Download Section 1.2 - Gordon State College

Section 1.2
Linear Functions
and Graphs
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EXAMPLE
The following table shows the cost per day for a
rental car depending on how many miles you
drive a day.
x
(Miles)
100
C
(Cost)
$60
150
$75
200
$90
250
$105
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LINEAR FUNCTIONS
Definition: A linear function is a function
of the form
f (x) = mx + b.
NOTE: The letters “m” and “b” represent
coefficients (numbers).
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STRAIGHT LINES AND
LINEAR GRAPHS
The graph of the linear function f (x) = mx + b
is the straight line consisting of all the points
(x,y) in the xy-plane that satisfy the equation
y = mx + b.
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SLOPE AND y-INTERCEPT
OF A LINE
• The y-value where a line crosses the yaxis is called the y-intercept.
• The slope of a line in the xy-plane is
defined by
rise
slope 
run
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SLOPE-INTERCEPT
EQUATION
In the equation y = mx + b,
• The constant b is the y-intercept of the line.
• The coefficient m of x is the slope of the line.
• This equation is called the slope-intercept
equation for a line.
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POINT-SLOPE FORMULA
FOR A LINE
If (x0, y0) is a fixed point on a line, and (x, y)
is any other point on the line, we can find the
slope by
y  y0
m  slope 
.
x  x0
The equation y  y 0  m( x  x 0 )
is called the point-slope equation for the line with
slope m that passes through the point (x0, y0).
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GRAPHS OF EQUATIONS
Definition: The graph of an equation involving
two variables x and y consists of all points in the
xy-plane whose coordinates (x, y) satisfy the
equation.
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GRAPHS OF LINES
There are three types of equations whose graphs are
straight lines:
• Equations whose graphs are vertical lines.
These have the form x = constant.
• Equations whose graphs are horizontal lines.
These have the form y = constant.
• Equations whose graphs are slanted lines (lines
that are neither vertical or horizontal). These
have the form y = mx + b.
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GRAPH OF A FUNCTION
Definition: The graph of the function f is the
graph of the equation y = f (x).
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THE VERTICAL LINE TEST
Recall that a function assigns to each number x a
specific number f (x). Thus, a graph is the
graph of a function if and only if no vertical
line intersects the graph at more than one
point.
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EXAMPLE
The following table shows the cost per day for a
rental car depending on how many miles you
drive a day.
(a)How many miles did
x
C
you drive if the cost was
(Miles) (Cost)
$80?
100
$60
150
200
250
$75
$90
$105
(b)What do the slope and
y-intercept mean in
practical terms?
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