This document discusses linear functions and their key characteristics: - A linear function is one that can be written in the form y = mx + b, where m is the slope and b is the y-intercept. Its graph is a straight line. - The slope of a line is defined as the rise over the run between two points and indicates whether a line increases, decreases, or is horizontal. - Lines can be graphed using their slope and a point, two points, or their x- and y-intercepts. The slope-intercept form y = mx + b allows easy graphing of a line from its slope and y-intercept. - Applications include finding the rate of change from points

1. 2.1 Linear Functions
• Linear Function
A function f defined by where
a and b are real numbers, is called a linear
function.
Its graph is called a line.
Its solution is an ordered pair, (x,y), that makes
the equation true.
baxxf +=)(

2. 2.1 Linear Functions
Example
The points (0,6) and (–1,3) are solutions of
since 6 = 3(0) + 6 and 3 = 3(–1) + 6.
63)( += xxf
3 6y x= +

3. 2.1 Graphing a Line Using Points
• Graphing the line 63 += xy
x y
−2 0
−1 3
0 6
1 9
Connect with a straight line.
(0,6) and (–2,0) are the y- and x-intercepts of the
line y = 3x + 6, and x = –2 is the zero of the function.

4. 2.1 Graphing a Line with the TI-83
• Graph the line with the TI-83
Xmin=-10, Xmax=10, Xscl=1
Ymin=-10, Ymax=10,Yscl=1
63 += xy

5. 2.1 x- and y-Intercepts
Locating x- and y-Intercepts
To find the x-intercept of the graph of
y = ax + b, let y = 0 and solve for x.
To find the y-intercept of the graph of
y = ax + b, let x = 0 and solve for y.

6. 2.1 Zero of a Function
Zero of a Function
Let f be a function. Then any number c
for which f (c) = 0 is called a zero of the
function f.

7. 2.1 Graphing a Line Using the Intercepts
Example: Graph the line .52 +−= xy
x y
x-intercept 0 5
y-intercept 2.5 0

8. 2.1 Application of Linear Functions
A 100 gallon tank full of water is being drained at a rate of
5 gallons per minute.
a) Write a formula for a linear function f that models the number of
gallons of water in the tank after x minutes.
b) How much water is in the tank after 4 minutes?
c) Use the x- and y-intercepts to graph f. Interpret each intercept.
1005)(
amountinitialchange)ofrateconstant()(
+−=
+=
xxf
xxf
gallons80100)4(55)4( =+−=f
-intercept, let 0 5(0) 100 100
meaning that the tank initially has 100 gallons in it.
-intercept, let 0 0 5 100 20 minutes
meaning that the tank takes 20 minutes to empty.
y x y
x y x x
= ⇒ =− + =
= ⇒ =− + ⇒ =

9. 2.1 Constant Function
• Constant Function
b is a real number.
The graph is a horizontal line.
y-intercept: (0,b)
Domain
range
Example:
bxf =)(
),( ∞−∞=
}{b=
3)( −=xf

10. 2.1 Constant Function
Constant Function
A function defined by f(x) = b, where b is a real number,
is called a constant function. Its graph is a horizontal
line with y-intercept b. For b not equal to 0, it has no
x-intercept. (Every constant function is also linear.)

11. 2.1 Graphing with the TI-83
• Different views with the TI-83
• Comprehensive graph shows all intercepts
63)( += xxf 63)( += xxf

12. 2.1 Slope
• Slope of a Line
$20,082 $5991 $14,091
$705.
2004 1984 20
−
= ≈
−
In 1984, the average annual cost
for tuition and fees at private four-
year colleges was $5991. By
2004, this cost had increased to
$20,082. The line graphed to the
right is actually somewhat
misleading, since it indicates that
the increase in cost was the same
from year to year.
The average yearly cost was $705.

13. The slope m of the line passing through the points
(x1, y1) and (x2, y2) is
x
y
0
),(
11
yx
),(
22
yx
12
xxx −=∆
12
yyy −=∆
2.1 Formula for Slope
2 1
2 1
2 1
, where 0.
y y y
m x x x
x x x
∆ −
= = = − ≠
∆ −
V
)
1
,
2
( yx

14. 2.1 Example: Finding Slope Given Points
7
4
7
4
25
)1(3
12
12
−=
−
=
−−
−−
=
−
−
=
xx
yy
m
Determine the slope of a line passing through
points (2, −1) and (−5, 3).

15. 2.1 Graph a Line Using Slope and a Point
• Example using the slope and a point to graph a
line
– Graph the line that passes through (2,1) with slope
3
4
−
x
y
0
(2,1)
down 4
x
y
0
(2,1)
right 3
(5,-3)
x
y
0
(2,1)
(5,-3)

16. 2.1 Slope of a Line
Geometric Orientation Base on Slope
For a line with slope m,
1. If m > 0, the line rises from left to right.
2. If m < 0, the line falls from left to right.
3. If m = 0, the line is horizontal.

17. 2.1 Slope of Horizontal and Vertical Lines
• The slope of a horizontal line is 0.
• The slope of a vertical line is undefined.
• The equation of a vertical line that passes through
the point (a,b) is
(0,4)
x
y
0
(1,4)
0
4
0
01
44
==
−
−
=m
x
y
0
(4,4)
4
undefinedm ==
−
−
=
0
4
44
04
.x a=

18. 2.1 Vertical Line
Vertical Line
A vertical line with x-intercept a has an
equation of the form x = a. Its slope is
undefined.

19. 2.1 Slope-Intercept Form of a Line
Slope-Intercept Form
The slope-intercept form of the equation of a
line is where m is the slope and
b is the y-intercept.
,y mx b= +

20. 2.1 Matching Examples
Solution:
1) 2 3y x= + 2) 2 3y x= − + 3) 2 3y x= − −
A. B. C.
1) C, 2) A, 3)B

21. 2.1 Application of Slope
• Interpreting Slope
In 1980, passengers traveled a total of 4.5 billion miles
on Amtrak, and in 2007 they traveled 5.8 billion miles.
a) Find the slope m of the line passing through the
points (1980, 4.5) and (2007, 5.8).
Solution:
b) Interpret the slope.
Solution:
5.8 4.5 1.3 13 0.05
2007 1980 27 270
m −= = = ≈
−
The average number of miles traveled on
Amtrak increased by about 0.05 billion, or
50 million miles per year.