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Chapter 2 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 of 17


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 of 17


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 of 17


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 of 17


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 of 17


2.3 #23 Determine if

the relation defines

as a function of .



as a function of .



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 of 17


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 of 17


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 of 17


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 of 17


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 of 17


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


( )

.

x

y


( ) √

.

x

y

Chapter 2 of 17



( ) | | .

x

y

2.6 #40 Use the graph of ( ) to graph

( ).

x

y

2.6 #54 Use the graph of ( ) to graph

( ).

x

y


( ) ( ) .

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 of 17



( ) √ .

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 of 17


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 of 17


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 of 17


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 of 17


For exercises 54-60, refer to the functions and

. Evaluate the function and write the domain in

interval notation.

( ) ( ) √ ( )

2.8 #54 ( ( ))

2.8 #60 ( )( )

For exercise 70, refer to the functions and .

Evaluate the function and write the domain in

interval notation.

( ) ( )

2.8 #70 ( )( )

2.8 #78 The cost to buy tickets online for a

dance show is $60 per ticket.

a. Write a function that represents the cost

( ) (in $) for tickets to the show.

b. There is a sales tax of 5.5% amd a processing

fee of $8.00 for a group of tickets. Write a

function that represents the total cost ( ) for

dollars spent on tickets.

c. Find ( )( ).

d. Find ( )( ) and interpret its meaning in

the context of the problem.

2.8 #92 the graphs of and are shown. Find

the function values of , if possible.

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