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Course Title:
Advanced Honors Algebra II, Grades 9 & 10, Level 5
Length of Course:
One Year (5 credits)
Prerequisites:
Algebra, Geometry
Description:
Although traditional when considering topics, this course must be
taught with any eye towards the future. It is in the study of
Algebra II that key concepts and problem solving techniques need
to be introduced and refined. Well placed questions, initiated now,
will allow the students to begin to consider, on an intuitive basis,
Calculus, Trigonometry and Discrete Math.
It is also a goal of this course to provide students with the skills
necessary to manipulate algebraic expressions expediently and
accurately.
This class must be taught following the Rule of Three. That is
each topic must be considered numerically, graphically and
algebraically. For example, we obtain solutions algebraically
when that is the most appropriate technique to use, and we obtain
solutions graphically or numerically when algebra is difficult to
use. Students must be urged to solve problems by one method and
then confirm their solutions by an alternative method. Students
must begin preparing to see the value of each of these methods and
must learn to choose the one that is most appropriate. This
approach reinforces the idea that to understand the problem fully
students need to understand it algebraically, as well as graphically
and numerically. In order for students to pursue this investigation,
we must provide them with the technology that will allow it.
Therefore, in addition to teaching Algebra II, it is critical that
students receive instruction in the graphing calculator. This class
is the place to introduce the power of this instrument. We again
must emphasize that even on a graphing calculator it is possible to
consider algebraic as well as numerical thinking.
This course strives to give students a proper balance between the
mastery of skills and the comprehension of key concepts. With
that in mind, this curriculum guide clearly defines the learning
objectives for each unit in terms of the key skills and key concepts
that must be mastered within each unit.
Evaluation:
Student performance will be measured using a variety of
instructor-specific quizzes and chapter tests as well as a common
departmental Quarter, Midterm and Final Exams. Assessments
will equally emphasize measurement of the degree to which
required skills have been mastered as well as how well key
concepts have been understood.
Text:
Algebra 2, Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff,
McDougal Littell, 2001
Topic
What is the complex
Number System
Prerequisite Unit: The Complex Number System
Learning Objectives: Key Definitions, Skills and Concepts
What are natural (counting), whole, integers, rational, irrational, real, pure
imaginary and complex numbers?
Skills check, ability to:
Identify the set(s) to which a given number belongs
Identify subset relations among the sets of numbers
Concept check:
What is closure and how is this concept related to the expansion of the
complex numbers?
What is density? Which numbers are more dense: rationals or wholes?
Explain.
Topic
2.1 Functions & Their
Graphs
Unit 2: Linear Equations & Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is the definition of a function? What is the domain & range of a
function? What is a linear function?
Skills check, ability to:
Identify functions
Graph relations
Use the vertical line test
Evaluate functions
Graph a linear function
Concept check:
What does it mean to say f is a function of x?
Is f ( x) the same as f  x ?
Are all functions
Linear functions?
Are all lines functions?
2.2 Slope & Rate of
Change
What is the slope of a line? What kinds of slopes do parallel and
perpendicular lines have?
Skills check, ability to:
Correctly find the slope of a line given two points on the line, its equation, the
equation of a parallel or perpendicular line.
Concept Check:
Do all lines have a slope?
What causes a line to have no slope?
Is slope a concept unique to a line?
2.3 Quick Graphs of
Linear Equations
How can a line be graphed using the slope-intercept method? How can a line
be graphed using the intercept method?
Skills check, ability to:
Graph a line using slope-intercept and intercept methods
Graph horizontal and vertical lines
Concept check:
Can all lines be graphed using slope-intercept or intercept methods? Explain.
2.4 Writing Equations
of Lines
What is the symbolic appearance of a linear equation? What is direct
variation?
Skills check, ability to:
Write the equation of a line given its graph, two points, the equation of a
parallel or perpendicular line, an anecdote
Identify a linear function that is direct variation
Concept check:
What, in the equation of a line, distinguishes an oblique line from one that
is horizontal or vertical?
2.5 Correlation & Best
Fitting Lines
What is a line of best fit? What does it mean to say that two quantities have a
correlation?
Skill check, ability to:
Describe a correlation given a scatterplot
Approximate a line of best fit given a set of data
Concept check:
Is it always appropriate to model data with a line of best fit? Explain.
What is extrapolation and why should this be considered in all modeling?
Graphing Calculator
Scatterplots and Linear
Regression
2.6 Linear
Inequalities
How can a set of data be entered? How can a scatterplot be shown? How can
the graphing calculator be used to obtain the linear model of the data? When
is this reliable?
How is a linear inequality graphed?
Skill check, ability to:
Graph a linear inequality
Create a linear inequality from a real world problem
Concept check:
What is it about a linear inequality that determines the kind of boundary
that is appropriate?
Topic
2.7 Piecewise
Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is a piecewise function? How are these represented symbolically
and graphically? What is a step function?
Skill check: ability to:
Recognize when a problem will result in a piecewise function
Correctly model an anecdote with a piecewise function
Correctly graph a piecewise function given its equation
Correctly evaluate a piecewise function
Concept check:
Are all piecewise functions step functions?
2.8 Absolute Value
Functions
What is the absolute value function? What is its domain and range?
Skills check, ability to:
Identify the vertex of an absolute value function given its equation
Identify whether the vertex is a “high” or “low” point on the graph given
its equation
Graph an absolute value function given its equation or an anecdote
Concept check:
Is an absolute value function a linear function? Explain.
What characteristic of the equation controls the width of an absolute value
function? Explain.
Unit 3: Systems of Linear Equations & Inequalities
Topic
Learning Objectives: Key Definitions, Skills and Concepts
3.1 Solving Systems
What is a system of equation? How is it solved graphically?
By Graphing
Skills check, ability to:
Solve a system of linear equations by graphing
Identify the graph of a system that has no solutions
Identify the graph of a system that has an infinite number of solutions
Correctly check the solution to a system of linear equations
Concept check:
Are all systems of equations linear?
What is a 2x2 system, a 3x3 system, etc?
What is the “dimension requirement” so that a unique solution exists?
5 x  6 y  24
Consider this system: 
. Explain how slopes can be used to
 2 x  3 y  15
show that the system has only one solution.
Graphing Calculator
How is a linear system of equations solved on a graphing calculator?
Skills check, ability to:
Correctly enter equation
Correctly identify a window that is appropriate
Correctly use the intersect feature
Correctly use the technology to determine when the lines are coinciding
3.2 Solving Linear
Systems Algebraically
What are the algebraic methods for solving s linear system?
Skills check, ability to:
Solve a linear system by substitution and elimination
Concept check:
Will algebraic methods always work when solving a linear system? Explain.
What is the algebraic equivalent to a system of parallel lines, a system of
coinciding lines?
What about a given system should be considered when determining a method
of solving?
Write a system of linear equations that has (-4.3, 8) as its only solution
3.3 Graphing &
Solving Systems of
Linear Inequalities
What does the solution to a system of linear inequalities look like?
How is graphing used to solve a system of linear inequalities?
Skills check, ability to:
Correctly solve a system of linear inequalities by graphing
Concept check:
Is there an algebraic equivalent to a graphing solution for a system of
linear inequalities?
3.4 Linear
Programming
What is an optimization problem? How is it solved using linear
programming? What is an objective function? What are constraints?
What is a feasible region?
Skill check, ability to:
Correctly interpret a real world situation into constraints and an objective
function
Correctly use linear programming techniques to optimize the objective
function
Concept check:
Are all graphing solutions acceptable solution?
What kinds of solutions are mandated by the context of the problem?
How do you interpret an unbounded feasible region?
Why is the optimal solution found at a vertex of the feasible region?
3.5 Graphing Three
Equations in three
Variables
How do we graph in three dimensions?
Skills check, ability to:
Correctly plot points in three dimensions
Correctly graph linear equations in three variables
Correctly model an real world problem in three variables
Concept check:
What does the equation ax  by  cz  d look like in three dimensions?
3.6 Solving Systems of
Linear Equations in
Three Variables
What are the graphing and algebraic methods used to solve a 3x3 System?
Skills check, ability to:
Correctly solve a 3x3 system of equations
Concept check:
Topic
Unit 4: Matrices & Determinants
Learning Objectives: Key Definitions, Skills and Concepts
What is a matrix? How is it described? When are two matrices considered to
be equal? What are matrix operations? What is a scalar?
4.1 Matrix Operations
Skills check, ability to:
Correctly identify the dimensions of a matrix
Correctly identify amn
Correctly add and multiply a matrix by a scalar
Concept check:
Is it always informative to add rows or columns of a matrix?
What conditions must be met so that two matrices can be added?
Is matrix addition commutative and associative? Explain.
True/False: If A and B have the same number of elements, then A+B exists
True/False: If A+B exists then A-B exists.
What is matrix multiplication?
4.2 Multiplying
Matrices
Skills check, ability to:
correctly identify when two matrices can be multiplies
correctly multiply matrices
Concept check:
Is matrix multiplication commutative and associative? Explain.
Graphing Calculator
How are matrices entered on a graphing calculator? How are matrices added
and multiplied on a graphing calculator? What error messages will appear if
matrices cannot be combined or multiplied?
4.3 Determinants &
Cramer’s Rule
What is the determinant of a matrix? How can Cramer’s rule be used to
solve systems of equations?
Skills check, ability to:
Correctly calculate the determinant of a 2x2 and 3x3 matrix
Correctly use Cramer’s Rule to solve a system of equations
Concept check:
Do all matrices have determinants? Explain the circumstances in which a
matrix fails to have a determinant.
What is the “Cramer’s Rule” equivalent to a system that has either no
solution or an infinite number of solutions?
What about a system of equations would lead one to use Cramer’s rule as
a method of solution?
4.4 Identity & Inverse
Matrices
What is the identity matrix? What is the inverse of a matrix?
Skill check, ability to:
Identify whether two matrices are inverses
Find the inverse of a 2x2 matrix
Solve matrix equations
Concept check:
Explain how you could tell if two matrices are inverses.
How are inverse matrices used to solve systems of equations?
4.5 Solving Systems
Using Inverse Matrices
Skills check, ability to:
Create a matrix equation from a system of equations
Correctly use inverse matrices to solve the system of equations
Concept check:
Suppose you try to solve a system of equations using inverse matrices and
your calculator. When you are done entering the coefficient matrix (A) and
the constant matrix (B), you call for A 1 * B and your calculator reads:
ERR: Singular Matrix. What is your calculator telling you?
True/False: Every matrix has an inverse.
Topic
5.1 Graphing Quadratic
Functions
Unit 5: Quadratic Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is a quadratic function? How can a quadratic function be graphed?
Skills check, ability to:
Identify a quadratic function given its graph or equation
Identify the vertex and line of symmetry of a parabola given its graph
Identify whether a parabola opens up or down given its graph or equation
Identify a quadratic equation in standard, intercept and vertex form
Quickly sketch a quadratic function given in vertex form
Quickly sketch a quadratic function given in intercept form
Change from intercept form and vertex form to standard form
Correctly solve a quadratic equation by graphing the associated quadratic
function.
Concept check:
In what ways are quadratic functions the same/different than linear functions?
Which coefficient(s) control opening up or down?
By expanding f ( x)  a( x  h)2  k knowing that the vertex is at  h, k  ,
demonstrate that it is reasonable for the vertex of f ( x)  ax 2  bx  c to be
b 
 b
found at  , f ( )  .
2a 
 2a
Why are the solutions to a quadratic equation found at the x-intercepts of the
associated quadratic function?
If a quadratic equation has no solutions, what does its graph look like?
If a quadratic equation has one solution, what does its graph look like?
Topic
5.2 Solving Quadratic
Equations By Factoring
Learning Objectives: Key Definitions, Skills and Concepts
How is a quadratic equations solved by factoring?
Skills check, ability to:
Factor out the greatest common factor from a polynomial
Factor the difference of two perfect squares
Factor the sum/difference of two perfect cubes
Group factor
Factor quadratic trinomials
Use the zero product property to solve a quadratic equation by factoring
Concept check:
Is it true that every quadratic equation has a solution? Explain.
Is it true that every quadratic equation has two solutions? Explain.
Is factoring a method that will work for all quadratic equations? Explain
5.3 Solving Quadratic
Equations By Square
Roots
What are radicals, radicands and index numbers? How can a quadratic
equation be solved using square roots?
Skills check, ability to:
Simplify radicals
Identify which quadratic equations can be solved by square roots
Solve a quadratic equation by square roots
Concept check:
What kinds of numerical solutions can be found using square roots that
cannot be found using graphing or factoring?
Graphing Calculator
How can a quadratic equation be entered into the calculator? How can a
graphing calculator be used to identify the vertex of a quadratic function?
How can the zero feature be used to estimate the roots of a quadratic
equation. After trying to use the zero feature to estimate the roots of a
quadratic equation your calculator says: “ERR: NO SIGN CHNG”. What
is the calculator telling you?
Topic
Learning Objectives: Key Definitions, Skills and Concepts
What is i ? What is an imaginary number? What is a complex number?
5.4 Complex Numbers
Skills check, ability to:
Simplify pure imaginary numbers using i
Add/subtract/multiply/divide complex numbers
Solve a quadratic equation having complex roots
Concept check:
Describe the powers of i . Distinguish the powers of i from the powers of a
real number.
Describe how you can evaluate i n , n  whole numbers .
1
What is the standard form of ?
i
Describe the nature of the graph of f ( x)  ax 2  bx  c when a,b, and c are
real numbers and the equation ax 2  bx  c  0 has nonreal complex solutions.
How can the process of completing the square be used to solve quadratic
equation?
5.5 Completing The
Square
Skills check, ability to:
Correctly complete the square
Solve a quadratic equation by completing the square
Change the form of a quadratic function from standard or intercept into vertex
Concept check:
Of all the methods used to solve a quadratic equation outline the pros and
cons of each. Are there some that can be used in all cases and some that have
special requirements?
Topic
5.6 The Quadratic
Formula and the
Discriminant
Learning Objectives: Key Definitions, Skills and Concepts
What is the discriminant of a quadratic equation? How is it used? How is a
quadratic equation solved using the quadratic formula?
Skills check, ability to:
Solve a quadratic equation using the quadratic formula
Calculate the discriminant of a quadratic equation
Correctly interpret the nature of the roots based upon the value of the
discriminant
Concept check:
If b 2  4ac  0 , what can you say about the zeros of the quadratic function
f ( x)  ax 2  bx  c ?
5.7 Graphing &
Solving Quadratic
Inequalities
What does the solution to a quadratic inequality with two variables look like?
How can a quadratic inequality in one variable be solved by graphing and
algebraically (using a number line)?
Skills check, ability to:
Graph a two variable quadratic inequality
Graph of system of quadratic inequalities
Solve a quadratic inequality by graphing and algebra
Concept check:
5.8 Modeling With
Quadratic Functions
What is a quadratic model? How can a data set be modeled by a quadratic
function?
Skills check, ability to:
Write a quadratic function in standard, vertex or intercept form given its
graph, equation (in an alternate form), or three points on the graph
Find a quadratic model for a given set of data
Concept check:
Is it always possible to create a quadratic model for a given data set?
When is this appropriate and when is it not? Explain.
Topic
6.1 Using Properties of
Exponents
Unit 6: Polynomials & Polynomial Functions
Learning Objectives: Key Definitions, Skills and Concepts
What are the properties of exponents and how can they be used to simplify
expressions?
Skills check, ability to:
Evaluate numerical expressions
Simplify algebraic expressions
Use scientific notation
6.2 Evaluating &
Graphing Polynomial
Functions
Concept check:
Which of the properties of exponents can be called the ‘distributive
property”?
What is a polynomial function? What do the graphs of polynomial functions
look like?
Skills check, ability to:
Identify polynomial functions, including degree, leading coefficient and
constant
Evaluate polynomial functions
Identify end behavior for a polynomial function
Graph polynomial functions
Concept check:
Are all linear functions polynomial functions?
Are all polynomial functions linear functions?
Is it possible for a third degree polynomial function to have no real zeros?
Explain.
Is it possible for a fourth degree polynomial function to have no real zeros?
Explain.
T/F: All polynomial functions have an x-intercept
T/F: All polynomial functions have a y-intercept
What are even and odd functions? Are all functions even or odd?
6.3 Adding,
Subtracting and
Multiplying
Polynomials
What are the operations on polynomials?
Skills check, ability to:
Add, subtract and multiply polynomials
Recognize the sum/difference of perfect cubes
Concept check:
What are like terms? Must they have the same coefficients?
6.4 Factoring &
Solving Polynomial
Equations
How are polynomial equations solved algebraically?
How are polynomial equations solved graphically?
Skill check, ability to:
Factor polynomial expressions using special patterns
Factor polynomial expressions using group factoring
Factor polynomial expressions in quadratic form
Solve a polynomial equation by factoring
Concept check:
Explain which factoring techniques should be used for polynomials that are
binomials, trinomials and 4-term polynomials.
6.5 The Remainder &
Factor Theorems
How are polynomials divided?
What is synthetic division and when is it appropriate?
What is the remainder theorem and what does it reveal?
Skills check, ability to:
Divide polynomials using long division
Divide polynomials using synthetic division
Decide if one polynomial is a factor of another polynomial
Find zeros of polynomial functions using division
Concept check:
Consider a polynomial function. What is the difference between the xintercepts and zeros?
Consider a polynomial equation. What is the difference between the roots of
the equation and the real zeros of the function?
Fill-in: -4 is a __________of x 2  3 x  4 =0
-4 is an _________of f ( x)  x 2  3 x  4
-4 is a __________of f ( x)  x 2  3 x  4
x+4 is a _________of x 2  3 x  4
Explain how to carry out the following division using synthetic division.
Work through the steps with complete explanations. Interpret and check your
4 x3  5 x 2  3x  1
result.
2x 1
Topic
6.6 Finding Rational
Zeros
Learning Objectives: Key Definitions, Skills and Concepts
How are the rational zeros of a polynomial function found algebraically?
What is the rational zero theorem?
Skills check, ability to:
List the possible rational zeros of a polynomial function
Identify the rational zeros of a polynomial function using the Rational Zero
Theorem
Solve polynomial equations using the rational zero theorem
Concept check:
How can the graph of a polynomial function help to identify the rational
zeros?
1
Write a paragraph describing how the zeros of f ( x)  x 3  x 2  2 x  3 are
3
3
2
related to the zeros of g ( x)  x  3x  6 x  9. In what ways does this
example illustrate how the Rational Zeros Theorem can be applied to find the
zeros of a polynomial with rational number coefficients?
6.7 Using the
Fundamental Theorem
of Algebra
What is the Fundamental Theorem of Algebra and how does it relate to the
number of zeros of a polynomial?
Skills check, ability to:
Find the number of solutions to a polynomial equation or the number of zeros
of a polynomial function
Find the zeros of a polynomial function
Write a polynomial function/equation given the zeros/roots
Concept check:
If the coefficients of a polynomial function are real and rational, why must
imaginary zeros come in pairs?
Topic
6.8 Analyzing Graphs
of Polynomial
Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is a turning point of a polynomial function?
How many turning points will a polynomial function have?
Skills check, ability to:
Graph a polynomial function given its equation in factored form
Graph a polynomial function given its equation in standard form
Concept check:
Are all turning points maximums?
Do all polynomial functions have turning points?
What is meant by multiplicity of factors?
Suppose -3, 2 and 5 are the solution to a cubic polynomial equation. Sketch a
graph of the polynomial function. Could there be more than one sketch?
How many more? Explain.
What is a local minimum? How is this different from an absolute minimum?
6.9 Modeling With
Polynomial Function
What does finite difference refer to and how does it help decide on an
appropriate polynomial model?
Skills check, ability to:
Write the equation of a polynomial function given its graph
Correctly identify the degree of a polynomial model given a set of data
Create a polynomial model of a set of data
Utilize the regression feature on a graphing calculator to find a polynomial
model
Concept check:
Topic
7.1 nth Roots and
Rational Exponents
Unit 7: Powers, Roots & Radicals
Learning Objectives: Key Definitions, Skills and Concepts
What is the nth root of a number? How is it written?
How are nth roots written using rational exponents?
Skills check, ability to:
Evaluate the real nth roots of numbers
Evaluate a constant raised to a rational power
Solve equations using nth roots
Concept check:
What is the principal root of a number?
T/F: 9  3
1
T/F: 9 2  3
7.2 Properties of
Rational Exponents
What properties of exponents also apply to rational exponents? How are
these properties used to simplify expressions?
Skills check, ability to:
Simplify expressions that have rational exponents
Multiply radicals
Rationalize an irrational denominator
Add/subtract radicals
Concept check:
Under what circumstances can two radicals be added?
Under what circumstances can two radicals be multiplied?
7.3 Power Functions &
Function Operations
What is a power function? What are function operations?
Skill check, ability to:
Identify a power function
Distinguish a power function from a polynomial function
Identify domain & range of a power function
Compose functions
Identify the domain of a composed function
Concept check:
Is f ( g ( x))  g ( f ( x)) ? Explain
7.4 Inverse Functions
What makes two functions inverses?
How do we create the inverse of a given function?
Skills check, ability to:
Identify when two functions are inverses given the equations and/or
graphs
Create the inverse of a given function
Identify when the inverse will not be a function
Use the horizontal line test to predict whether a function is invertible
Concept check:
In the expression f 1 ( x) is -1 an exponent?
T/F: All functions have inverses
Under what circumstances will a function be invertible? Explain
T/F: All linear functions are invertible
7.5 Graphing Square
Root & Cube Root
Functions
What does the graph of f ( x)  x and f ( x)  3 x look like? What are their
domain and range?
Skills check, ability to:
Graph the parent square root and cube root functions
Graph the transformations of these parent curves
Graph the transformations of other basic function (e.g.
f ( x)  x, f ( x)  x2 , f ( x)  x )
Identify domain and range of a transformed function
Concept check:
Given the graph of f ( x)  sin x, sketch f ( x)  3sin( x  4)  7
Consider g ( x)  af ( x  b)  c . Describe the effects a, b and c have on the
graph of f ( x ).
Topic
7.6 Solving Radical
Equations
Learning Objectives: Key Definitions, Skills and Concepts
What is a radical equation and how are they solved? How are equations with
rational exponents solved? What is an extraneous root?
Skills check, ability to:
Solve a radical equation algebraically
Solve an equation containing rational exponents algebraically
Identify extraneous root
Concept check:
How can a graph help identify when equations will have extraneous
solutions?
Is it always important to check solutions to equations? When is it necessary?
(Turkey into a Pig)
Topic
8.1 Exponential
Growth
Unit 8: Exponential & Logarithmic Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is an exponential function? What is an asymptote? What does
exponential growth look like algebraically and graphically? What is
concavity?
Skills check, ability to:
Recognize an exponential function algebraically & graphically
Identify the base and coefficient from an equation
Identify an asymptote from a graph and equation
State the domain and range of an exponential function given its equation
and/or graph
Recognize an exponential growth function from a graph, equation and
anecdote
Create an exponential growth function from an anecdote
Graph a transformed exponential faction
Use the compound interest formula
Concept check:
Are all exponential functions ?
Are all exponential functions concave up?
Why can’t the base of an exponential function be a negative number?
Write an problem that would have the following as its answer: P  100(5) x
Topic
8.2 Exponential Decay
Learning Objectives: Key Definitions, Skills and Concepts
What does exponential decay look like algebraically and graphically?
Skills check, ability to:
Identify the base and coefficient from an equation
Identify an asymptote from a graph and equation
State the domain and range of an exponential decay function given its
equation and/or graph
Recognize an exponential decay function from a graph, equation and anecdote
Create an exponential decay function from an anecdote
Graph a transformed exponential function
Concept check:
2 1
You are asked to sketch f ( x)  ( ) x  4 . Explain how you would do this.
3 8
Describe the effects of different values of a, b and k in the function
f ( x)  abkx .
Write two exponential functions, having the same base, with one being
growth and one being decay.
Without using formulas or graphs, compare and contrast exponential and
linear functions.
8.3 The Number e
What is e? What is it used for?
Skill check, ability to:
Sketch f ( x)  e x
Sketch transformed functions using e
Simplify expressions using powers of e
Use the continuous interest formula
Concept check:
What kind of number is e?
Consider the function f ( x)  e x . Identify its domain, range, asymptotes,
increasing/decreasing, concavity.
8.4 Logarithmic
Functions
What is a logarithmic function and why is it necessary?
How are logarithmic and exponential form related?
What are common and natural logs?
Skills check, ability to:
Change from exponential form to logarithmic form
Change from logarithmic form to exponential form
Evaluate logarithmic expressions (all bases)
Graph logarithmic functions (all bases)
Graph transformed logarithmic functions
Find the inverse of a logarithmic function
Concept check:
Evaluate: log 3 9 . Explain your answer.
Consider the function f ( x)  ln x . Identify its domain, range,
asymptotes, increasing/decreasing and concavity.
T/F: the logarithm of a positive number is positive.
Topic
8.5 Properties of Logs
Learning Objectives: Key Definitions, Skills and Concepts
What are the properties of logs and how are they related to the rules of
exponents?
Skills check, ability to:
Expand logarithmic expressions using the properties of logs
Condense logarithmic expressions using the properties of logs
Use the change of base formula
Concept check:
Write log 64 in four different ways
Can you write log3 6  log 2 5 as a single log? Explain.
Can you expand log3 (2 x  1) ? Explain.
If you knew the value of log 2 and log 5, how could you evaluate log 20
without using a calculator?
log b x
 log b x  log b y
T/F:
log b y
You wish to graph f ( x)  log5 x on your graphing calculator. You know the
calculator will not work using a base of 5. Write two other functions that are
equivalent and can be used on your calculator.
Describe how to transform the graph of
f ( x)  log x into the graph of g ( x)  log.1 x.
Topic
8.6 Solving
Exponential &
Logarithmic Equations
Learning Objectives: Key Definitions, Skills and Concepts
How do you solve exponential & logarithmic equations algebraically?
Skills check, ability to:
Solve exponential equations when the bases are or can be the same
Solve exponential equations when the bases cannot be the same
Solve a logarithmic equation with only one log
Solve a logarithmic equation with two or more logs that have the same base
Concept check:
Suppose you are solving an equation containing exponents. You find that
5
x 3  k . What is the next step?
Suppose you are solving a radical equation. You find that 6 10 y  3 . What is
the next step?
Why is it important to check the solutions to exponential and logarithmic
equations?
Your friend solves a logarithmic equation as follows:
2 log x  log 3  2
log(
x2
)2
3
x2
 102
3
x2
 100
3
x 2  300
x   300
x  10 3
Is he correct? If not explain the error.
Topic
8.7 Modeling With
Exponential & Power
Functions
Learning Objectives: Key Definitions, Skills and Concepts
How can you write an exponential function given two points?
How can you write a power function given two points?
How can you decide if a data set should be modeled by an exponential or
power function?
How can you use a graphing calculator to find a model?
Skills check, ability to:
Create an exponential and power model given two points
Decide which is an appropriate model
Use a calculator to find a model
Use a model to predict (interpolation v. extrapolation)
Concept check:
8.8 Logistic Growth
Functions
What is a logistic curve and what type of data does it accurately model?
What is a point of inflection?
Skills check, ability to:
Evaluate a logistic function
Graph a logistic function given its equation
Create a logistic model given a carrying capacity and two points
Solve a logistic equation
Concept check:
What does the inflection point indicate about the growth?
The formula for the inflection point of the function
c
 ln a c 
f ( x) 
is 
,  . Explain how this formula is derived.
 bx
1  ae
 b 2
Is it important to check the solutions to a logistic equation? Explain.
Without using formulas or graphs, compare and contrast exponential and
logistic functions.
Topic
9.1 Inverse & Joint
Variation
Unit 9: Rational Equations & Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is inverse and joint variation and what situations do they model?
What is the constant of variation?
Skills check, ability to:
Classify variation as direct or inverse
Write an inverse variation model given two points
Write an inverse or joint variation model given an anecdote
Write a direct or inverse variation model using powers
State the domain of an inverse variation function
Concept check:
Explain the differences among data that vary directly, inversely, and inversely
with the square. Include an example of each.
9.2 Graphing Simple
Rational Functions
What is a rational function? What is the graph of a rational function called?
What is a discontinuous function? What is a limit?
Skills check, ability to:
1
x
State the domain, range, equations of asymptotes of a rational function
Graph a rational function (numerator and denominator of degree 1) by finding
its vertical and horizontal asymptotes and then a few key points
Identify when a rational function will have no asymptotes
Identify when a rational function will have holes
Create a rational function that models an anecdote
Graph a simple rational function using transformations on f ( x ) 
Concept check:
Write any rational function that has a vertical asymptote at x=-7 and a
horizontal asymptote at y=4. Is this unique?
Polynomial functions have a characteristic known as end behavior. Rational
functions have limits as x   . How are these the same and how are they
different?
Topic
9.3 Graphing General
Rational Functions
Learning Objectives: Key Definitions, Skills and Concepts
How will the degrees of numerator and denominator affect the graph of a
rational function? What are local extrema?
Skills check, ability to:
Identify x and y intercepts, vertical and horizontal asymptotes, holes,
domain and range of rational functions
Graph rational functions
Create rational functions given an anecdote
Identify local and absolute extrema given a graph
Concept check:
Write a rational function that has a vertical asymptote at x=8. Write
another rational function that has a hole at x=8.
How are rational expressions multiplied and divided?
9.4 Multiplying &
Dividing Rational
Expressions
Skills check, ability to:
Multiply and divide rational expressions
Concept check:
x 2  10 x  25 x  5
T/F: 2

x  9 x  20 x  4
How can you tell when a rational expression is in simplest form?
x
Write three rational expressions that simplify to
x 1
9.5 Addition,
Subtraction and
Complex Fractions
How are rational expressions combined? How are complex fractions
simplified?
Skills check, ability to:
Combine rational expressions
Simplify complex fractions
Concept check:
T/F: The least common denominator of two rational expressions is always the
product of their denominators
T/F: The LCD of two rational expressions will have a degree greater than or
equal to that of the denominator with the highest degree
Explain how factoring is used in finding the LCD when combining two
rational expressions.
Topic
9.6 Solving Rational
Equations
Learning Objectives: Key Definitions, Skills and Concepts
What is a rational equation and how is it solved algebraically?
Skills check, ability to:
Solve rational equations and identify extraneous solutions
Concept check:
Explain why the equation
x
1
2
 
has no solutions.
x2 2 x2
Topic
10.1 The Distance &
Midpoint Formulas
Unit 10: Quadratic Relations & Conic Sections
Learning Objectives: Key Definitions, Skills and Concepts
How can you find the distance between two points, the midpoint of a line
segment and the equation of a perpendicular bisector of a line segment?
Skills check, the ability to:
Find the distance between two points
The midpoint of a line segment
The equation of the perpendicular bisector of a lone segment
Concept check:
How is the distance from a point to a line measured?
10.2 Parabolas
(In this section, the
vertex of the parabola
is at the origin)
What is the ‘conic section’ definition of a parabola? What is a focal point and a
directrix?
Skills check, ability to:
Graph a parabola given its equation
Identify the focus and directrix of a parabola given its equation/graph
Write the equation of a parabola given its graph or focus/directrix
Concept check:
Are all parabolas functions? Explain.
Consider the parabola f ( x)  ax 2 . What is the effect of increasing a ?
Explain how to find the distance from the focus to the directrix for the
parabola f ( y)  2 y 2 .
What is the reflective property of a parabola and what is its primary use?
Topic
10.3 Circles
(In this section the
center of the circle is at
the origin)
Learning Objectives: Key Definitions, Skills and Concepts
What is the equation of a circle? What is a tangent line drawn to a circle?
Skills check, ability to:
Graph a circle given its equation
Write the equation of a circle given its graph or radius
Write the equation of a tangent line drawn to a circle given the circle and
point of tangency
Concept check:
Can a circle have a radius of -5?
How can a circle be graphed on a graphing calculator?
Explain why x 2  y 2  0 does not represent a circle. What does the equation
represent?
Write the equation of a circle that does not intersect the x-Axis.
10.4 Ellipses
(In this section, the
center of the ellipse is
at the origin)
What is an ellipse? What is its equation? What are its vertices, covertices, focal points, center, major and minor axes?
Concept check, ability to:
Find the vertices of an ellipse given its equation
Find the equation of an ellipse given its focal points and another piece of
information that fixes the ellipse (e.g. minor axis length, a point on the
ellipse, etc.)
Graph ellipses given either equation and/or enough information to fix the
ellipse
Concept check:
What is a whispering gallery and how is this related to an ellipse?
What does the eccentricity of an ellipse measure? What is the eccentricity
of a circle? What is the maximum value of the eccentricity of an ellipse?
What is the least it can be?
Explain why a circle is also an ellipse.
The area of a circle is  r 2 . The area of an ellipse is  ab . Explain the
connection.
Topic
10.5 Hyperbolas
(In this section, the
center of the hyperbola
is at the origin)
Learning Objectives: Key Definitions, Skills and Concepts
What is a hyperbola? What is its equation? What are its vertices, focal
points, center, transverse axis and asymptotes?
Skills check, ability to:
Graph a hyperbola given its equation
Write the equation of a hyperbola given enough information to fix the
hyperbola
Concept check:
Are any hyperbolas functions? If yes, sketch one.
10.6 Graphing &
Classifying Conics
What is the equation of a parabola with vertex (h,k)?
What is the equation of a circle, ellipse and hyperbola with center (h,k)?
Skills check, ability to:
Write the equation of and graph a parabola whose vertex is not at the origin
Write the equation of and graph circles, ellipses and hyperbolas whose center
is not at the origin
Classify a conic given its equation in standard form
Graph a conic given its equation in standard form
Concept check:
How does the translation of an ellipse or hyperbola from center (0,0), to
center (h,k) affect the coordinates of the vertices and foci?
How does the translation of an ellipse affect the length of it major and minor
axes?
Describe how the translation of a hyperbola will affect its asymptotes.
10.7 Solving Quadratic
Systems
How can a quadratic system of equations be solved algebraically?
Skills check, ability to:
Solve a quadratic system of equations using substitution and elimination
Concept check:
Graph a nonlinear system that has three solutions
Describe the circumstance under which substitution would be better than
elimination to solve a nonlinear system.
Describe the circumstances under which elimination would be better than
substitution to solve a nonlinear system.
Recommended Unit Sequencing and Pacing Guide
Timeframe
Q1
Unit Prerequisite: The Complex Number System
Unit 1: Linear Equations & Functions
1.1 Functions and their Graphs
1.2 Slope & Rate of Change
1.3 Quick Graphs of Linear Equations
1.4 Writing Equations of Lines
1.5 Correlation & Best Fitting Lines
1.6 Linear Inequalities in Two Variables
1.7 Piecewise Functions
1.8 Absolute Value Functions
Unit 2: Systems of Linear Equations & Inequalities
2.1 Solving Linear Systems by Graphing
2.2 Solving Linear Systems Algebraically
2.3 Graphing & Solving systems of Linear Inequalities
2.4 Linear Programming
2.5 Solving Systems of Linear Equations in Three Variables
Unit 3: Matrices & Determinants
3.1 Matrix Operations
3.2 Multiplying Matrices
Timeframe
Q2
Unit 3: Matrices & Determinants
3.3 Determinants and Cramer’s Rule
3.4 Identity & Inverse Matrices
3.5 Solving Systems Using Matrices
Unit 4: Quadratic Functions
4.1 Graphing Quadratic Functions
4.2 Solving Quadratic Equations By Factoring
4.3 Solving Quadratic Equations by Finding Square Roots
4.4 Complex Numbers
4.5 Completing The Square
4.6 The Quadratic Formula & The Discriminant
4.7 Graphing & Solving Quadratic Inequalities
4.8 Modeling With Quadratic Functions
Unit 5: Polynomials & Polynomial Functions
5.1 Using Properties of Exponents
Midterm
Timeframe
Q3
Unit 5: Polynomials & Polynomial Functions
5.2 Evaluating & Graphing Polynomial Functions
5.3 Adding, Subtracting and Multiplying Polynomials
5.4 Factoring & Solving Polynomial Equations
5.5 The Remainder & Factor Theorems
5.6 Finding Rational Zeros
5.7 Using The Fundamental Theorem of Algebra
5.8 Analyzing Graphs of Polynomial Functions
5.9 Modeling With Polynomial Functions
Unit 6: Powers, Roots & Radicals
6.1 nth Roots and Rational Exponents
6.2 Properties of Rational Exponents
6.3 Power Functions and Function Operations
6.4 Inverse Functions
6.5 Graphing Square Root & Cube Root Functions
6.6 Solving Radical Equations
Unit 7: Exponential and Logarithmic Functions
7.1 Exponential Growth
7.2 Exponential Decay
7.3 The Number e
7.4 Logarithmic Functions
7.5 Properties of Logarithms
7.6 Solving Exponential & Logarithmic Equations
7.7 Modeling With Exponential & Power Functions
7.8 Logistic Growth Functions
Timeframe
Q4
Unit 8: Rational Equations & Functions
8.1 Inverse & Joint Variation
8.2 Graphing Simple Rational Functions
8.3 Graphing General Rational Functions
8.4 Multiplying & Dividing Rational Expressions
8.5 Addition, Subtraction and Complex Fractions
8.6 Solving Rational Equations
Unit 9: Quadratic Relations & Conic Sections
9.1 The Distance & Midpoint Formula
9.2 Parabola
9.3 Circles
9.4 Ellipses
9.5 Hyperbolas
9.6 Graphing & Classifying Conics
9.7 Solving Quadratic Systems
Final