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Linear Functions,Slope and Applications
•A function f is a linear function if it can be written
as
f(x) = mx + b
where m and b are constants.
If m = 0, the function is the constant function f(x) = b. If m = 1 and b = 0,
the
function is the identity function f(x) = x.
•Vertical and Horizontal Lines:
1. Horizontal lines are given by equations of the type y = b or f(x) = b. Are
horizontal lines functions ? What is the slope of a horizontal line?
2. Vertical lines are given by equations of the type x = a. Are vertical lines
functions? What is the slope of a vertical line?
• Linear Functions and Slope: Mathematically we define a
line’s steepness or slope,
as the ratio of its vertical change (rise) to the corresponding horizontal
change (run).
The slope m of a line containing the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by:
Example: Find the slope of the line containing the points (3, 7) and (5,1)
•Slope Intercept Equation : The linear function f given by
f(x) = mx + b has a
graph with slope m and yintercept (0, b). May also be written as y = mx + b.
Example: Write the slope intercept equation for a line with m = 2 and passing
through (5, 1).
•The pointslope equation of the line with slope m
passing through the point (x_{1}, y_{1})
is
y  y_{1} = m(x  x_{1})
Example: Write the pointslope equation of the line that
passes through (3, 7) and
(1,5).
•Vertical lines are parallel. nonvertical lines are
parallel if and only if they have the
same slope and different y intercepts.
• Two lines with slopes m_{1} and m_{2} are perpendicular if and only if the product
of
their slopes is 1.
m_{1} * m_{2} = 1
In other words, if m_{1} is the opposite reciprocal of m_{2}
then the two lines are perpendicular.
Lines are also perpendicular if one is vertical (x = a) and the other horizontal
(y = b).
Example: Write an equation for a line passing through the point (1, 6) and
parallel
to the line Then write a second equation for
the line perpendicular to
and passing through (1, 6).
Try these: pp9394: 12, 28, 38, 48
Answers:
12.) slope
28.) slope m = 2/3 and y intercept (0,4)
38.)
48.) neither
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