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Chapter 3 of 23

College Algebra Lecture Notes – Videos to accompany these notes can be found at www.mathvideos.net

Lecture Guide Math 105 - College Algebra

Chapter 3

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: 2/16/13

Chapter 3 of 23


3.1 – Quadratic Functions and Applications

Quadratic functions are of the form

( ) .

It is easiest to graph quadratic functions when

they are in the form ( ) ( )

using transformations. Here, the parabola has

the vertex at __________________.

3.1 #10 Use ( ) ( ) .

a. Determine whether the graph of the prabola

opens upward or downward.

b. Identify the vertex.

c. Determine the -intercept(s).

d. Determine the -intercept.

e. Sketch the function.

f. Determine the axis of

symmetry.

g. Determine the minimum

or maximum value of the function.

h. Determine the domain and range.

Completing the Square

1. Write your equation in the form:

2. If there's a leading coefficient, factor it out

of the first two terms on the right.

3. Cut the number in front of in half; write

this new value on the line below.

4. Square this new value and write the product

in the blank on the line above; add/subtract

this product at the right to keep the equation

balanced.

5. Insert an x, parentheses and exponent on

the left to complete the square.

6. Add together the values at the right

7. Write the function in vertex form.

Chapter 3 of 23


3.1 #18 Use ( ) .

a. Write the function in vertex form.


c. Identify the -intercept(s).

d. Identify the -intercept(s).


f. Determine the axis of symmetry.

g. Determine the minimum and maximum

values of the function.

h. State the domain and range.

3.1 #20 Use ( ) .

a. Write the function in vertex form.


c. Identify the -intercept(s).



f. Determine the axis of symmetry.




Chapter 3 of 23


3.1 #30 Use ( ) .

a. State whether the graph of the parabola

opens upward or downward.

b. Determine the vertex of the parabola.

c. Determine the -intercept(s).


e. Sketch the graph.

f. Determine the axis of

symmetry.




3.1 #36 A long jumper leaves the ground at an angle of above the horizontal, at a speed of 11 m/sec. The height of the jumper can be modeled by ( ) , where is the jumper’s height in meters and is the horizontal distance from the point of launch. a. At what horizontal distance from the point of launch does the maximum height occur? Round to 2 decimal places. b. What is the maximum height of the long jumper? Round to 2 decimal places. c. What is the length of the jump? Round to 1 decimal place.

3.1 #42 Two chicken coops are to

be built adjacent to one another

from 120 ft of fencing.

a. What dimensions should be used to

maximize the area of an individual coop?

b. What is the maximum area of an individual

coop?

Chapter 3 of 23


3.2 – Introduction to Polynomial Functions

Key Ideas:

The domain of a polynomial function is

___________________.

Informally, a polynomial function can be

drawn________________

___________________.

Chapter 3 of 23


3.2 #30 Determine the end behavior of the

graph of ( )

.

3.2 #34 Determine the end behavior of the

graph of ( ) ( )( ) ( ).

3.2 #38 Find the zeros of the function and state

the multiplicities.

( )


the multiplicities.

( )


the multiplicities.

( ) ( ) ( )

The Intermediate Value Theorem: If ( ) is a

polynomial and , if ( ) and ( ) have

opposite signs, then ( ) has at least one zero

in the interval .

picture:

3.2 #50 Determine whether the intermediate

value theorem guarantees that the function has

a zero on the given interval.

( )

a.

b.

c.

d.

Chapter 3 of 23


Key Ideas:

Let represent a polynomial function of

degree . Then the graph of has at most

_________ turning points.

Let be a polynomial function and let be

a zero of . Then the point ( ) is an -

intercept of the graph of . Furthermore,

o If is a zero of __________

multiplicity, then the graph crosses the

-axis at . The point ( ) is called a

cross point.

o If is a zero of __________

multiplicity, then the graph touches the

-axis at and turns back around (does

not cross the -axis). The point ( ) is

called a touch point.

For exercises 58 and 60, determine if the graph

can represent a polynomial function. If so,

assume that the end behavior and all turning

points are represented in the graph.

3.2 #58 a. Determine

the minimum degree of

the polynomial based on

the number of turning

points.

b. Detemine whether the leading coefficient is

positive or negative based on the end behavior

and whether the degree of the polynomial is

odd or even.

c. Approximate the real zeros of the function

and determine their multiplicities.

3.2 #60 a. Determine the

minimum degree of the

polynomial based on the

number of turning points.

b. Detemine whether the leading coefficient is

positive or negative based on the end behavior

and whether the degree of the polynomial is

odd or even.

c. Approximate the real zeros of the function

and determine their multiplicities.

Chapter 3 of 23


3.2 #64 Sketch ( ) .

3.2 #68 Sketch ( ) .

3.3 – Divison of Polynomials and the

Remainder and Factor Theorems

Division vocabulary: ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

q x r x

d x f x

( ) is called the _______________.

( ) is called the _______________.

( ) is called the _______________.

( ) is called the _______________.

3.3 #17 ( ) ( )

a. Use long division to divide.

b. Identify the dividend, divisor, quotient, and

remainder.

Dividend:

Divisor:

Quotient:

Remainder:

c. Check the result from part (a) with the

division algorithm.

Chapter 3 of 23


3.3 #24 Use long division to divide:

( ) ( )

3.3 #28 Use long division to divide:

3.3 #40 Use synthetic division to divide.

( ) ( )

3.3 #44 Use synthetic division to divide

.

3.3 #50 Use the remainder theorem to

evaluate ( ) for the

given values of

a. ( ) b. (

) c. (√ ) d. ( )

Chapter 3 of 23


3.3 #58 Use the remainder theorem to

determione if the given numer is a zero of the

polynomial. ( )

a. ( ) b. ( )

3.3 #64 a. Use synthetic division and the factor

theorem to determine if ( ) is a

factor of ( ) .

b. Use synthetic division and the factor

theorem to determine if ( ) is a

factor of ( ) .

c. Use the quadratic formula to solve the

equation .

d. Find the zeros of the polynomial

( ) .

3.3 #66

a. Factor ( ) , given

that is a zero.

b. Solve

3.3 #76 Write a degree 2 polynomial with zeros

√ and √ .

Chapter 3 of 23


3.4 – Zeros of Polynomials

The Rational Zero Theorem

If ( )

has integer coefficients and , and if

(written in lowest terms) is a rational zero of ,

then

is a factor of the constant term

is a factor of the leading coefficient

The rational zero theorem does not guarantee

the existence of rational zeros. Rather, it

indicates that if a rational zero exists for a

polynomial, then it must be of the form

( )

( ).

3.4 #18 List the possible rational zeros.

( )

3.4 #26 Find all the zeros.

( )

Theorem: If the sum of the coefficients is ____,

then is a zero (and is a factor).

If after changing the signs of the coefficients of

the odd-degreed terms, the sum of the "new"

coefficients is zero, then is a zero.

(Note: This theorem is not in your textbook.)


( )

Chapter 3 of 23



( )

3.4 #40 ( )

has as a zero.

a. Find all the zeros.

b. Factor ( ) as a product of linear factors.

c. Solve the equation ( ) .

3.4 #50 Write a polynomial ( ) of lowest

degree with zeros of

(mulitplicity 2) and

(mulitiplicity 1) and with ( ) .

Chapter 3 of 23


3.4 #56 Determine the number of possible

positive and negative real zeros for the given

function.

( )

3.4 #68 a. Determine if the upper bound theorem

identifies as an upper bound for the real zeros

of ( ).

b. Determine if the lower bound theorem

identifies as a lower bound for the real zeros

of ( ).

( )

3.4 #76 Find the zeros and their multiplicities.

Consider using Descartes’ rule of signs and the

upper and lower bound theorem to limit your

search for rational zeros. (Hint: see exercise 68.)

( )

Chapter 3 of 23


3.5 – Rational Functions

Rational functions are of the form

( ) ( )

( ) where ( ) and ( ) have no

factors in common.

If ( ) and ( ) DO have factors in common,

you get removable discontinuities in your

graph. For example:

( )

Three Types of Discontinuities:

removable discontinuities

nonremovable (gap) discontinuities

nonremovable (asymptotic) discontinuities

We want to analyze graphs and behaviors of

rational functions, but need some notation:

Example: Graph ( )

and describe the

behaviors as approaches the -axis from both

sides. Also describe the behavior of as

increases and decreases without bound.

Identifying Vertical and Horizontal Asymptotes

Consider ( ) ( )

( ) where ( ) and ( )

have no factors in common. If is a zero of

( ), then ______ is a vertical asymptote

of the graph of ( ).

If ( )

is a

rational function with numerator of degree

and denominator of degree , then…

1. If , then has no horizontal asymptote.

2. If , then the line (the -axis) is the

horizontal asymptote of .

3. If , then the line

is the horizontal

asymptote of .

Chapter 3 of 23


3.5 #18 Write the domain of ( )

in

interval notation.

3.5 #24

Refer to

the graph

of the

function

and

complete

the

statement.

a. As ( ) _____________.

b. As ( ) _____________.

c. As ( ) _____________.

d. As ( ) ______________.

e. The graph in increasing over the interval(s)

____________________________.

f. The graph in decreasing over the interval(s)

____________________________.

g. The domain is _________________.

h. The range is __________________.

i. The vertical asymptote is the line _____.

j. The horizontal asymptote is the line ____.

3.5 #28 Determine the vertical asymptote of

the graph of ( )

.

3.5 #30 Determine the vertical asymptotes of

the graph of ( )

.

For exercises 36-40, (a) identify the horizontal

asymptote (if any), and (b) if the graph of the

function has a horizontal asymptote, determine

the point where the graph corsses the

horizontal asymptote.

3.5 #36 ( )

3.5 #38 ( )

3.5 #40 ( )

Chapter 3 of 23


3.5 #48 Identify the asymptotes.

( )

3.5 #60 Graph ( )

( ) by using a

transformation of the graph of

.

3.5 #74 Graph ( )

.

3.5 #82 Graph ( )

.

Chapter 3 of 23


3.5 #86 Graph ( )

.

3.5 #89 Graph ( )

.

3.6 – Polynomial and Rational Inequalities

3.6 #18 The graph of

( ) is given.

Solve the inequalities.

a. ( )

b. ( )

c. ( )

d. ( )

3.6 #26 Solve the equations and inequalities.

a.

b.

c.

d.

e.

Chapter 3 of 23


3.6 #28 Solve. ( )

3.6 #42 Solve.

3.6 #56 The graph of

( ) is given.

Solve the inequalities.

a. ( )

b. ( )

c. ( )

d. ( )

3.6 #62 Solve the inequalities.

a.

b.

c.

d.

3.6 #74 Solve.

Chapter 3 of 23


The vertical position ( ) of an object moving

upward or downward under the influence of

gravit is given by ( )

,

where

is the acceleration due to gravity (32 ft/sec2 or 9.8

m/sec2).

is the time of travel.

is the initial velocity.

is the initial vertical position.

3.6 #86 Suppose that a basketball player jumps

straight up for a rebound.

a. If his initial velocity is 16 ft/sec, write a

function modeling his vertical position ( ) (in

ft) at a time seconds after leaving the ground.

b. Find the times after leaving the ground

when the player will be at a height of more

than 3 ft in the air.

3.6 #94 Write the domain in interval notation.

( ) √

3.6 #102 Write the domain in interval notation.

( ) √

Find the domain graphically…

( ) √

Chapter 3 of 23


3.7 – Variation

Direct Variation

Inverse Variation

In exercises 12-20, write a variation model

using as the constant of variation.

3.7 #12 Simple interest on a loan or

investment varies directly as the amount of

the loan.

3.7 #14 The time of travel is inversely

proportional to the rate of travel .

3.7 #18 The variable is directly proportional

to the square of and inversely proportional to

the square of .

3.7 #20 The variable varies jointly as and

and inversely as the cube root of .

3.7 #30 The number of

people that a ham can serve

varies directly as the weight

of the ham. An 8-lb ham feeds 20 people.

a. How many people will a 10-lb ham serve?

b. How many people will a 15-lb ham serve?

c. How many people will an 18-lb ham serve.

d. If a ham feeds 30 people, what is the weight

of the ham?

3.7 #40 The resistance of a wire varies directly as its length and inversely as the square of its diameter. A 50-ft wire with a 0.2-in. diameter has a resistance of 0.0125 . Find the resistance of a 40-ft wire with a diameter of 0.1-in.

Chapter 3 of 23


Some Chapter 3 Review Problems

1. Given that is a zero, find the other

zeroes of ( ) .

2. Sketch: ( ) ( )( )

3. Solve: ( ) ( )( )

4. Find the vertex: ( )

5. Graph:

( )

Chapter 3 of 23


6. Graph: ( )

7. Find all the zeroes of

.

8. Solve:

Chapter 3 of 23


9. Use the Rational Zero Theorem and

Descartes' Rule of Signs to find all the zeroes of

( ) .

10. Given ( ) ,

a. Determine the end behavior of the graph of the

function.

b. List all possible rational zeros.

c. Find all the zeros of (and state the multiplicities

of) ( ) .

d. Determine the -intercepts.

e. Determine the -intercepts.

f. Is ( ) even,

odd, or neither?

g. Graph ( ) .

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