Section 2.2: Lines and Rates of Change
Learning Objectives:
1.
2.
3.
4.
5.
Graph linear equations using the intercept method
Find the slope of a line and interpret it as a rate of change
Graph horizontal and vertical lines
Identify parallel and perpendicular lines
Apply linear equations in context
The Graph of a Linear Equation
Example: Graph 4 x − 7 y =
28 using intercepts
Example: Graph =
y 3x − 4
Slope
The Slope Formula
Given two points P1 = ( x1 , y1 ) and P2 = ( x2 , y2 ) , the slope of the nonvertical line
through P1 and P2 is m =
y2 − y1
, where x2 ≠ x1 .
x2 − x1
∆y
, between the quantities measured
∆x
along each axis. The slope formula is often read as “the change in y over the change in
x”.
The slope value expresses a rate of change,
Example: Find the slope of the line through the given points, then use m =
∆y
to find an
∆x
additional point on the line.
a.
( 3, 0 )
and ( 0, 4 )
b.
( −2, −4 )
and (1,8 )
Example: Although leaning made the Tower of Pisa famous, it could also one day bring
it crashing to the ground. To help protect the tower, restorations were made in the
1990s to shift the tower more upright. The top of the building is now 55.86 m from the
ground and has a horizontal displacement of 3.9 m. Find the slope ratio for the Leaning
Tower of Pisa and discuss its meaning in this context.
Horizontal and Vertical Lines
Horizontal Lines
Vertical Lines
The equation of a horizontal line is
The equation of a vertical line is
y = k,
x = h,
where (0, k) is the y-intercept.
where (h, 0) is the x-intercept.
The Slope of a Horizontal Line
The Slope of a Vertical Line
The slope of any horizontal line
The slope of any vertical line
is zero.
is undefined.
Example: Compute the slope of the line through the given points.
a.
( 3, −4 ) , ( −5, 2 )
b.
( −2,3) , ( 5,3)
Parallel and Perpendicular Lines
Parallel Lines
Perpendicular Lines