Course: Algebra II Year: 2016-2017 Teacher: various Unit 1: Functions Standards Essential Questions Enduring Understandings A.REI.D.10 1) What is a function A function is a special Understand that the and how can it be kind of relation where graph of an equation in represented? each member of the two variables is the set domain of the of all its solutions 2) What is the function is associated plotted in the meaning of the with exactly one coordinate plane, often domain and range of a member of the range forming a curve (which function? of the function. could be a line). Different types of A.CED.2 Create 3) What is a family of functions can be equations in two or functions? transformed in similar more variables to ways. represent relationships 4) What is the effect between quantities; of a transformation on graph equations on the dependent and coordinate axes with independent variables labels and scales. of a function? F.IF.1 Understand that a function from one set 5) How can functions (called the domain) to be used to model real another set (called the world situations, make range) assigns to each predictions, and solve element of the domain problems? exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the Approximate Time Frame: 8-9 Weeks Skills Content Vocabulary Determine if a given relation is a function or not. Given a function, determine its domain and range. Represent a function by an equation, table, graph, or verbal description and move comfortably from one representation to another. Determine the reasonableness of the domain and range of a function in a realistic context. Provide examples of functions that are not smooth and in one piece, such as piecewise functions and step functions. Be able to graph piecewise and step functions, and to Relations and Functions (includes domain and range) Parent Functions and Transformations Piecewise Functions (including even and odd functions) Absolute value function Asymptote Axis of symmetry Boundary line Constant Function Constant interval Constraint Decreasing interval Dependent variable Domain Equivalent equation Even function Exponential function Family of functions Floor function Function Function notation Greatest integer function Horizontal and vertical axes Horizontal shift/translation Increasing interval Independent variable Input and output of a function Inside change Interval notation
input x. The graph of f is the graph of the equation y = f(x). F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and describe their properties. Determine if a function has odd or even symmetry. Represent a verbal description of a function transformation symbolically. Understand the difference between a transformation of an independent variable and a dependent variable. Given a function f(x), be able to describe the effects of the transformations f(x) + k, f(x + k), and kf(x) for a constant k. Given a graph of a function and a transformation of that function, be able to determine the transformation that is represented in the graph. Given a graph of a function, be able to graph (by hand) a transformation of that function. Isometry Line symmetry Linear growth Maximum/minimum Natural domain Nonlinear growth Objective function Odd function Ordered pair Origin Outside change Parabola Parameter Parent function Piecewise function Quadratic function Range Reflection Relation Root function Rotational symmetry Slope Step function Symmetric with respect to the origin Symmetric with respect to the y-axis Transformation Vertex Vertical stretch Vertical shift/translation
absolute value functions. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F.IF.B.4 For a function Explain the meaning of the value of an output from a function Use functions to answer questions in realistic contexts.
that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. A.REI.11b Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equations f(x) = g(x); find the solutions approximately, e.g. using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Unit 2: Quadratic Functions Extended Topics Standards Essential Questions Enduring Understandings A.SSE.3 Choose and 1) How can I The study of quadratics produce an equivalent determine the nature transcends the real form of an expression of the roots of a number system and to reveal and explain quadratic equation includes complex properties of the based on the value of solutions quantity represented by the discriminant? Dynamic software, the expression. graphing calculators, A.SSE.3a Factor a 2) What is the and other technology quadratic expression to definition of the can be used to explore reveal zeros of the imaginary number i? and deepen our function it defines. understanding of A.REI.A.2 Solve 3) What is the mathematics. simple rational and structure of the radical equations in one complex numbers? variable, and give examples showing how 4) What is the extraneous solutions Fundamental Theorem may arise. of Algebra, and what A.REI.4 Solve is its relation to quadratic equations in quadratic functions? one variable. A.REI.4b Solve 5) How do I solve quadratic equations by radical equations? inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write Approximate Time Frame: 4-5 Weeks Skills Content Vocabulary Identify types of numbers: natural (counting), whole, integers, rational, irrational, real, complex. Determine whether a set of numbers is closed under an operation. Determine whether or not a number, or sum or product of numbers is rational or irrational. State the definition of i. Express the square root of a negative number in terms of i. Evaluate i n for any natural number n. Identify the real and imaginary parts of a complex number, using the a+bi notation. State the conjugate of a complex number. Add, subtract, and multiply complex numbers. Complex Numbers (including number sets, powers of i, operations, complex conjugates, solving quadratics with complex solutions) Real and Complex Zero s Radical Equations Closed Sets Closure of sets under an operation Complex Conjugate Complex Number Discriminant Extraneous Root Extraneous Solution Imaginary Number Monomial Radical Equation Radicand Root Sets of Numbers (Natural (aka counting), Whole, Integers, Rational, Irrational, Real, Complex) Square Root Equation Standard Form of a Quadratic Function Zeros of a Function (Real & Complex)
them as for real numbers a and b. BF.A.1 Write a function that describes a relationship between two quantities. CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F.IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and Solve quadratic equations that have complex solutions. State the two solutions in a+bi form. Given a single complex solution to a quadratic equation, give the other solution the complex conjugate. This is true only if the quadratic equation has coefficients that are real numbers. Determine the number of roots of a quadratic equation based on the discriminant. Determine the sign of the discriminant of a quadratic equation given the graph of the corresponding quadratic function, making the connection between the x-intercepts of the graph of the function and the solutions of the quadratic equation. Add, subtract, and
absolute value functions. N.CN.1 Know there is a complex number i such that, and every complex number has the form a + bi with a and b real. N.CN.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (Note that only functions with real coefficients are considered in this investigation.) N.RN.3 Explain why the sum or product of two rational numbers is rational and the sum of a rational and an irrational number is multiply complex numbers, often in the context of verifying the sum and product of the zeros of a quadratic equation. Demonstrate that the Fundamental Theorem of Algebra is true for any quadratic function with real coefficients, i.e., any function of the form f(x) = ax 2 + bx + c has at least one zero within the complex numbers, by finding the solutions to the equation 0 = ax 2 + bx + c and using these solutions to factor the quadratic. Solve equations involving one and multiple square root terms. Solve radical equations graphically. Determine if a solution obtained from an equationsolving process is extraneous.
irrational; and that the product of a nonzero rational number and an irrational number is irrational. Unit 3: Polynomial Functions Standards Essential Questions Enduring Understandings F.IF.7c Graph 1) What are the basic Polynomial functions polynomial functions, features of a can be analyzed to identifying zeros when polynomial function determine their suitable factorizations based on the degree of unique characteristic are available, and the polynomial? s including degree, showing end behavior. end behavior, zeros, F.IF.7 Graph functions 2) How is the division and number of expressed symbolically of a polynomial, P(x), extrema and show key features by a binomial of the When comparing an of the graph, by hand in form (x a) connected exponential model simple cases and using to the polynomial with a polynomial technology for more function, y = P(x). model, the question is complicated cases. not if the exponential F.IF.4 For a function 3) What is the model will generate that models a connection between very large or very relationship between the zeros of a small inputs, but two quantities, interpret polynomial function, rather when. key features of graphs the x-intercepts of the Polynomials are closed and tables in terms of graph of the under addition, the quantities, and polynomial function, subtraction, and sketch graphs showing and the factors of the multiplication. key features given a polynomial? verbal description of the relationship. Key 4) How can features include: polynomial functions intercepts; intervals be applied to where the function is applications and increasing, decreasing, mathematical models? Model and solve contextual problems involving radical equations. Approximate Time Frame: 8-9 Weeks Skills Content Vocabulary Determine the basic shape and end behavior of the graph of a polynomial function based on the term of highest degree of the polynomial. Determine the x- and y-intercepts of a polynomial function by inspection of the equation of the function when the polynomial is in factored form. Identify extrema of a polynomial function given the graphic, symbolic, or numerical representations of the polynomial. Interpret the meaning of intercepts and extrema of a polynomial in the context of a realworld problem. Determine if a graph Properties of Polynomials Operations on Polynomials including Remainder and Factor Theorems Factoring Polynomials including the Conjugate Roots Theorem Applications of Polynomials Exponential versus Polynomial Growth Algebraic Identity Binomial Coefficient/Leading Coefficient Decreasing Function Degree Division Algorithm Divisor End behavior Exponential Function Exponential Growth Extrema Factor Theorem Increasing Function Linear Factor Mathematical Model Maximum Minimum Polynomial (monomial, binomial, trinomial) cubic quartic Power Quadratic Factor Quotient Regression Equations Remainder Remainder Theorem Root
positive, or negative; relative maxima and minima; multiplicity of roots; symmetries; end behavior; and periodicity. A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A.APR.2 Know and 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). A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A.APR.6 Rewrite simple rational 5) How does polynomial growth compare to exponential growth? of a polynomial function is tangent to the x-axis or crosses the x-axis depending on the multiplicity of the corresponding linear factor of the function. Add, subtract, and multiply polynomials. Divide polynomials using long division. State the Remainder Theorem and its implications. Determine if (x a) is a factor of P(x) by either calculating the remainder using long division or finding the value of P(a). Use the Factor Theorem to factor a polynomial given the x-intercepts or zeros of the polynomial function. Determine the zeros of a polynomial based on the factored form of the polynomial. Approximate the graph of a polynomial based on the zeros of the polynomial and end Scatter plots X-intercepts Y-intercepts Zero Zero of multiplicity Zero Product Property
expressions in different forms; write a(x)/b(x) in the form q(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), using inspection, long division, or, for more complicated examples, a computer algebra system. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 ) (x 2 + y 2 ). A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. A.CED.2 Create equations in two or more variables to behavior of the polynomial. Determine the exact equation of a polynomial function given its degree, its zeros (real or complex) and one other point on the graph. Use known polynomial identities to factor higher order polynomials of similar structure. Create a polynomial function model from real-life data using regression features of a graphing utility. Interpret the meaning of characteristics of a polynomial function in the context of a realworld problem. Construct and compare exponential and polynomial function models. Solve problems based on exponential and polynomial function models. Interpret whether a polynomial function or an exponential function best models a real-world
represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.) For example given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. A.REI.D.11 Explain why the x-coordinate of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the relationship. Recognize that quantities growing exponentially will exceed the growth of functions modeled with polynomial functions as x increases to.
solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include where f(x) and/or g(x) are linear, rational, absolute value, exponential and logarithmic functions. F.IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. Unit 4: Rational Expressions Standards Essential Questions Enduring Understandings A.APR.6 Rewrite 1) How does the The arithmetic of simple rational denominator of a rational expressions expressions in different rational function affect uses the same rules as forms; write a(x)/b(x) the domain of the the arithmetic of in the form q(x) + function? rational numbers. r(x)/b(x), where a(x), Basic concepts of b(x), q(x), and r(x) are 2) How do you rational numbers can polynomials with the perform operations on be applied to rational degree of r(x) less than and simplify rational expressions. the degree of b(x), expressions? using inspection, long division, or, for more complicated examples, a computer algebra system. Approximate Time Frame: 3-4 Weeks Skills Content Vocabulary Determine if a rational expression is fully factored and if not, factor it. Determine the domain of a rational expression. Write a rational expression in simplified form. Determine the LCD of multiple rational expressions and rewrite the expressions if Simplifying Rational Expressions (including stating domain) Multiplying and Dividing Rational Expressions Adding and Subtracting Rational Expressions Degree Domain Equivalent expressions Excluded values Factors Least common denominator Proportional Proportion Rational Expression Ratio Values excluded from the domain
A.APR.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. necessary. Extend the operational rules of rational numbers to rational expressions. Simplifying Complex Fractions Unit 5: Exponential Functions Standards Essential Questions Enduring Understandings A-CED-2 Create 1) How can you solve Some exponential equations in two or equations involving equations can be more variables to exponents? solved using represent relationships comparable between quantities; 2) What is the effect exponential graph equations on of a transformation on expressions. coordinate axes with the dependent and Exponential functions labels and scales. independent variables can be transformed in F-IF. 7e Graph of an exponential ways similar to other exponential and function? functional logarithmic functions, relationships. showing intercepts and 3) How do you find Exponential functions end behavior, and the sum of a geometric are closely related to trigonometric functions, series? geometric series. showing period, midline, and amplitude. F-IF-8 Write a function defined by an expression in different but equivalent forms to Approximate Time Frame: 3-4 Weeks Skills Content Vocabulary Rewrite exponential expressions to solve equations Represent a verbal description of a function transformation symbolically. Understand the difference between a transformation of an independent variable and a dependent variable. Given a function f(x), be able to describe the effects of the transformations f(x) + k, f(x + k), and Solving Exponential Equations (not requiring use of logarithms) Graphing Exponential Functions Transformations of Exponential Functions Finite Geometric Series Common Ratio Equivalent Equations Finite Sum Geometric Sequences Geometric Series Horizontal Asymptote Horizontal Reflection Horizontal/Vertical Shifts Horizontal/Vertical Stretches/Shrinks Summation Notation
reveal and explain different properties of the function. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. A-SSE-4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Unit 6: Right Triangle Trigonometry Standards Essential Questions Enduring Understandings G-SRT.C.8 Use 1) How are the The concept of trigonometric ratios and trigonometric ratios similarity enables us to the Pythagorean related to similarity? explore geometric kf(x) for a constant k. Given a graph of a function and a transformation of that function, be able to determine the transformation that is represented in the graph. Given a graph of a function, be able to graph (by hand) a transformation of that function. Recognize a geometric series. Use summation notation to represent a finite geometric series. Approximate Time Frame: 3-4 Weeks Skills Content Vocabulary Determine trigonometric ratios for a given angle in Trigonometric Ratios Alpha Ajacent Angle of Depression
Theorem to solve right triangles in applied problems.* G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. 2) How is trigonometry used to find unknown values and solve real world problems? relationships and apply trigonometric ratios to solve real world problems. a right triangle. Use a scientific calculator appropriately to find missing values in a right triangle. Find the missing side lengths and angle measures in a right triangle. Solve real world problems using right triangle trigonometry. Solving Right Triangles Applications of Right Triangle Trigonometry Angle of Elevation Complementary Angles Cosine Degree Mode Hypotenuse Inverse Cosine, Sine, Tangent Leg Opposite Pythagorean Theorem Ratio Sine Tangent Theta Unit 7: Probability Standards Essential Questions Enduring Understandings S-ID-5 Summarize 1) What is Understand that it is categorical data for two probability? not possible to predict categories in two-way with certainty shortterm frequency tables. 2) How are events behavior of Interpret relative defined? random phenomena frequencies in the but it is possible to context of the data 3) What is meant by use probability to (including joint, independent / predict long-run marginal, and dependent events? patterns. conditional relative Probability models are frequencies) Recognize 4) How are useful tools for possible associations probabilities, making decisions, and trends in the data. including compound choices, or S-CP-1. Describe probabilities, predictions. events as subsets of a calculated? sample space (the set of outcomes) using 5) What is conditional Approximate Time Frame: 5-6 Weeks Skills Content Vocabulary Given the description of a random process (or a sequence random processes), specify the sample space. Use set notation to specify an event. Determine the union and intersection of two (or more) events. Determine the complement of an event. Use probability to Sample Spaces Theoretical and Experimental Probability Independent Events and the Multiplication Rule Conditional Probability Interpreting Two-Way Frequency area probability model column variable combination conditional probability conditional relative frequencies dependent events event false negative false positive fundamental counting principle general multiplication rule grand total
characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP-2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S-CP-3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP-4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being probability and how does conditional probability apply to real-life events? 6) How are two-way frequency tables used to model real-life data? assess the likelihood that a particular event will occur. Use the General Addition Rule to find the probability of the union of two events. Use the Complement Rule to find the probability that an event does not occur. Create a probability model in which all simple outcomes in a finite sample space are equally likely. Use counting techniques to determine the number of ways an event can occur then determine the probability that the event occurs. Estimate the probability that an event will occur from the relative frequency that the event occurs in many trials. Tables joint relative frequencies/percent ages independent events intersection of two events marginal relative frequencies / percentages multiplication rule for independent events mutually exclusive events probability probability model random process relative frequency of an event row variable sample space true negative true positive two-way table union of two events variable
classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S-CP-5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have S-CP-6 Find the conditional probability of A given B as a Represent a sample space and events with an area probability model and use it to find probabilities. Use simulation to estimate the probability that an event will occur. Use the Multiplication Rule for Independent Events to determine whether two events are independent or dependent. Use the Multiplication Rule for Independent Events to determine the probability that two independent events occur simultaneously. Given two events A and B from an everyday situation, assess whether is larger than, smaller than, or equal to P(A). In situations in
fraction of B s outcomes that also belongs to A, and interpret the answer in terms of the model. S-CP-7. 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. which outcomes are equally likely, determine P(B A) as the ratio of the number of outcomes in A Bto the number of outcomes in A. Use an area probability model for two events A and B to calculate P(A B) by determining the fraction of B s area that overlaps with A. More generally, determine P(B A) as the ratio of P( A B) to P(A). Use the General Multiplication Rule to find P( A B). Understand how the General Multiplication Rule can be used with tree diagrams. Organize data from two survey questions into a two-way
frequency table. Given data organized in a two-way frequency table, calculate the marginal relative frequencies/percen tages. Given data organized in a two-way frequency table, calculate the joint relative frequencies/percen tages. Given data organized in a two-way table, calculate conditional relative frequencies/percen tages. Represent marginal and conditional percentages with bar graphs. Interpret marginal, joint, and conditional percentages in the context of the data. Describe associations and trends of data in
two-way tables. Determine the expected value of a probability model with numeric outcomes.