# 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 y-intercept (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 point-slope 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 point-slope equation of the line that
passes through (-3, 7) and

(-1,-5).

•Vertical lines are parallel. non-vertical 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: pp93-94: 12, 28, 38, 48

Answers:

12.) slope

28.) slope m = 2/3 and y- intercept (0,-4)

38.)

48.) neither

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