Advanced+Algebra+S2

** Advanced Algebra Lesson Objectives **

 * Semester 2 Semester 1 **

** Chapter 8 ** **In this chapter you will:**
 * **use a third variable to write parametric equations that separately define //x// and //y//**
 * **simulate objects in motion with parametric equations**
 * **review the trigonometric ratios - sine, cosine, and tangent**

8.3 The student will


 * Learn about right angle trigonometry.
 * Define the trigonometric ratios sine, cosine, and tangent
 * Use trigonometric ratios to find the unknown lengths of the sides of a right triangle
 * Use inverse trigonometric ratios to find the unknown angle measures of a right triangle
 * Use trigonometric ratios and parametric equations to model and solve problems about motion
 * Find bearings corresponding to angles with the //x//- and //y//-axes, and find angles corresponding to given bearings.

**In this chapter you will:**
 * Chapter 5 **
 * **write explicit equations for geometric sequences**
 * **use exponential functions to model real-world growth and decay scenarios**
 * **review the properties of exponents and the meaning of rational exponents**
 * **learn how to find the inverse of a function**
 * **apply logarithms, the inverses of exponential functions**

5.1 The students will


 * Learn about exponential functions
 * Write explicit equations for geometric sequences
 * Define // exponential function // and recognize // y=ab x // as the parent function
 * See real-world growth and decay situations, and recognize that the exponential function models growth when // b // is greater than 1 and models decay when // b // is less than 1
 * Learn about half-life and doubling time
 * Evaluate an exponential function using either the explicit equations or graphical methods.

5.2 The students will


 * Learn about properties of exponents and power functions
 * Introduce algebraic proof
 * Review the properties of exponents, specifically, rewriting powers with the same base
 * Introduce the parent power function //y = ax b //, and distinguish it from the exponential function // y = ab x //
 * Find some solutions to power equations using the properties of exponents
 * Introduce rational exponents as a means of solving equations

5.3 The students will


 * Learn about rational exponents and roots
 * Discover that x ½ is equivalent to [[image:http://upload.wikimedia.org/math/2/d/a/2daa5f53f1bccd91041fe1506cc9f47e.png]] Define rational exponents as equivalent roots or roots raised to powers
 * Introduce the root notation
 * Formally define the point-ratio form of an exponential function
 * Use the point-ratio form to find an exponential equation through two points.

5.4 The students will


 * Learn about applications of exponential and power equations
 * Find solutions to real-world applications of rational-exponential, exponential, and power functions

5.5 The students will


 * Learn about building inverses of functions
 * Define the inverse relation of a function and recognize that the inverse may not be a function or may be a function over a restricted domain
 * Given several points in a function, find inverse points
 * See relation-inverse symmetry across the line //y = x//
 * Find the composition of functions with their inverses
 * Apply inverses in real-world situations such as time-distance and temperature-pressure

5.6 The students will


 * Learn about logarithmic functions
 * Review solving equations for exponents
 * Define logarithm with base b and common logarithm with base 10
 * Discover, using a calculator, that //log(10 x ) = x//
 * Solve logarithmic equations with base 10 and with bases other than 10
 * Establish that the inverse of an exponential function is a logarithmic function

5.7 The students will


 * Learn about properties of logarithms
 * Use the properties of exponents to multiply, divide, and exponentiate with logarithms
 * Learn to change bases so that all logarithms can be calculated using common logarithms
 * Formally define the properties of logarithms

5.8 The students will


 * Learn about applications of logarithms
 * Take the logarithm of both sides of an exponential equation to solve it
 * Use logarithms to straighten a curve so that an algebraic model of the data can be found

** Chapter 6 ** **In this chapter you will:**
 * **use matrices to organize information**
 * **add, subtract, and multiply matrices**
 * **solve systems of linear equations with matrices**
 * **graph two-variable inequalities on a coordinate plane and solve systems of inequalities**
 * **write and graph inequalities that represent conditions that must be met simultaneously**

6.1 The students will
 * Learn about matrix representations
 * Create transition diagrams and transition matrices
 * Relate transition diagrams to matrices
 * Understand matrices as a way to organize information
 * Learn vocabulary associated with matrices: //dimension, row, column, entry//
 * Find subsequent totals from the given initial value and percentages

6.2 The students will
 * Learn about matrix operations
 * Learn these matrix operation: addition, scalar multiplication, and multiplication
 * Analyze matrix dimensions to determine whether they can be added or multiplied
 * Use matrices as a tool for describing transformations of geometric shapes
 * Understand dilations as stretches or shrinks by the same factor both horizontally and vertically

6.3 The students will
 * Learn about the row reduction method
 * Represent a system of equations as an augmented matrix
 * Interpret reduced row-echelon form
 * Understand and use row operations in a matrix
 * Use matrix row reduction to solve systems of equations

6.4 The students will
 * Learn about solving systems with inverse matrices
 * Find the inverse of a matrix if it exists
 * Use inverse matrices to solve systems of equations
 * Understand the characteristics of an identity matrix

6.5 The students will
 * Learn about systems of linear inequalities
 * Review linear inequalities and the solution of an inequality, including multiplying and dividing by a negative number
 * Write inequalities to describe given real-world constraints
 * Graph systems of inequalities
 * Interpret the meaning of the points within a feasible region
 * Find the vertices of a feasible region

6.6 The students will
 * Learn about linear programming
 * Learn about uses of linear programming and optimization
 * Apply linear programming to situations with two variables
 * Optimize a function over a feasible region

** Chapter 7 ** **In this chapter you will:**
 * **find polynomial functions that fit a set of data**
 * **study quadratic functions in general form, vertex form, and factored form**
 * **find roots of a quadratic equation from a graph, by factoring, and by using the quadratic formula**
 * **define complex numbers and operations with them**
 * **identify features of the graph of a polynomial function**
 * **use division and other strategies to find roots of higher-degree polynomials**

7.1 The students will
 * Learn about polynomial degree and finite differences
 * Define //polynomial, monomial, binomial, and trinomial//
 * Determine the degree of a polynomial by using finite differences
 * Explore falling objects
 * Write polynomials in general form
 * Use finite differences and systems of equations to find a polynomial function that fits a data set

7.2 The students will
 * Learn about equivalent quadratic forms
 * Understand the correspondence between the zeros of a polynomial function and the roots of an equation
 * Use the zero-product property to find the roots of equations
 * Comprehend the relationships among the general form, the factored form, and the vertex form of a quadratic equation
 * Relate the vertex form of a quadratic equation to the parent function //y=x 2 //

7.3 The students will
 * Learn about completing the square
 * Explore projectile motion
 * Understand completing the square as one way to convert the general form of a quadratic equation to vertex form
 * Use formulas to convert the general form of a quadratic equation to vertex form
 * Use the vertex form of a quadratic equation to solve problems involving maximums or minimums

7.4 The students will
 * Learn about the quadratic formula
 * Use the vertex form of a quadratic equation to find the equation's roots
 * Derive the quadratic formula from the vertex form of a quadratic equation
 * Use the quadratic formula to solve application problems

7.5 The students will
 * Learn about complex numbers
 * Define complex numbers as numbers of the form //a + bi//, where //i 2 = -1//
 * Identify and find the conjugate of a complex number
 * Find nonreal solutions as conjugate pairs
 * Explore arithmetic computations with complex numbers
 * Introduce a complex plane with a real axis and an imaginary axis

7.6 The students will
 * Learn about factoring polynomials
 * Explore functions defined by 3rd-degree polynomials (cubic functions)
 * Use graphs of polynomial equations to find the roots and write the equation in factored form
 * Relate the graphs of polynomial equations to the number and types of roots

7.7 The students will
 * Learn about higher-degree polynomials
 * Investigate extreme values and the end behavior of polynomial functions
 * Identify possible degrees of a polynomial function by looking at its graph
 * Identify and find the lowest-degree polynomial that has given roots
 * Find additional roots when given one complex root
 * Relate the graphs of polynomial equations to the number and types of roots

7.8 The students will
 * Learn more about finding solutions
 * Learn to find zeros of higher-degree polynomial functions when only a few of the zeros can be found from its graph
 * Use the Rational Root Theorem to identify rational numbers that might be zeros
 * Confirm that a number is a zero of a polynomial function by using the Factor Theorem and use long division to factor and find zeros of higher-degree polynomial functions
 * Understand and use synthetic division as a shortened form of long division for polynomials

** Chapter 9 ** **In this chapter you will:**
 * **use the distance formula to find the distance between two points on a plane and to solve distance and rate problems**
 * **study rational functions and learn special properties of their graphs**
 * **add, subtract, multiply, and divide rational expressions**

9.1 The students will
 * Learn about using the distance formula
 * Derive the distance formula from the Pythagorean Theorem
 * Apply the distance formula to problems involving a locus of points

9.6 The students will
 * Learn about rational functions
 * Investigate inverse variations
 * Define rational function
 * Graph and find the equations of transformations of the parent function //y = 1/x//
 * Use rational functions to solve mixture problems and for other applications

9.7 The students will
 * Learn about graphs of rational functions
 * Identify characteristics of the graph of a rational function from its graph
 * Learn to write the equation of a rational function from its graph
 * Rewrite a function as a rational function
 * Review factoring
 * Identify holes, vertical asymptotes, //x//-intercepts, //y//-intercepts, and horizontal asymptotes of rational functions

9.8 The students will
 * Learn about operations with rational expressions
 * Learn to add, subtract, multiply, and divide rational expressions
 * Review factoring
 * Simplify rational expressions in order to find //x//-intercepts, vertical asymptotes, and holes