Download §1.3 Lines and Linear Functions

Slope and equations
Linear models
Parallel and perpendicular lines
§1.3
Lines and Linear Functions
Tom Lewis
Spring Semester
2017
Slope and equations
Linear models
Outline
Slope and equations
Linear models
Parallel and perpendicular lines
Parallel and perpendicular lines
Slope and equations
Linear models
Parallel and perpendicular lines
2
3
6
9
Figure: The slope is invariant: 2/3 = 6/9.
The slope of a line
Informally, the slope of a line is the ratio of the change in y to a
corresponding change in x .
Slope and equations
Linear models
Parallel and perpendicular lines
Definition
The slope of the line passing through the points A(x1 , y1 ) and
B (x2 , y2 ) is given by
m=
∆y
y2 − y1
=
,
∆x
x2 − x1
provided x2 6= x1 . If x2 = x1 , then the line is vertical and its slope
is undefined.
Slope and equations
Linear models
Parallel and perpendicular lines
Definition (Point-slope form)
The equation
y − y1 = m(x − x1 )
is an equation of a line passing through the point A(x1 , y1 ) and
having slope m. It is called the point-slope form of the equation of
the line.
Slope and equations
Linear models
Parallel and perpendicular lines
Definition (Slope-intercept form)
The equation
y = mx + b
is an equation of a line having slope m and passing through the
point (0, b), called the y-intercept of the line. It is called the
slope-intercept form of the equation of the line.
Slope and equations
Linear models
Parallel and perpendicular lines
Theorem (General form)
Any line can be represented by an equation of form
Ax + By = C ,
where A and B are not both equal to 0.
Slope and equations
Linear models
Parallel and perpendicular lines
Problem
1. Find the equation of the line passing through the point
P (−1, 5) and having slope m = −1/2.
2. Find the equation of the line passing through the points
P (2, 4) and Q(5, 2).
Slope and equations
Linear models
Parallel and perpendicular lines
Problem
Graph the lines 2x + 3y = 12 and 5x + 2y = 20 and find their
point of intersection.
Slope and equations
Linear models
Parallel and perpendicular lines
Problem (The meaning of slope)
Show that m is the change in y given a unit change in the variable
x.
Slope and equations
Linear models
Parallel and perpendicular lines
Definition
A function that can be put in the form y = f (x ) = mx + b is
called a linear function.
Slope and equations
Linear models
Parallel and perpendicular lines
Problem (A price function)
AAA Flooring will install x square feet of hardwood flooring for a
price of p = 5.43x + 138 dollars.
1. How much will it cost to install 1100 square feet of flooring?
2. What is the meaning of the slope?
3. What is the meaning of the y-intercept (or p-intercept)?
Slope and equations
Linear models
Parallel and perpendicular lines
Problem
At ACME Rent-A-Car it costs $65.00 to drive a car 100 miles and
$155.00 to drive a car 300 miles. Assuming that the cost of renting
a car is a linear function of the distance the car is driven, express
the cost of renting a car as a function of the mileage.
1. What does the slope represent?
2. What does the y-intercept represent?
Slope and equations
Linear models
Theorem
1. Two lines are parallel if and only if
1.1 they are both vertical or
1.2 they have the same slope.
2. Two lines are perpendicular if and only if
2.1 one is vertical and the other is horizontal or
2.2 the product of their slopes is -1.
Parallel and perpendicular lines
Slope and equations
Linear models
Parallel and perpendicular lines
Problem
A line L has equation 2x + 3y = 8.
1. Find the equation of the line that is parallel to L and passing
through the point P (2, 5).
2. Find the equation of the line that is perpindicular to L and
passing through the point P (2, 5).