Sullivan Algebra and Trigonometry: Section 2.3

LinesObjectives

• 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.

•Identify the Slope and the y Intercept of a Line from its Equation.

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 1P = ( , )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

Some Important Facts about slope:

1. When the slope of a line is positive, the line slants upward from left to right. (L1)

2. When the slope of a line is negative, the line slants downward from left to right. (L2)

3. When the slope is zero, the line is horizontal. (L3)

4. When the slope is undefined, the line is vertical. (L4)

L1L2

L3

L4

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

m32

b 3