Sullivan Algebra and Trigonometry: Section 2.3

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 From and in General Form.

Let and be two distinct points with . The slope m of the non-vertical line L containing P and Q is defined by the formula

( )11, yxP = ( )22 , yxQ =

21 xx ≠

my yx x

x x= −−

≠2 1

2 11 2

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

21 xx =

x x2 1−

y y2 1−P = ( , )x y1 1

Q = ( , )x y2 2

y

x

Slope can be though of as the ratio of the vertical change ( ) to the horizontal change ( ), often termed “rise over run”.

y y2 1−

x x2 1−

P = ( , )x y1 1

Q = ( , )x y1 2

Ly

x

If , then is zero and the slope is undefined. Plotting the two points results in the graph of a vertical line with the equation .

21 xx = x x2 1−

1xx =

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 = -3, horizontal change = 2).

y

x

(1,4)

2

-3

(3,1)

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

y

x

x = 2

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

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

Example: Graph the line y=4.

y

x

y = 4

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 real numbers and A and B are not both 0.

The equation of a line L is in slope-intercept form with it 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

m=32

b =3