Download Unit 1: Basic Concepts

Last update: July 2013
College Algebra K Level Scope and Sequence
2013-14
First Semester
Sections in
Book
MA0314
SLOs
Time
(days)
Unit 1: Review of Basic Concepts and
Polynomials
R.2, R.3, R.4
NA
18
Unit 2: Rational and Radical Expressions
R.5, R.6, R.7
NA
13
1.1, 1.2
NA
10
Unit 4: Quadratic Equations with Applications
1.3, 1.4, 1.5
MA1314-2
12
Unit 5: Other Equations and Inequalities with
Applications
1.6, 1.7, 1.8
MA1314-2
MA1314-6
(includes
finals week)
Unit Name
Unit 3: Linear Equations with Applications
Total
25
78
Second Semester
Unit Name
Sections in
Book
Unit 6: Linear Functions
2.1 - 2.8
Unit 7: Polynomial and Rational Functions
3.1 - 3.6
Unit 8: Exponential and Logarithmic
Functions
4.1 - 4.6
Unit 9: Systems and Matrices
5.1, 5.2, 5.6,
5.7
MA0314
SLOs
MA1314-1
MA1314-3
MA1314-1
MA1314-2
MA1314-3
MA1314-4
MA1314-6
MA1314-1
MA1314-2
MA1314-3
Time
(days)
MA1314-5
(includes
finals week)
Total
26
22
25
26
99
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 1: Review of Basic Concepts and Polynomials
Textbook Sections: R.2, R.3, R.4
Enduring Understandings:
Time Spent: 20 Days
The student understands that mathematics is a structure with rules and definitions that
can be applied to solve problems.
The student understands the importance of the skills required to manipulate symbols in
order to solve problems.
The student understands that expressions can be mathematically manipulated to create
equivalent forms of the expression.
The student understands that polynomials can be written in a variety of ways and still
be equivalent.
Vocabulary:
natural numbers, whole numbers, integers, rational numbers, irrational numbers, real
numbers, simplify, evaluate, base, power, exponent, monomial, binomial, trinomial,
polynomial, degree, term, coefficient, constant, leading coefficient, factor
Student Learning Outcomes:
No MATH1314 SLOs apply here.
The student will know…
•
•
•
the characteristics of the
number lines and the
coordinate plane for all real
numbers.
that numbers can be
categorized into sets
(Natural, Whole, Integers,
Rational, Irrational, Real).
that properties of real
numbers can be used to
simplify expressions.
The student will be able to…
•
•
•
identify the set to which a given number belongs.
 Ex: π is an irrational number
apply properties of real numbers and order of operations
to evaluate and/or simplify expressions.
apply properties of real numbers and order of operations
to evaluate and/or simplify expressions.
 2 1   5  1 
 Ex:  − +  − 
−  − 
 9 4   18  2  
3  16
32
40 
 Ex:
y+
z−


8 9
27
9 
 Ex:
−5y + x
− xy
when x = −4 and y = 2
•
use exponent rules to simplify monomial expressions.
•
 3x 2 y 
 Ex: 
 2 x −1y 3 


simplify expressions, including +, −, ×, ÷.
5
2
5
 Ex:
+ 2 + 3
p p
p
2
(
 Ex: m (5m − 2 ) + 9 (5 − m ) − 2 m2 − 2m + 5
•
)
factor polynomials of various types.
 GCF (Ex: 28 x 4 y 2 + 7 x 3 y − 35x 4 y 3 )
 Grouping (Ex: 10ab − 6b + 35a − 21 )
 Trinomials (Ex: 6k 2 + 5kp − 6 p2 )
 Perfect square trinomials (Ex: 9m2 n2 + 12mn + 4 )
 Difference of two squares (Ex: 36 z 2 − 81y 4 )
 Sum/Difference of two cubes (Ex: 8m3 − 27n3 )
 Factoring with substitution (Ex: (2n + 1)2 − 9n2 )
2
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 2: Rational and Radical Expressions
Textbook Sections: R.5, R.6, R.7
Enduring Understandings:
Time Spent: 13 Days
The student understands that expressions can be mathematically manipulated to create
equivalent forms of the expression.
The student understands that concepts involving simple expressions (such as numerical
fractions) can be applied to more complex expressions.
The student understands that there are rules that define how an expression can be
mathematically manipulated, including rules involving exponents, rational expressions,
and radicals.
Vocabulary:
rational expression, domain (of a rational expression), zero-factor Property,
fundamental principle of fractions, least common denominator, complex fraction,
negative exponent, rational exponent, product rule, quotient rule, power to a power,
radical notation, radical sign, radicand, index, nth root, principal nth root, like radicals,
unlike radicals
Student Learning Outcomes:
No MATH1314 SLOs apply here.
The student will know…
•
•
•
•
•
•
•
that there are some value(s)
for which a rational
expression may be
undefined.
the fundamental principle of
fractions,
ac
a
for b ≠ 0, c ≠ 0 , and
=
bc
b
how to apply it.
the properties of exponents,
particularly as they apply to
negative and rational
exponents.
that the properties of
exponents may be combined
to simplify certain types of
expressions.
that complex fractions may
be simplified, in some cases,
from the standpoint of
negative exponents.
the product, quotient, and
power rules for radicals, and
how to use them, in
combination if necessary, to
simplify radical expressions.
that radical notation and
rational exponents of an
expression are related.
The student will be able to…
•
determine the domain over which a rational expression is
defined.
( x + 6 ) ( x + 4) is x | x ≠ −2, − 4
 Ex: the domain of
{
}
( x + 2) ( x + 4)
•
apply the fundamental principle of fractions to
multiplication, division, addition, subtraction, and
simplification of rational expressions and complex
fractions.
2 p2 + 7 p − 4
 Ex:
5p2 + 20 p
 Ex:
x3 − y 3
2 x + 2y + xz + yz
•
2
2
x −y
2 x 2 + 2y 2 + zx + zy 2
 Ex:
3p2 + 11p − 4
9 p + 36
÷
3
2
24 p − 8 p
24 p4 − 36 p3
y
8
+
y −2 2−y
3
1
− 2
 Ex: 2
x + x −2
x − x − 12
5
6−
k
 Ex:
5
1+
k
apply exponent rules to simplify expressions and solve
problems, including those with rational and negative
exponents.
25r 7 z 5
 Ex:
10r 9 z
 Ex:
•
3
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
(3x ) (3x )
(3 x )
−1
2
 Ex:
5
−1
−2
2
−2
1
 Ex: −1296 4
 Ex: 16
 Ex:
−
27
3
4
1
5
• 27
273
3
 Ex: ( y − 2 )
 Ex:
•
(x + y)
2
+ ( y − 2)
3
−1
4
5
(
=5 −32
(p
p2 + q =
 Ex:
2
)
4
+q
)
=
16
( −2) =
4
1
2
simplify radical expressions
64
 Ex: 6
729
 Ex:
6
( −2)
 Ex:
3
m • 3 m2
 Ex:
7 3
 Ex:
3
 Ex:
6
2
81x 5 y 7 z 6
98 x 3 y + 3x 32 xy
 Ex:
•
3
x −1 + y −1
convert between radical notation and the rational
exponent form of an expression.
 Ex: ( −32 )
•
−1
3
9
 Ex:
(
 Ex:
3
 Ex:
(
63
2 +3
5
−
x6
3
)(
3
8 −5
)
4
x9
)
2
11 − 1  3 11 + 3 11 + 1 


 Ex: 4 −64
rationalize the denominator of a rational expression with
a radical in the denominator.
7
 Ex:
3− 7
4
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 3: Linear Equations with Applications
Textbook Sections: 1.1, 1.2
Enduring Understandings:
Time Spent: 10 Days
The student understands the basic terminology of linear equations and how to solve
them.
The student understands that certain principles and methods can be applied to finding
solutions to real-world problems.
The student understands that there are many practical real-world applications that can
be modeled with linear equations.
Vocabulary:
equation, solution, root, solution set, variable, first-degree equation, identity, conditional
equation, contradiction, empty set, literal equation, independent variable, dependent variable
Student Learning Outcomes:
No MATH1314 SLOs apply here.
The student will know…
•
•
•
•
•
•
the basic terminology of
equations and the definition
of a linear equation in one
variable.
that there is a variety of
methods than can be used
to solve linear equations.
that the process of solving
equations can be broken
down into steps.
that there are different types
of linear equations.
the addition and
multiplication principles of
equality.
that linear equations can be
used to model real-world
situations.
The student will be able to…
•
solve linear equations on one variable.
 Ex: 3 (2 x − 4 ) =7 − ( x + 5)
•
remove fractions from an equation by multiplying both
sides of the equation by the common denominator.
2x + 4 1
1
7
 Ex:
+ x = x−
3
2
4
3
 Ex: 0.3(x + 2) – 0.5(x + 2) = -0.2x – 0.4
identify different types of linear equations (i.e. identities,
conditional equations, contradictions).
 Ex: 0 = 0 (Identity)
 Ex: x = 3 (Conditional)
 Ex: -3 = 7 (Contradiction)
solve a literal equation for a specified variable.
 Ex: Solve A − P =
P r t for P
•
•
•
solve certain types of geometry, simple interest, motion,
and mixture problems that can be modeled with linear
equations.
 Ex: If the length of each side of a square is increased by 3 cm, the
perimeter of the new square is 40 cm more than twice the length of
each side of the original square. Find the dimensions of the
original square.
 Ex: Latoya borrowed $5240 for new furniture. She will pay it off in
11 months at an annual simple interest rate of 4.5%. How much
interest will she pay?
 Ex: Maria and Eduardo are traveling to a business conference. The
trip takes 2 hours for Maria and 2.5 hours for Eduardo, since he
lives 40 miles further away. Eduardo travels 5 mph faster than
Maria. Find their average rates.
 Ex: Charlotte is a chemist. She needs a 20% solution of alcohol.
She has a 15% solution on hand as well as a 30% solution. How
many liters of the 15% solution should she add to 3 L of the 30%
solution to obtain her 20% solution?
 Ex: An artist has sold a painting for $410,000. He needs some of
the money in 6 months and the rest in 1 year. He can get a treasury
bond for 6 months at 4.65% and one for a year at 491%. His
broker tells him the total investment will earn a total of $14,961.
How much should be invested at each rate to obtain that amount of
interest?
5
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 4: Quadratic Equations with Applications
Textbook Sections: 1.3, 1.4, 1.5
Enduring Understandings:
Time Spent: 12 Days
The student understands that the number system used to solve problem scan be
extended to include nonreal numbers (i.e. complex numbers).
The student understands the basic terminology of quadratic equations and how to solve
them.
The student understands that certain principles and methods can be applied to finding
solutions to real-world problems.
The student understands that there are many practical real-world applications that can
be modeled with quadratic equations.
Vocabulary:
imaginary unit, complex number, real number, pure imaginary, complex conjugate,
standard form (a+bi), quadratic equation, second-degree equation, standard form
( ax 2 + bx + c =
0 ), zero-factor property, double root, square root property, completing
the square, quadratic formula, cubic equation, discriminant, Pythagorean theorem
Student Learning Outcomes:
MA1314-2. Recognize, graph, and apply polynomial, [rational, radical, exponential, logarithmic
and absolute value functions] and solve related equations.
The student will know…
•
•
•
•
•
•
•
the definition and
characteristics of the
imaginary unit and complex
numbers.
the definition of a quadratic
equation.
the zero-factor property and
how to apply it.
the square root property and
how to apply it.
that there are various
methods of solving a
quadratic equation,
including: factoring, taking
the square root, completing
the square, and the
quadratic formula.
the significance of the
discriminant in a quadratic
equation.
that there are real-world
applications problems that
can be modeled with
quadratic equations.
The student will be able to…
•
•
rewrite an expression with a negative radicand using the
imaginary unit
4i
 Ex: −16 =
perform operations (+, -, ×, ÷) on complex numbers,
including using a complex conjugate for division and/or
simplification.
−48
 Ex:
24
−8 + −128
4
 Ex: 3 − ( 4 − i ) − 4i + ( −2 + 5i )
 Ex:
 Ex:
(
2 + 4i
)(
2 + 4i
)
−3 + 4i
2−i
simplify powers of i.
 Ex: i15
solve quadratic equations using various methods (zerofactor property, square root property, completing the
square, quadratic formula).
−4 x2 + x =
−3
 Ex:
 Ex:
•
•
−4 x2 + x + 3 =
0
0 etc.
( −4x − 3)( x − 1) =
6
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
 Ex:
−3
(5x − 3)2 =
(5x − 3)2
=−3
5x − 3 =±i 3
5x= 3 ± i 3
=
x
3±i 3
3 i 3
or
±
5
5
5
7
−4 x2 + 8 x =
 Ex:
−4( x2 − 2 x +
)=
7
−4( x2 − 2 x + 1) =7 − 4
2
3 etc.
−4 ( x − 1) =
 Ex: 2 x 2= x − 4
1 ± 12 − 4(2)(4)
etc.
2(2)
solve cubic equations.
 Ex: x 3 + 8 =
0
solve literal quadratic equations for a specified variable.
 Ex: rt 2 − st
= k (r ≠ 0) Solve for t.
x =
•
•
•
•
identify the discriminant of a quadratic equation and use
it to identify the type of solutions for that equation.
 Ex: 5x 2 + 2 x − 4 =
0 has a discriminant of
22 − 4(5)(−4) =
84 so the equation has 2 irrational
solutions.
solve application problems modeled by quadratic
equations (area and volume, Pythagorean theorem,
height of a projectile).
 Ex: A piece of machinery is capable of producing rectangular
sheets of metal such that the length is three times the width. If
squares measuring 5 in by 5 in are cut from each of the four
corners, the resulting piece of metal can be shaped into a box by
folding up the sides. If the volume of the box must be 1435 in3 ,
what should the dimensions of the original piece of metal be?
 Ex: Erik finds a piece of land in the shape of a right triangle. The
longer leg of the triangle is 20 m longer than twice the length of the
shorter leg. Find the lengths of the sides of the triangular lot.
 Ex: If a projectile is shot vertically upward from the ground with
an initial velocity of 100 ft/sec, neglecting air resistance, its height
s, in feet above the ground t seconds after projection is given by
s=
−16t 2 + 100t . a) After how many seconds will the projectile
be 50 ft above the ground? b) How long will it take for the
projectile to return to the ground?
 Ex: See textbook (Ex 4, p. 125)
7
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 5: Other Equations and Inequalities with Applications
Textbook Sections: 1.6, 1.7, 1.8
Enduring Understandings:
Time Spent: 25 Days
The student understands the basic terminology of radical equations and inequalities and
how to solve them.
The student understands the basic terminology of rational equations and inequalities
and how to solve them.
The student understands that certain principles and methods can be applied to finding
solutions to real-world problems.
The student understands that there are many practical real-world applications that can
be modeled with radical and rational equations and inequalities.
Vocabulary:
rational equation, linear inequality, interval notation, open interval, closed interval,
quadratic inequality, rational inequality, absolute value equation, absolute value
inequality
Student Learning Outcomes:
MA1314-2. Recognize, graph, and apply polynomial, rational, radical, [exponential, logarithmic]
and absolute value functions and solve related equations.
MA1314-6. Solve absolute value, polynomial and rational inequalities.
The student will know…
•
•
•
•
•
•
•
there are real-world
situations that can be
modeled with rational or
radical equations.
the concepts and
terminology of inequalities
and their solution sets.
there are various methods
for solving linear, three-part,
quadratic, absolute value,
and rational inequalities.
that the solution to an
inequality can be expressed
in various ways, including
set-builder notation and
interval notation.
the definition and properties
of absolute value
expressions.
there are real-world
situations that can be
modeled with absolute value
equations and inequalities.
The student will be able to…
•
•
•
•
solve rational equations that lead to linear equations.
3x − 1
2x
 Ex:
−
=
x
x −1
3
solve rational equations that lead to quadratic equations.
3x + 2 1
−2
 Ex:
+
=2
x −2
x
x − 2x
solve real-world problems modeled by rational equations
such as work/rate problems.
 Ex: One computer can do a job twice as fast as the other. Working
together, both computers can do the job in 2 ours. How long would
it take each computer, working alone, to do the job?
solve equations containing a radical including one square
root, two square roots, or one or more cube roots.
0
 Ex: x − 15 − 2 x =
2x + 3 − x + 1 =
1
 Ex:
3
•
0
 Ex: 4 x 2 − 4 x + 1 − 3 x =
interpret rational exponents to create and solve a radical
equation.
2
 Ex: ( x + 1)
(
 Ex: 5x 2 − 6
•
•
− ( x + 1)
3
)
1
4
1
3
−2 =
0
=
x
solve higher degree equations
 Ex: 12 x 4 − 11x 2 + 2 =
0
solve inequalities, including: linear, quadratic, threepart, and rational.
2x − 1
 Ex:
<5
3x + 4
8
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
1
2
1
1
x + x − ( x + 3) ≤
3
5
2
10
x−4
 Ex: −3 ≤
<4
−5
solve real-world problems modeled by inequalities.
 Ex: If the revenue and cost of a certain product are given by
C 2 x + 1000 , where x is the number of units
R = 4 x and =
produced and sold, at what production level does R at least equal
C?
 Ex: If a projectile is launched from ground level with an initial
velocity of 96 ft per sec, its height in feet t seconds after launching
is s feet, where s =
−16t 2 + 96t . When will the projectile be
greater than 80 ft above ground level?
use absolute value to describe distances.
 Ex: “k is less than 5 units from 8” becomes k − 8 ≥ 5
 Ex:
•
•
•
solve absolute value equations and inequalities.
2x + 3
 Ex:
=1
3x − 4
5 1
2
 Ex:
− x >
3 2
9
1
 Ex: 5x +
−2 <5
2
 Ex: 3x − 7 + 1 < −2

9
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 6: Linear Functions
Textbook Sections: 2.1 - 2.8
Enduring Understandings:
Time Spent: 26 Days
The student understands the relationship between the algebraic and geometric
representations of a circle and parts of a circle.
The student understands the basic terminology of relations and linear functions.
The student understands that often the relationship between two quantities can be
modeled by a mathematical relation, and in some of those cases, a linear function.
The student understands that linear functions can be represented in a variety of ways.
The student understands that the graphs of parent functions have characteristics such
as continuity and symmetry which set them apart from other functions.
The student understands that functions can be combined through performing operations
(+, -, ×,÷) or by composition.
Vocabulary:
ordered pair, rectangular coordinate system, distance formula, midpoint formula, xaxis, y-axis, Cartesian coordinate system, coordinate plan, quadrants, coordinates,
graph, x-intercept, y-intercept, circle, center-radius form, general form, relation,
function, domain, range, function notation, increasing function, decreasing function,
constant function, linear function, standard form, slope, point-slope form, parallel,
perpendicular, horizontal, vertical, continuous function, identity, square, square root,
cube, cube root, greatest integer function, expansion, compression, reflection,
symmetry, even function, odd function, translations, composite function
Student Learning Outcomes:
MA1314-1. Demonstrate and apply knowledge of properties of functions, including domain and
range, operations, compositions, inverses and piecewise defined functions.
MA1314-3. Apply graphing techniques.
The student will know…
•
•
•
•
•
•
•
•
•
the distance and midpoint
formulas.
that the equation of a circle
describes the set of points on
the circle.
that an ordered pair is a
solution of an equation if its
values make the equation
true when substituted in for
the variables.
the definition of a relation.
that a function is a particular
kind of relation.
how to interpret functional
notation.
the concept of slope in a
linear function.
that the domain of a function
represents the independent
(x) values and the range
represents the dependent (y)
values of the function.
the unique equations for a
vertical line (x=a) and a
The student will be able to…
•
calculate the distance between two points.
 Given P ( −5, − 7 ) ; Q ( −13, 1) ,
PQ=
( −13 − −5)
2
+ (1 − −7 ) = 8 2
2
) (
(
 Given P 3 2, 4 5 ; Q
•
state the domain and range for a function.

•
•
•
)
2, − 5 , find PQ.
Ex: =
y
7 − 2x
determine if three given points can be the vertices of a
right triangle.
 Ex: Are ( −6, 4 ) , ( 0, − 2 ) and ( −10, 8 ) vertices of a
right triangle?
find the midpoint of a segment defined by two points or
an endpoint when given the midpoint and the other
endpoint.
 Ex: Given midpoint M (5, 8 ) and endpoint (13, 10 ) ,
find the coordinates of the other endpoint.
write the equation of a circle in center-radius from given
information.
 Ex: Given center (2, 0) and radius 6
 Ex: Write the equation of the circle from a graph.
10
College Algebra K Level Scope and Sequence 2013-14
•
•
•
•
•
•
•
horizontal line (y=b) through
a point (a, b).
that transforming the graph
of a parent function will
create infinitely many new
functions.
the definition and
characteristics of even and
odd functions.
a linear function can be
written in different forms.
the concept of continuity.
functions can be added,
subtracted, multiplied, or
divided to create a new
function.
the concept of composition of
functions.
the graphs and
characteristics of the
following parent functions:
 identity (y=x)
 square (y=x2)






square root (y =
3
•
convert between the center-radius form and general
form of the equation of a circle and vice versa.
 Ex:
0
x2 + y 2 + 6 x + 8y + 9 =
x2 + 6 x
2
+ y 2 + 8y
=
−9
2
16
x + 6 x + 9 + y + 8y + 16 =
16
( x + 3)2 + ( y + 4)2 =
•
•
x)
cube (y=x )
absolute value (y=|x|)
1
inverse (y = )
x
inverse squared
1
(y = 2 )
x
greatest integer
(y=||x||)
Last update: June 2013
•
determine if a given relation is a function.
 Ex: {(5, 1), (3, 2), (4, 9), (7, 8)}
Ex:
Ex:
 Ex:
 Ex: x + y < 3
−7
 Ex: y =
x −5
identify the domain and range and intervals of
increasing and decreasing of a function.
5
 Ex: y =
x −1
find the values of functions.
 Ex: Find f (−1) for f =
( x ) 2x 2 − 9
{
}
 Ex: Find f (−1) for f =
( −4, 0) , ( −1, 6 ) , (0, 8)
 Ex: Find f (−1) for
•
•
•
•
write a linear function in various forms (i.e. slopeintercept, point-slope, standard) and convert between
the forms.
1
=
 Ex: Write the function y − 3
( x + 4) in slope2
intercept form and in standard form.
recognize graphs of basic functions and identify
discontinuities, if they exist.
represent intervals of increasing and decreasing in a
function.
 Ex: Identify the intervals of increasing and
decreasing for the function shown.
graph functions by hand and on a graphing calculator.
3
 Ex: through (-1, 3) with slope
2
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College Algebra K Level Scope and Sequence 2013-14
•
•
Last update: June 2013
 1

 Ex: through  − , 4  , m =
0
2


find the slope of a linear function.
 Ex: through (2, -1) and (-3, -3)
 Ex: vertical line through (4, -7)
 Ex: 4x+3y=12
write the equation of a line in standard and slopeintercept form.
 Ex: through ((1, 3) and slope is 2
 Ex: through (-1, 3) and (3, 4)
 Ex: slope is 5 and y-intercept is 15
 Ex: vertical line through (-6, 4)
 Ex: horizontal line through (-7, 4)
 Ex: through (-1, 4) and parallel to x + 3y =
5
 Ex: through (-5, 6) and perpendicular to x = −2
 Ex: find k so that the line through (4, -1) and (k, 2)
is perpendicular to the line 2y − 5x =
1
•
•
•
•
determine the slope and y-intercept of each line.
 Ex: x+2y=-4
graph piece-wise functions with domain restrictions.
if −5 ≤ x < −1
 2x

 Ex: f ( x ) =  −2
if −1 ≤ x < 0
 x 2 − 2 if
0≤ x ≤2

write a piece-wise function given a graph.
 Ex:
transform the graphs of functions, including
translations, reflections, and dilations and write the
equations of transformed graphs.
 Ex: Describe the transformations on the parent
function for the following:
y x2 − 1
 =
 y = x +2 −3
 f (x) =
− ( x + 1)
3
 f=
(x) 2 x + 1
•
determine symmetry of functions.
 Ex: Describe the symmetry of each of the following:
 =
y x2 + 4
y
 =
x −3
 2x − y =
6
 x2 + y 2 =
25
•
determine algebraically whether a function is even, odd,
or neither.
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College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
 Ex: g ( x ) =x 5 + 2 x 3 − 3x
•
 Ex: 2 x 2 − 3
 Ex: x 2 + 6 x + 9
perform operations on functions and the composition of
functions.
−2 x 2 − 1 , find:
 Ex: For f ( x=
) x − 3 and g ( x ) =



( f + g ) (3)
( f − g ) ( −1)
( fg ) (2)
f 
   ( −1)
g
 Ex: Let f ( x )= 2 − x , find:
 f ( x + h)
 f ( x + h) − f ( x )

•
f ( x + h) − f ( x )
h
find the composition of two given functions.
2
5
and g ( x ) = , find ( f  g ) ( x )
 Ex: Given f ( x ) =
x
x+4
and ( g  f ) ( x )
1
( x − 2) , prove
4
that ( f  g ) ( x ) = x and ( g  f ) ( x ) = x .
x ) 4 x + 2 and g=
 Ex: For f ( =
(x)
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College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 7: Polynomial and Rational Functions
Textbook Sections: 3.1 - 3.6
Enduring Understandings:
Time Spent: 22 Days
The student understands the basic terminology and characteristics of polynomial
functions and how to apply them.
The student understands the basic terminology and characteristics of rational functions
and how to apply them.
The student understands that the relationship between quantities is often important in
applied mathematics and can often be represented as variations.
Vocabulary:
polynomial function, leading coefficient, zero, quadratic function, parabola, vertex,
synthetic division, remainder theorem, factor theorem, rational zeros theorem,
fundamental theorem of algebra, multiplicity of the zero, conjugate zeros theorem,
turning point, end behavior, reciprocal function, asymptote, discontinuous, rational
function, oblique, direct variation, inverse variation, combined variation, joint variation
Student Learning Outcomes:
MA1314-1. Demonstrate and apply knowledge of properties of functions, including domain and
range, operations, compositions, inverses and piecewise defined functions.
MA1314-2. Recognize, graph, and apply polynomial, rational, [radical, exponential, logarithmic]
and absolute value functions and solve related equations.
MA1314-3. Apply graphing techniques.
MA1314-4. Evaluate all roots of higher degree polynomial and rational functions.
MA1314-6. Solve [absolute value,] polynomial and rational inequalities.
The student will know…
•
•
•
•
•
•
•
the definition and
characteristics of a
polynomial function.
that linear and quadratic
functions are polynomial
functions.
the division algorithm.
that synthetic division is a
shortcut for long division of
polynomials.
the remainder theorem.
the definition and
characteristics of a rational
function.
some relationships between
quantities can be expressed
as a variation.
The student will be able to…
•
•
graph and analyze a quadratic function using
transformation techniques.
2
1
 Ex: f ( x ) =
− x − 4) + 5
(
2
transform the general form of a quadratic function to the
vertex (graphing) form using completing the square.
 Ex: f ( x ) = 2x 2 − 4x + 5
=
2( x 2 − 2 x
)+5
2
= 2( x − 2 x + 1) + 5 − 2
= 2 ( x − 1) + 3
2
•
•
•
•
analyze the quadratic models of real-world situations.
 Ex: Sum and product of two numbers
 Ex: Area problems
 Ex: Projectiles
divide polynomials using long division or synthetic
division.
5x 3 − 6 x 2 − 28 x − 2
 Ex:
x +2
use the remainder theorem to determine values and
zeros of polynomial function.
 Ex: Determine if 2+i is a zero of
f ( x ) = x 3 + 3x 2 − x + 1
use the factor theorem and/or conjugate zeros theorem
to factor polynomials.
 Ex: Factor f ( x ) = 24 x 3 + 80 x 2 + 82 x + 24
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College Algebra K Level Scope and Sequence 2013-14
•
Last update: June 2013
find all complex zeros of a polynomial in exact value
 Ex: The zeros of f ( x ) = 5x 3 − 9 x 2 + 28 x + 6 are −
•
•
•
•
1
5
and 1 ± i 5 .
identify the characteristics of the graph of a polynomial
function, including zeros, turning points, and end
behavior.
 Ex: f ( x ) =x 4 + 3x 3 − 3x 2 − 11x − 6
graph and analyze the function f ( x ) = ax n where n ∈
{natural numbers}, and its transformations
graph and analyze the reciprocal (inverse) functions
1
1
f (x) =
and f ( x ) = 2 and their transformations.
x
x
−1
 Ex:
=
+4
g (x)
2
( x + 5)
identify the characteristics of the graph of a rational
P (x)
where P(x) and Q(x) are
function, f ( x ) =
Q (x)
polynomial functions: including intercepts, asymptotes
(vertical, horizontal, and oblique), removable
discontinuities, and end behavior.
x 2 − 16 ( x + 3)
 Ex: f ( x ) =
x2 − 9 ( x + 4)
(
(
•
)
)
identify a variation problem and set up an equation to
represent the situation, including direct variation, inverse
variation, joint variation, and combined variation.
 Ex: Simple interest varies jointly as principal and time. If $1000
left in an account for 2 years earned $110, find the amount of
interest earned by $5000 for 5 years.
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College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 8: Exponential and Logarithmic Functions
Textbook Sections: 4.1 - 4.6
Enduring Understandings:
Time Spent: 25 Days
The student understands the basic terminology and characteristics of exponential
functions and how to apply them.
The student understands the basic terminology and characteristics of logarithmic
functions and how to apply them.
The student understands the inverse relationship between exponential and logarithmic
functions.
The student understands the equivalence relationship between exponential and
logarithmic expressions and can utilize this relationship to solve problems.
Vocabulary:
inverse operation, one-to-one function, horizontal line test, inverse function,
exponential function, compound interest, e, compounding continuously, logarithm,
base, argument, logarithmic function, properties of logarithms, common log, natural
log, Change-of- Base Theorem, exponential growth, exponential decay, carbon dating
Student Learning Outcomes:
MA1314-1. Demonstrate and apply knowledge of properties of functions, including domain and
range, operations, compositions, inverses and piecewise defined functions.
MA1314-2. Recognize, graph, and apply [polynomial, rational, radical,] exponential, logarithmic
[and absolute value functions] and solve related equations.
MA1314-3. Apply graphing techniques.
The student will know…
•
•
•
•
•
•
the definition of the inverse
of a function.
that an exponent can have a
real irrational value
the definition and
characteristics of an
exponential function.
the number e and its value in
application problems.
the definition and
characteristics of a
logarithmic function.
the properties of logarithms,
including common and
natural logarithms, and the
change of base formula.
The student will be able to…
•
•
determine if a function is one-to-one by using the
horizontal line test and/or algebraically.
determine the inverse of a function and if the inverse is
a function.
x −3
4x + 3
and g ( x ) =
,
 Ex: Given f ( x ) =
x+4
1− x
determine if f −1 ( x ) = g ( x )
•
graph and analyze exponential functions
=
f ( x ) ax , a > 0 .
•
graph and analyze exponential function
transformations.
− 23 x +6 + 5
 Ex: f ( x ) =
•
use the properties of exponents to solve exponential
equations.
 Ex: 32− x + 4 = 8x −6
2
•
 Ex: r 3 = 4
use exponential functions to calculate values involving
compound interest, including continuously compounded
interest.
r

 Ex:=
A P 1 + 
n

•
nt
 Ex: A = Pert
identify the logarithmic function as the inverse of a
given exponential function and vice versa.
16
College Algebra K Level Scope and Sequence 2013-14
•
Last update: June 2013
apply properties of logarithms evaluate and/or simplify
expressions and solve equations.
 Ex: Simplify logm
5r 3
25
2
1
log5 5m2 + log5 25m2
3
2
7
 Ex: Solve log4 x =
2
apply the change of base formula to evaluate
expressions.
graph and analyze logarithmic functions with
transformations.
x ) 3log2 ( − x ) − 4
 Ex: f (=
 Ex: Simplify −
•
•
•
•
use common logs and natural logs to solve problems
involving logarithms.
 Ph of solutions
 Loudness of sound
 Earthquake intensity
use the properties of exponents and logarithms to solve
equations.
 Ex: 32 x +1 = 5x −3
3
 Ex: log2 ( x + 6 ) − log2 ( x + 2 ) =
•
solve problems involving exponential growth and decay.
 Finance
 Population growth
 Radioactive decay
 Carbon dating
 Newton’s Law of Cooling
17
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
Unit 9: Systems and Matrices
Textbook Sections: 5.1, 5.2, 5.6, 5.7
Enduring Understandings:
Time Spent: 26 Days
The student understands that linear systems can be solved by different methods to
achieve the same solution.
The student understands that a system containing non linear equations can contain 1,
2, 3, 4 or no solutions.
The student understands that there are many practical real-world applications that can
be modeled with systems of equations and inequalities.
The student understands that functions representing real-world situations can be
optimized by representing a region of feasible solutions.
Vocabulary:
linear systems, independent, dependent, inconsistent, matrix, augmented matrix,
Gauss-Jordan method, optimization, optimal amount, region of feasible solutions
Student Learning Outcomes:
MA1315-5. Recognize, solve and apply systems of linear equations using matrices.
The student will know…
•
•
•
•
the definition and
characteristics of systems of
equations.
the basic terminology and
structure of matrices.
that matrices can be added,
subtracted and multiplied.
that the multiplication of
matrices is not commutative
and is dependent on the
dimensions of the matrices.
The student will be able to…
•
solve systems of linear equations by a variety of
methods: substitution method, elimination method, or
matrices.
 Ex: 4 x + 3y =
−13
−x + y =
5
 Ex:
•
x+y+z =
2
2x + y − z =
5
x − y + z =−2
determine is a system is one of the following:
 inconsistent (with no solution)
 independent (with one solution)
 dependent (with infinitely many solutions)
 Ex:
9 x − 5y =
1
−18 x + 10y =
1
•
•
determine the dimensions of a matrix and identify any
square, column, or row.
perform operations on matrices, including:
 addition
 −4 3  2 −8
 Ex: 
=

12 −6  5 10 


•
•
subtraction
 Ex:  −10 −4 6  −  −2 5 3
multiplication
1 2   −1
 Ex: 
 
3 4  7 
 scalar multiplication
use equivalent matrices to find indicated values.
 −3 a  c 0 
 
=

 b 5 4 d 
convert a system of linear equations into an
18
College Algebra K Level Scope and Sequence 2013-14
Last update: June 2013
augmented matrix and vice versa.
2 x + 3y =
11
2 3 11
 Ex:
becomes 

x + 2y =
8
1 2 8 
•
3x + 2y + z =
1
3 2 1 1


 Ex:  0 2 4 22 becomes
2y + 4 z =
22
 −1 −2 3 15 
− x − 2y + 3z =
15
solve a system using the Gauss-Jordan method on a
graphing calculator.
 Ex: x + y − 5z =
−18
3x − 3y + z =
6
x + 3y − 2 z =
−13
•
give all solutions of nonlinear systems of equations.
 Ex: x 2 − y =
0
x+y =
2
 Ex: 2 x 2 + 3y 2 =
5
•
3x 2 − 4y 2 =
−1
graph two-variable inequalities on a coordinate plane.
 Ex: x + 2y ≤ 6
 Ex: y ≤ x 2 − 4
•
graph the solution set of a system of inequalities.
 Ex: x + y ≥ 0
2x − y ≥ 3
 Ex: x + 2y ≤ 4
y ≥ x2 − 1
•
use linear programming to solve real-world
applications.
 Ex: Robin takes vitamin pills each day. She wants at least 16
units of Vitamin A, at least 5 units of Vitamin B, and at least 20
units of Vitamin C. She can choose between red pills, costing 10
cents each, that contain 8 unit of Vitamin A, 1 unit of Vitamin B,
and 2 units of Vitamin C; and blue pills costing 20 cents each
that contain 2 unit of Vitamin A, 1 unit of Vitamin B, and 7 units
of Vitamin C. How many of each pill should she buy to minimize
her cost and yet fulfill her daily requirements?
19