite reciprocalsslopes are opposite reciprocals of each other like

•The product of their slopes is -1product of their slopes is -1. 121 mm

21

1m

m

Example: Find the equation of the line perpendicular to y = -3x + 10 passing through (1,5).

Slope of perpendicular line: 1 1

3 3m

y x 513

1

1 15

3 3y x

y x 13

143

l

m

n 1 1y y m x x

Isn’t thisSlope-

intercept form?

Definition: Oblique Lines

o Oblique lines are lines that intersect but are Oblique lines are lines that intersect but are not perpendicular to each other.not perpendicular to each other.

o Oblique lines do not form right angles.Oblique lines do not form right angles.o This means that the slopes of oblique lines This means that the slopes of oblique lines

are not the same and they are not opposite are not the same and they are not opposite reciprocals. reciprocals.

l

m

n

Which of the following pairs of slopes are slopes of Which of the following pairs of slopes are slopes of o Oblique lines? Oblique lines? o Parallel lines? Parallel lines? o Perpendicular lines? Perpendicular lines?

How do you know?How do you know?

a. 2/3 and 3/2 a. 2/3 and 3/2 c. 2 and - 0.5 c. 2 and - 0.5 e. 1.25 and -1.25 e. 1.25 and -1.25 g. ¾ and 0.75g. ¾ and 0.75

Slopes and Oblique, Parallel, and Perpendicular Lines

a. 2/3 and 3/2a. 2/3 and 3/2 b. -2/3 and -3/2b. -2/3 and -3/2d.d. 5/4 and 2/35/4 and 2/3f.f. 7/4 and 3/77/4 and 3/7h. 0 and undefinedh. 0 and undefined

b. -2/3 and -3/2b. -2/3 and -3/2d.d. 5/4 and 2/35/4 and 2/3

e. 1.25 and -1.25e. 1.25 and -1.25 f.f. 7/4 and 3/77/4 and 3/7g. ¾ and 0.75g. ¾ and 0.75

c. 2 and - 0.5c. 2 and - 0.5

h. 0 and undefinedh. 0 and undefined