1 section 1.2 linear functions and graphs. 2 example x (miles) c (cost) 100$60 150$75 200$90 250$105...
TRANSCRIPT
1
Section 1.2
Linear Functions
and Graphs
2
EXAMPLE
x
(Miles)
C
(Cost)
100 $60
150 $75
200 $90
250 $105
The following table shows the cost per day for a rental car depending on how many miles you drive a day.
3
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 ANDLINEAR 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.
5
SLOPE AND y-INTERCEPTOF A LINE
• The y-value where a line crosses the y-axis is called the y-intercept.
• The slope of a line in the xy-plane is defined by
slo p erise
ru n
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SLOPE-INTERCEPTEQUATION
• 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.
In the equation y = mx + b,
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POINT-SLOPE FORMULAFOR 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
my y
x x
slo p e 0
0
.
The equation y y m x x 0 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
• 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.
There are three types of equations whose graphs are straight lines:
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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
x
(Miles)
C
(Cost)
100 $60
150 $75
200 $90
250 $105
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 you drive if the cost was $80?
(b)What do the slope and y-intercept mean in practical terms?