1-3 Linear Functions and Straight Lines
Linear functions




are of the form f(x) = mx + b
e.g
f(x) = -3x + 4 (m = -3, b = 4)
called linear because they graph as straight lines
sometimes written y = mx + b (slope-intercept form)
Graphing a linear function using intercept method
Example: or f(x) = 2x + 4
(1) convert to equation form: y = 2x + 4
(2) Find intercepts: set x = 0, solve to get y-intercept = 4
set y = 0, solve to get x-intercept = -2
(3) Plot the intercepts and draw the line:

y



   





x




1-3
p.1
Graphing a function having restricted domain
Most real-world functions will have restricted domains, e.g.
A = 6t + 10, 0 ≤ t ≤ 100
The “0 ≤ t ≤ 100” is a domain restriction, meaning that
the function is valid only for values of t between 0 and 100,
inclusive.
To graph it, calculate the points at the extreme left and
right:
if t = 0, A = 10  point (0, 10)
if t = 100, A = 610  point (100, 610)
Graph the points and draw the line:
A







1-3









t

p.2
Slope of a line
slope of line is a ratio:
vertical rise
y
=
horizontal run
x
Computing slope, given two points:
In general, given two points (x1, y1) and (x2, y2), the slope
of the line passing through them is
y  y1
m= 2
(slope formula)
x2  x1
1-3
p.3
Kinds of slope
When the slope is zero, we have a constant function.
When a function is written in slope-intercept form
f(x) = mx + b or
y = mx + b
 the coefficient m of x will be the slope
 the constant term b will be the y-intercept
e.g. the graph of f(x) = - ¾ x + 12 has slope -3/4, and yintercept 12.
1-3
p.4
Interpretation of slope
The following graph represents the value of an investment
(in $’s) over time (in years):

slope = rate = 100 $/yr



100



1




x
As you can see, it is a linear function, and has slope = 100.
By looking at the graph, you can see that the investment
grows by $100/year, so the interpretation of “slope = 100”
for this linear function is:
“The investment increases by $100 per year ($100/year)”
Notice the form of this statement: “Y per X” or “Y/X”
 Y is the slope expressed in y-axis units
 X is the x-axis unit
Slope can also be thought of as a rate of change (of the yvariable with respect to the x-variable). If the x-variable is
time in hours and the y-variable is distance in miles, then
the slope would be expressed as e.g. 50 miles per hour.
Indeed the familiar rate of speed!
1-3
p.5
Point-slope form of a line
This form is used to find the equation of a line when you
know a point (x1, y1) on the line, and its slope m:
y – y1 = m(x – x1)
(point-slope form)
Finding the equation of a line, given two points
Example: points: (1, 3) (3, 6)
63
(1) find the slope: m =
= 3/2
3 1
(2) use the point-slope form:
y - 3 = 3/2(x – 1)
simplify:
y = (3/2)x – 3/2 + 3
y = (3/2)x + 3/2
(stated using slope-intercept form)
f(x) = (3/2)x + 3/2 (stated using functional notation)
To state in standard form (Ax + By = C):
y = (3/2)x + 3/2
2y = 3x + 3 (clear of fractions by multiplying by 2)
-3x + 2y = 3
or 3x – 2y = -3 (either way)
1-3
p.6