Chapter 3 Page 1 of 23
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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 Page 2 of 23
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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 Page 3 of 23
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3.1 #18 Use ( ) .
a. Write the function in vertex form.
b. Identify the vertex.
c. Identify the -intercept(s).
d. Identify the -intercept(s).
e. Sketch the function.
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.
b. Identify the vertex.
c. Identify the -intercept(s).
d. Identify the -intercept(s).
e. Sketch the function.
f. Determine the axis of symmetry.
g. Determine the minimum and maximum
values of the function.
h. State the domain and range.
Chapter 3 Page 4 of 23
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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).
d. Identify the -intercept(s).
e. Sketch the graph.
f. Determine the axis of
symmetry.
g. Determine the minimum and maximum
values of the function.
h. State the domain and range.
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 Page 5 of 23
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3.2 – Introduction to Polynomial Functions
Key Ideas:
The domain of a polynomial function is
___________________.
Informally, a polynomial function can be
drawn________________
___________________.
Chapter 3 Page 6 of 23
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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.
( )
3.2 #40 Find the zeros of the function and state
the multiplicities.
( )
3.2 #44 Find the zeros of the function and state
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 Page 7 of 23
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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 Page 8 of 23
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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 Page 9 of 23
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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 Page 10 of 23
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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 Page 11 of 23
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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.)
3.4 #30 Find all the zeros.
( )
Chapter 3 Page 12 of 23
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3.4 #34 Find all the zeros.
( )
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 Page 13 of 23
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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 Page 14 of 23
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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 Page 15 of 23
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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 Page 16 of 23
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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 Page 17 of 23
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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 Page 18 of 23
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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 Page 19 of 23
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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 Page 20 of 23
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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 Page 21 of 23
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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 Page 22 of 23
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6. Graph: ( )
7. Find all the zeroes of
.
8. Solve:
Chapter 3 Page 23 of 23
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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 ( ) .