5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

46 STEP 4. Review the Knowledge You Need to Score High

5.1 Lines

Main Concepts:Slope of a Line, Equations of a Line, Parallel and Perpendicular Lines

Slope of a Line

Given two pointsA(x 1 ,y 1 ) andB(x 2 ,y 2 ), theslopeof the line passing through the two given

points is defined as

m=

y 2 −y 1

x 2 −x 1

where (x^2 −x 1 )= 0

Note that if (x 2 −x 1 )=0, thenx 2 =x 1 , which implies that pointsAandBare on a vertical

line parallel to they-axis, and thus, the slope isundefined.

Example 1

Find the slope of the line passing through the points (3, 2) and (5,−4).

Using the definitionm=

y 2 −y 1

x 2 −x 1

, the slope of the line ism=

− 4 − 2

5 − 3

=

− 6

2

=−3.

Example 2

Find the slope of the line passing through the points (−5, 3) and (2, 3).

The slopem=

3 − 3

2 −(−5)

=

0

2 + 5

=

0

7

=0. This implies that the points (−5, 3) and (2, 3) are

on a horizontal line parallel to thex-axis.

Example 3

Figure 5.1-1 is a summary of four different orientations of lines and their slopes:

y

m > 0 m < 0 m = 0

Horizontal line

Parallel to x-axis

m is undefined

Vertical line

Parallel to y-axis

0

y

0 x

y

0 x

y

x 0 x

Figure 5.1-1

Equations of a Line

y=mx+b Slope-intercept formof a line wheremis its slope andbis they-intercept.

y−y 1 =m(x−x 1 )Point-slope formof a line wheremis the slope and (x 1 ,y 1 ) is a point on

the line.

Ax+By+C= 0 General formof a line whereA,B, andCare constants andAandB

are notbothequal to 0.