Download Sections 2.1-2.2

Equation of a Line
Thm. A line has the equation y = mx + b,
where m = slope and b = y-intercept.
 This is called Slope-Intercept Form
Ex. Find the slope and y-intercept:
a) y = 3x + 1
b) 2x + 3y = 1
Ex. Graph:
a) y

21
x
4
2
-5
5
-2
-4
Ex. Graph:
b) y2
4
2
-5
5
-2
-4
Ex. Graph:
c) xy

2
4
2
-5
5
-2
-4
Slope
Thm. The slope between (x1,y1) and (x2,y2) is
y2 y1
m
x2 x1
Ex. Find the slope between (-2,0) and (3,1).
• Rising line has positive slope:
2
5
,4
3

8
,

• Falling line has negative slope:
7
• Horizontal line as slope of 0:
0 0
5 3
,
• Vertical line has undefined slope:
2 8
0 0
,
A vertical line has an equation like x = 3,
and can’t be written as y = mx + b
Thm. Parallel lines have the same slope.
Thm. Perpendicular lines have slopes that
are negative reciprocals.
Ex. Are the lines parallel, perpendicular, or
neither?
(3,-1) to (-3,1) and (0,3) to (-1,0)
Thm. A line with m = slope that passes through
the point (x1,y1) has the equation
y – y1 = m(x – x1)
This is called Point-Slope Form
Ex. Write the equation in Slope-Intercept Form:
a) slope = 3, contains (1,-2)
Ex. Write the equation in Slope-Intercept Form:
b) contains (2,5) and (4,-1)
Ex. Find equation of the lines that pass through (2,-1) are:
a) parallel to, and
b) perpendicular to,
the line 2x – 3y = 5
1
12
Ex. The maximum slope of a wheelchair ramp is . A
business is installing a ramp that rises 22 in. over a
horizontal distance of 24 ft. Is the ramp steeper than
required?
Ex. An appliance company determines that the total cost, in
dollars, of producing x blenders is
C = 25x + 3500
Explain the significance of the slope and y-intercept.
Ex. A college purchases exercise equipment worth
$12,000. After 8 years, the equipment is determined
to have a worth of $2000. Express this relationship as
a linear equation. How many years will pass before
the equipment is worthless?
Practice Problems
Section 2.1
Problems 9, 21, 41, 51, 69, 107, 109
Functions
Def. (formal) A function f from set A to set B
is a relation that assigns to each element x
in set A exactly one element y in set B.
Def. (informal) A set of ordered pairs is a
function if no two points have the same
x-coordinate
Set A
(the x’s)
The input
Set B
(the y’s)
The output
Domain
Range
x is called the independent variable
y is called the dependent variable
Ex. Determine whether the relation is a
function:
a) x 0 3 2 -1 3
y
b)
5
7
2
-2
-3
0
4
Ex. Determine whether the relation is a
function:
c) x is the number of representatives from a
state, y is the number of senators from the
same state
d) x is the time spent at a parking meter, y is
the cost to park
Ex. Determine whether the relation is a
function:
e) x2 + y = 1
f) x + y2 = 1
Rather than writing
y = 7x + 2
we can express a function as
f (x) = 7x + 2
 This is called Function Notation
Ex. Let g(x) = -x2 + 4x + 1, find
a) g(2)
b) g(t)
c) g(x + 2)
The next example is called a piecewise
function because the equation depends on
what we are plugging in.
2

x
1x

0
Ex. Let f
, find f (-1),
x

x

1x

0

f (0), and f (1).
Finding the domain means determining all
possible x’s that can be put into the function
Ex. Find the domain of the function
f : {(-3,0), (-1,4), (0,2), (2,2), (4,-1)}
Often, finding the domain means finding the
x’s that can’t be used in the function
Ex. Find the domain of the function
1
a) gx
 
x5

r
b) Volume of a sphere: V
4
3
3
Ex. Find the domain of the function
2
c) h
x

4x

Ex. You’re making a can with a height that is 4 times as
long as the radius. Express the volume of the can as a
function of height.
Ex. When a baseball is hit, the height of a baseball is
given by the function f (x) = -0.0032x2 + x + 3, where
x is distance travelled (in ft) and f (x) is height (in ft).
Will the baseball clear a 10-foot fence that is 300 ft
from home plate?
f
x

h

f
x



Ex. For f (x) = x2 – 4x + 7, find
h
Practice Problems
Section 2.2
Problems 9, 15, 29, 35, 59, 79, 87, 93