Chapter 2 Page 1 of 17 College Algebra Lecture Notes – Videos to accompany these notes can be found at www.mathvideos.net Lecture Guide Math 105 - College Algebra Chapter 2 to accompany “College Algebra” by Julie Miller Corresponding Lecture Videos can be found at Prepared by Stephen Toner & Nichole DuBal Victor Valley College Last updated: 2/5/13
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Chapter 2 Page 1 of 17
College Algebra Lecture Notes – Videos to accompany these notes can be found at www.mathvideos.net
Lecture Guide Math 105 - College Algebra
Chapter 2
to accompany
“College Algebra” by Julie Miller
Corresponding Lecture Videos can be found at
Prepared by
Stephen Toner & Nichole DuBal Victor Valley College
Last updated: 2/5/13
Chapter 2 Page 2 of 17
College Algebra Lecture Notes – Videos to accompany these notes can be found at www.mathvideos.net
2.1 – The Rectangular Coordinate System
A. Distance and Midpoint
Midpoint Formula: (
)
Distance Formula:
√( ) ( )
2.1 #14 (a) Find the exact distance between the
points. (b) Find the midpoint of the line
segment whose endpoints are given.
( ) and ( )
2.1 #22 Determine if the given points form the
vertices of a right triangle.
( ), ( ) and ( )
2.1 #26 Identify the set of values for which
will be a real number.
2.1 #27 Identify the set of values for which
will be a real number.
√
Chapter 2 Page 3 of 17
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2.1 #40 Graph the equation by plotting points.
x
y
2.1 #45 Estimate the - and -intercepts from
the graph.
2.1 #56 Find the - and -intercepts.
2.1 #69 (a) Determine
the exact length and
width of the rectangle
shown. (b) Determine
the perimeter and area.
2.1 #71 The endpoints of
a diameter of a circle are
shown. Find the center
and radius of the circle.
Chapter 2 Page 4 of 17
College Algebra Lecture Notes – Videos to accompany these notes can be found at www.mathvideos.net
2.2 – Circles
Definition: A circle is the set of all points in a plane that are equidistant from a fixed point
called the______________. The
fixed distance from any point on the circle to
the center is called the __________.
The Standard Form of an Equation of a Circle:
( ) ( ) has center ( )
with radius .
2.2 #16 Determine the center and radius of the
circle. ( ) ( )
2.2 #26 (a) Write the equation of the circle in
standard form. (b) Graph the circle.
Center: ( ); Radius √
x
y
2.2 #32 (a) Write the equation of the circle in
standard form. (b) Graph the circle.
The center is ( ) and another point on
the circle is ( ).
x
y
2.2 #44 Write the equation in standard form:
( ) ( ) . Then if possible,
identify the center and radius of the circle. If
the equation represents a degenerate case,
give the solution set.
Chapter 2 Page 5 of 17
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2.2 #52 Write the equation in standard form:
( ) ( ) . Then if possible,
identify the center and radius of the circle. If
the equation represents a degenerate case,
give the solution set.
2.2 #56 A radar transmitter on a ship has a range of 20 nautical miles. If the ship is located at a point ( ) on a map, write an equation for the boundary of the area within the range of the ship’s radar. Assume that all distances on the map are represented in nautical miles.
2.3 – Functions and Relations
Definition: A set of ordered pairs ( ) is
called a _____________ in and .
The set of values in the ordered pairs is
called the _________ of the relation.
The set of values in the ordered pairs is
called the _________ of the relation.
Definition: Given a relation in and , we say
that is a function of if for each value of in
the domain, there is ____________
value of in the range.
2.3 #18 a. Write a set of
ordered pairs ( ) that
define the relation.
b. Write the domain of the relation.
c. Write the range of the relation.
d. Determine if the relation defines as a
function of .
Chapter 2 Page 6 of 17
College Algebra Lecture Notes – Videos to accompany these notes can be found at www.mathvideos.net
2.3 #23 Determine if
the relation defines
as a function of .
2.3 #27 Determine if
the relation defines
as a function of .
2.3 #32 Determine if
the relation defines
as a function of .
2.3 #45 Given ( )
, find ( )
2.3 #54 Given ( ) , find ( )
2.3 #80 Determine the - and -intercepts for
the function ( ) | |.
2.3 #88 Determine
the domain and range
for the function.
2.3 #98 Write the domain in interval notation.
( )
2.3 #106 Write the domain in interval notation.
( )
√
2.3 #116 In an isosceles triangle, two angles are
equal in measure. If the third angle is
degrees, write a relationship that represents
the measure of one of the equal angles ( ) as
a function of .
Chapter 2 Page 7 of 17
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2.3 #110 Use the
graph of ( ) to
answer the following.
a. Determine ( ).
b. Determine ( ).
c. Find all for
which ( ) .
d. Find all for
which ( ) .
e. Determine the -
intercept(s).
f. Determine the -
intercept.
g. Determine the
domain of .
h. Determine the
range of .
2.4 – Linear Equations and Functions
Definition: A linear equation in and can be
written in the form . This is called
the _______________ form of the
equation of a line.
2.4 #20 Graph the equation and identify the -
and -intercepts.
x
y
2.4 #38 Determine the slope of the line passing
through the given points.
( ) and ( )
Chapter 2 Page 8 of 17
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Definition: Given a line with a slope and -
intercept ( ) the ___________
form of the line is given by .
2.4 #59 Given ,
a. Write the equation in slope-intercept form if
possible, and determine the slope and -
intercept.
b. Graph the equation using the slope and -
intercept.
x
y
2.4 #76 a. Use the slope-intercept form to
write an equation of the line that passes
through ( ) with slope .
b. Write the equation using function notation
where ( ).
2.4 #90 The
function given
by ( )
shows the
average monthly
temperature
( ) for Cedar
Key. The value
of is the
month number
and
represents January.
a. Find the average rate of change in
temperature between months 3 and 5 (March
and May).
b. Find the average rate of change in
temperature between months 9 and 11
(September and November).
c. Comparing the results in parts (a) and (b),
what does a positive rate of change mean in
the context of this problem? What does a
negative rate of change mean?
Chapter 2 Page 9 of 17
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2.4 #98 Determine the average rate of change
of the function ( ) √ on the given
interval.
a.
b.
c.
2.4 #100 Use the graph to solve the equation
and inequalities. Write the solutions to the
inequalities in interval notation.
2.5 – Applications of Linear Equations
Definition: The ___________ formula
for a line is given by ( ),
where is the slope of the line and ( ) is a
point on the line.
2.5 #12 Use the point-slope formula to write an
equation of the line which passes through
( ) with . Write the answer in slope-
intercept form (if possible).
2.5 #18 Use the point-slope formula to write an
equation of the line which passes through
( ) and ( ). Write the answer in
slope-intercept form (if possible).
Chapter 2 Page 10 of 17
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2.5 #29 Given the slope
,
a. Determine the slope of a line parallel to the
given line, if possible.
b. Determine the slope of a line perpendicular
to the given line, if possible.
2.5 #44 Write the equation of the line passing
through ( ) and parallel to the line defined
by . Write the answer in slope-
intercept form (if possible) and in standard
from with no fractional coefficients.
2.5 #50 Write the equation of the line passing
through ( ) and perpendicalar to the line
defined by . Write the answer in
slope-intercept form (if possible) and in
standard from with no fractional coefficients.
2.5 #68 A lawn service company charges $60
for each lawn maintenance call. The fixed
monthly cost of $680 includes telephone
service and depreciation of equipment. The
variable costs include labor, gasoline, and taxes
and amount to $36 per lawn.
a. Wrtie a linear cost function representing the
monthly cost ( ) for maintenance calls.
b. Write a linear revenue function representing
the monthly revenue ( ) for maintenance
calls.
c. Write a linear profit function representing
the monthly profit ( ) for maintenance
calls.
d. Determine the number of lawn maintenance
calls needed per month for the company to
make money.
e. If 42 maintenance calls are made for a given
month, how much money will the lawn service
make or lose?
Chapter 2 Page 11 of 17
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2.6 – Transformations of Graphs
You should already be familiar with the
following basic functions and their graphs.
"Code List" For Transformations
( )
( )
( )
( )
( )
( )
( )
2.6 #22 Use transformations to graph
( ) √ .
x
y
2.6 #24 Use transformations to graph
( )
.
x
y
2.6 #28 Use transformations to graph
( ) √
.
x
y
Chapter 2 Page 12 of 17
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2.6 #30 Use transformations to graph
( ) | | .
x
y
2.6 #40 Use the graph of ( ) to graph
( ).
x
y
2.6 #54 Use the graph of ( ) to graph
( ).
x
y
2.6 #62 Use transformations to graph
( ) ( ) .
x
y
To graph a function requiring multiple
transformations on the parent function, the
sequence of transformations is important.
Perform horzontal transformations first.
These are operations on .
Perform vertical transformations next.
These are operations on ( ).
Order of transformations:
1. Horizontal shrink/stretch/reflection “ ”.
2. Horizontal shift “ ”.
3. Vertical shrink/stretch/reflection “ ”.
4. Vertical shift “ ”.
Chapter 2 Page 13 of 17
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2.6 #70 Use transformations to graph
( ) √ .
x
y
2.6 #88 Use transformations on the basic
functions to write a rule ( ) that would
produce the given graph.
2.6 #90 Use transformations on the basic
functions to write a rule ( ) that would
produce the given graph.
2.7 – Analyzing Graphs of Functions and
Piecewise-Defined Functions
2.7 #13 Determine whether the graph of the
equation is symmetric with
respect to the -axis, -axis, origin, or none of
these.
x
y
A function is even if _______________________.
Even functions are symmetric about (the -
axis).
x
y
A function is odd if _______________________.
Odd functions are symmetric about the origin.
x
y
2.7 #28 Use the graph
to determine if the
function is even, odd,
or neither.
Chapter 2 Page 14 of 17
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2.7 #36 Given ( ) ,
a. find ( ).
b. Find ( ).
c. Is ( ) ( )?
d. Is the function even, odd, or neither?
2.7 #40 Determine if ( ) | |
is even, odd, or neither.
2.7 #50 Evaluate the function for the given
values of .
( ) { | |
a. ( )
b. ( )
c. ( )
d. ( )
e. ( )
At Wet Willy's Water World, infants under 2 are
free, then admission is charged according to
age. Children 2 and older but less than 13 pay
$2, teenagers 13 and older, but less than 20 pay
$5, adults 20 and older but less than 65 pay $7,
and senior citizens 65 and older get in at the
teenage rate. Write this information in the
form of a piecewise-defined function and state
the domain for each piece. Then sketch the
graph and find the cost of admission for a
family of nine which includes: one grandparent
(70), two adults (44/45), 3 teenagers, 2 children
and one infant.
Chapter 2 Page 15 of 17
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2.7 #62 Graph: ( ) {
√
x
y
2.7 #70 Graph:
( ) {
x
y
A function is ___________________ when it
"goes up" from left to right, and it is
_____________ when it "goes down" from left
to right. A function is ____________ where its
graph is horizontal.
Formal definition of an increasing function:
A function ( ) is said to be increasing on an
open interval ( ) if for all ( ) where
, __________________.
2.7 #90 Use interval
notation to write the
intervals over which
( ) is (a) increasing,
(b) decreasing, and (c)
constant.
2.7 #100 Identify the
location and values of
any relative maxima or
minima of the function.
2.7 #110 Produce a rule
for the function whose
graph is shown.
Chapter 2 Page 16 of 17
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2.8 – Algebra of Functions and Function
Composition
2.8 #14 Find ( )( ) and identify the graph
of , given ( ) and ( ) .
For exercises 16-24, evaluate the functions for
the given values of .
( ) | | ( )
2.8 #16 ( )( )
2.8 #18 ( )( )
2.8 #24 (
) ( )
The domains of and will all be
the same (the intersection of their separate
domains). The domain of
will be further
restricted so that ( )
For exercises 26-36, refer to the functions and
. Evaluate the function and write the domain
in interval notation.
( ) ( )
2.8 #26 ( )( )
2.8 #36 (
) ( )
2.8 #42 Find the difference quotient, ( ) ( )
and simplify for ( ) .
Chapter 2 Page 17 of 17
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