Standardized Test Preparation

New York Regents

Algebra II - Grade 11

Assessment Exam - NY Regents Algebra II
The Real Number System eTAP Lesson
Extend the properties of exponents to rational exponents.
Explore how the meaning of rational exponents follows from extending the properties of integer exponents.
Negative and Rational Exponents

The number e
Convert between radical expressions and expressions with rational exponents using the properties of exponents.
Exponential Functions and Data

Graphs of Exponential Functions
The Complex Number System eTAP Lesson
Perform arithmetic operations with complex numbers.
Know there is a complex numberisuch that i2= –1, and every complex number has the form a + bi with a and b real.
Complex Numbers

Square Root Functions
Use the relation i2= –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Complex Plane

Radical Functions

Radical Equations
Seeing Structure in Expressions eTAP Lesson
Interpret the structure of expressions.
Recognize and use the structure of an expression to identify ways to rewrite it.
Function Notation
Write expressions in equivalent forms to reveal their characteristics.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Factor quadratic expressions including leading coefficients other than 1 to reveal the zeros of the function it defines.
Direct Variation

Linear Parametric equations
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Use the properties of exponents to rewrite exponential expressions.
Comparing and Ordering Rational Numbers
Arithmetic with Polynomials and Rational Expressions eTAP Lesson
Understand the relationship between zeros and factors of polynomials.
Apply the Remainder Theorem: For a polynomial p(x)and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Multiplying and Dividing Polynomials

Graphs of Polynomial Functions
Identify zeros of polynomialfunctionswhen suitable factorizations are available.
Cubic Equations
Rewrite rational expressions.
Rewrite rational expressions in different forms: Write a(x)/b(x)in the formq(x)+ r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x).
Zeros of Polynomial Functions

Inverse Variation
Creating Equations eTAP Lesson
Create equations that describe numbers or relationships.
Create equations and inequalities in one variable to represent a real-world context.
Reasoning with Equations and Inequalities eTAP Lesson
Understand solving equations as a process of reasoning and explain the reasoning.
Explain each step when solving rational or radical equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Radical Equations
Solve rational and radical equations in one variable, identify extraneous solutions, and explain how they arise.
Solve equations and inequalities in one variable.
Solve quadratic equations in one variable. Solve quadratic equations by:
  1. inspection
  2. taking square roots
  3. factoring
  4. completing the square
  5. the quadratic formula
  6. graphing
Write complex solutions in a+bi form.
Simple Quadratic Functions


Quadratic Functions in Intercept Form

Factoring Quadratics
Solve systems of equations.
Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Graphs and Dipgrahs
Represent and solve equations and inequalities graphically.
Given the equations y=f(x)and y=g(x):
  1. recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x)
  2. find the solutions approximately using technology to graph the functions or make tables of values
  3. find the solution of f(x) < g(x)or f(x)≤g(x)graphically
  4. interpret the solution in context.

Dividing Rational Numbers

Slope Intercept Form of Equations
Interpreting Functions eTAP Lesson
Understand the concept of a function and use function notation.
Recognize that a sequence is a function whose domain is a subset of the integers.
Arithmetic and Geometric Series
Interpret functions that arise in applications in terms of the context.
For a function that models a relationship between two quantities:
  1. interpret key features of graphs and tables in terms of the quantities
  2. sketch graphs showing key features given a verbal description of the relationship.

Determine the Equation from a Relation
Calculate and interpret the average rate of change of a function over a specified interval.
Analyze functions using different representations.
Graph functions and show key features of the graphby hand and using technology when appropriate. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Graphs of Polynomial Functions

Zeros of Polynomial Functions
Graph cube root, exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.
Cubic Equations

Amplitude and Period
Analyze functions using different representations.
Write a function in different but equivalent forms to reveal and explain different properties of the function. Use the properties of exponents to interpret exponential functions, and classify them as representing exponential growth or decay.
The number e
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Sums of Geometric Series
Building Functions eTAP Lesson
Build a function that models a relationship between two quantities.
Write a function that describes a relationship between two quantities. Determine a function from context.Determine an explicit expression, a recursive process, or steps for calculation from a context.
Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations.
Sums of Arithmetic Series

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Build new functions from existing functions.
Using f(x) + k,kf(x), f(kx),and f(x+ k):
  1. identify the effect on the graph when replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)
  2. find the value of k given the graphs
  3. write a new function using the value of k
  4. use technology to experiment with cases and explore the effects on the graph. Include recognizing even and odd functions from their graphs.

Basic Operations


Functions as Graphs in the Coordinate System
Find the inverse of a one-to-one function both algebraically and graphically.
Inverse Variation
Understand inverse relationships between exponents and logarithms algebraically and graphically.
Inverses of Linear Functions
Represent and evaluate the sum of a finite arithmetic or finite geometric series, using summation (sigma) notation.
Geometric Mean, Harmonic Mean, and Weighted Arithmetic Mean
Explore the derivation of the formulas for finite arithmetic and finite geometric series. Use the formulas to solve problems.
Frequency Distribution Table, Summation Notation, and Mean Formula
Linear, Quadratic, and Exponential Models eTAP Lesson
Interpret expressions for functions in terms of the situation they model.
Construct a linear or exponential function symbolically given:
  1. a graph
  2. a description of the relationship
  3. two input-output pairs (include reading these from a table).

Inverses of Linear Functions

Inverses of Exponential Functions
Use logarithms to solve exponential equations, such as abct= d(where a, b, c,and d are real numbers and b > 0) and evaluate the logarithm using technology.
Exponential and Logarithmic Equations
Interpret the parameters in a linear or exponential function in terms of a context.
Using Logarithms to Model Data
Trigonometric Functions eTAP Lesson
Extend the domain of trigonometric functions using the unit circle.
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Sine and Cosine and the Unit Circle
Apply concepts of the unit circle in the coordinate plane to calculate the values of the six trigonometric functions given angles in radian measure.
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Amplitude and Period
Model periodic phenomena with trigonometric functions.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, horizontal shift, and midline.
Tangent Function
Prove and apply trigonometric identities.
Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. Find the value of any of the six trigonometric functions given any other trigonometric function value and when necessary find the quadrant of the angle.
Interpreting Categorical and Quantitative Data eTAP Lesson
Summarize, represent, and interpret data on a single count or measurement variable.
Recognize whether or not a normal curve is appropriate for a given data set.
Normal Distributions
If appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables.
Represent bivariate data on a scatter plot, and describe how the variables’ values are related. Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data.
Paired Data Sets and Scatterplots
Making Inferences and Justifying Conclusions eTAP Lesson
Understand and evaluate random processes underlying statistical experiments.
Determine if a value for a sample proportion or sample mean is likely to occur based on a given simulation.
Indirect Proof
Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
Recognize the purposes of and differences among surveys, experiments, and observational studies. Explain how randomization relates to each.
Discrete Probability Distribution
Given a simulation model based on a sample proportion or mean, construct the 95% interval centered on the statistic (+/- two standard deviations) and determine if a suggested parameter is plausible.
Standard Deviation of Random Variable
Use the tools of statistics to draw conclusions from numerical summaries.
Data Classification, Range, and Midrange
Use the language of statistics to critique claims from informational texts. For example, causation vs correlation, bias, measures of center and spread.
Mean and Variance of Random Variable
Conditional Probability and the Rules of Probability eTAP Lesson
Understand independence and conditional probability and use them to interpret data.
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
Multiple Events

Conditional Probability
Interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and calculate conditional probabilities.
Binomial Distributions

Normal Distributions
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
Apply the Addition Rule,P(A or B) = P(A) + P(B) -P(A and B), and interpret the answer in terms of the model.
Probability Data Plots

Dependent & Independent Variables