Algebra 2

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APRArithmetic With Polynomials and Rational Expressions
APRArithmetic With Polynomials and Rational Expressions
The symbol for a binomial squared. A plus B in parentheses squared.
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.1.1
The symbol for a binomial squared. A plus B in parentheses squared.
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.
A-APR.3.4
The symbol for a binomial squared. A plus B in parentheses squared.
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.2.2
The symbol for a binomial squared. A plus B in parentheses squared.
Rewrite simple rational 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 the more complicated examples, a computer algebra system.
A-APR.4.6
The symbol for a binomial squared. A plus B in parentheses squared.
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.2.3
CEDCreating Equations
CEDCreating Equations
The symbol for a binomial squared. A plus B in parentheses squared.
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. ★
A-CED.1.1
The symbol for a binomial squared. A plus B in parentheses squared.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★
A-CED.1.4
The symbol for a binomial squared. A plus B in parentheses squared.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ★
A-CED.1.2
The symbol for a binomial squared. A plus B in parentheses squared.
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. ★
A-CED.1.3
REIReasoning With Equations and Inequalities
REIReasoning With Equations and Inequalities
The symbol for a binomial squared. A plus B in parentheses squared.
Explain each step in solving a simple equation 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.
A-REI.1.1
The symbol for a binomial squared. A plus B in parentheses squared.
Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x² = 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 them as a ± bi for real numbers a and b.
A-REI.2.4.B
The symbol for a binomial squared. A plus B in parentheses squared.
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 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 cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. ★
A-REI.4.11
The symbol for a binomial squared. A plus B in parentheses squared.
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A-REI.1.2
The symbol for a binomial squared. A plus B in parentheses squared.
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.3.6
The symbol for a binomial squared. A plus B in parentheses squared.
Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
A-REI.2.4.A
The symbol for a binomial squared. A plus B in parentheses squared.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3.
A-REI.3.7
SSESeeing Structure in Expressions
SSESeeing Structure in Expressions
The symbol for a binomial squared. A plus B in parentheses squared.
Interpret expressions that represent a quantity in terms of its context. ★ a. Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.1.1.a
The symbol for a binomial squared. A plus B in parentheses squared.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ a. Factor a quadratic expression to reveal the zeros of the function it defines.
A-SSE.2.3.A
The symbol for a binomial squared. A plus B in parentheses squared.
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. ★
A-SSE.2.4
The symbol for a binomial squared. A plus B in parentheses squared.
Interpret expressions that represent a quantity in terms of its context. ★ b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.
A-SSE.1.1.b
The symbol for a binomial squared. A plus B in parentheses squared.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A-SSE.2.3.B
The symbol for a binomial squared. A plus B in parentheses squared.
Use the structure of an expression to identify ways to rewrite it. For example, see x^4- y^4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).
A-SSE.1.2
The symbol for a binomial squared. A plus B in parentheses squared.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^(1/12))^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A-SSE.2.3.c
BFBuilding Functions
BFBuilding Functions
Functions Logo. f(x)
Write a function that describes a relationship between two quantities. ★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.1.1.A
Functions Logo. f(x)
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★
F-BF.1.2
Functions Logo. f(x)
Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x³ or f(x) = (x+1)/(x–1) for x ≠ 1.
F-BF.2.4.A
Functions Logo. f(x)
Find inverse functions. d. Produce an invertible function from a non-invertible function by restricting the domain.
F-BF.2.4.d
Functions Logo. f(x)
Write a function that describes a relationship between two quantities. ★ b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F-BF.1.1.B
Functions Logo. f(x)
Use the change of base formula.
F-BF.2.a
Functions Logo. f(x)
Find inverse functions. b. Verify by composition that one function is the inverse of another.
F-BF.2.4.b
Functions Logo. f(x)
Write a function that describes a relationship between two quantities. ★ c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
F-BF.1.1.C
Functions Logo. f(x)
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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-BF.2.3
Functions Logo. f(x)
Find inverse functions. c. Read values of an inverse function from a graph or a table, given that the function has an inverse.
F-BF.2.4.c
IFInterpreting Functions
IFInterpreting Functions
Functions Logo. f(x)
For a function 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. ★
F-IF.2.4
Functions Logo. f(x)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.3.7.A
Functions Logo. f(x)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
F-IF.3.7.d
Functions Logo. f(x)
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = 1.02^t, y = 0.97^t, y = 1.01^12t, y = 1.2^(t/10), and classify them as representing exponential growth or decay.
F-IF.3.8.B
Functions Logo. f(x)
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 engines in a factory, then the positive integers would be an appropriate domain for the function. ★
F-IF.2.5
Functions Logo. f(x)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-IF.3.7.b
Functions Logo. f(x)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift.
F-IF.3.7.e
Functions Logo. f(x)
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-IF.3.9
Functions Logo. f(x)
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. ★
F-IF.2.6
Functions Logo. f(x)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F-IF.3.7.c
Functions Logo. f(x)
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
F-IF.3.8.A
LELinear, Quadratic, and Exponential Models
LELinear, Quadratic, and Exponential Models
Functions Logo. f(x)
For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. ★
F-LE.1.4
Functions Logo. f(x)
Interpret the parameters in a linear or exponential function in terms of a context. ★
F-LE.2.5
TFTrigonometric Functions
TFTrigonometric Functions
Functions Logo. f(x)
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle; Convert between degrees and radians.
F-TF.1.1
Functions Logo. f(x)
Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.
F-TF.3.8
Functions Logo. f(x)
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
F-TF.1.2
Functions Logo. f(x)
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. ★
F-TF.2.5
GPEExpressing Geometric Properties With Equations
GPEExpressing Geometric Properties With Equations
Geometry Logo. Includes four pictures of solids on an orange background.
Derive the equation of a parabola given a focus and directrix.
G-GPE.1.2
CNThe Complex Number System
CNThe Complex Number System
Number and Quantity Logo
Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
N-CN.1.1
Number and Quantity Logo
Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
N-CN.1.2
Number and Quantity Logo
Solve quadratic equations with real coefficients that have complex solutions.
N-CN.3.7
QQuantities
QQuantities
Number and Quantity Logo
Define appropriate quantities for the purpose of descriptive modeling. ★
N-Q.1.2
RNThe Real Number System
RNThe Real Number System
Number and Quantity Logo
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^3 = 5^((1/3)*3) to hold, so (5^(1/3))^3 must equal 5.
N-RN.1.1
Number and Quantity Logo
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N-RN.1.2
CPConditional Probability and the Rules of Probability
CPConditional Probability and the Rules of Probability
Statistics and Probability Logo. Includes four types of graphs.
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.1.1
Statistics and Probability Logo. Includes four types of graphs.
Construct and 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 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.1.4
Statistics and Probability Logo. Includes four types of graphs.
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. ★
S-CP.2.7
Statistics and Probability Logo. Includes four types of graphs.
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.1.2
Statistics and Probability Logo. Includes four types of graphs.
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 lung cancer. ★
S-CP.1.5
Statistics and Probability Logo. Includes four types of graphs.
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.1.3
Statistics and Probability Logo. Includes four types of graphs.
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. ★
S-CP.2.6
ICMaking Inferences and Justifying Conclusions
ICMaking Inferences and Justifying Conclusions
Statistics and Probability Logo. Includes four types of graphs.
Understand statistics as a process for making inferences about population parameters based on a random sample from that population. ★
S-IC.1.1
Statistics and Probability Logo. Includes four types of graphs.
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. ★
S-IC.2.4
Statistics and Probability Logo. Includes four types of graphs.
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
S-IC.1.2
Statistics and Probability Logo. Includes four types of graphs.
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. ★
S-IC.2.5
Statistics and Probability Logo. Includes four types of graphs.
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. ★
S-IC.2.3
Statistics and Probability Logo. Includes four types of graphs.
Evaluate reports based on data. ★
S-IC.2.6
IDInterpreting Categorical and Quantitative Data
IDInterpreting Categorical and Quantitative Data
Statistics and Probability Logo. Includes four types of graphs.
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. ★
S-ID.1.4
MDUsing Probability to Make Decisions
MDUsing Probability to Make Decisions
Additional Course Resources
Additional Course Resources
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FAIM Assessment Form 1A
FAIM Form 1A
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FAIM Assessment Form 2B
FAIM Form 2B
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FAIM Answer Key Form 2A
FAIM Answer Key 2A
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FAIM Assessment Form 1B
FAIM Form 1B
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FAIM Answer Key Form 1A
FAIM Answer Key 1A
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FAIM Answer Key Form 2B
FAIM Answer Key 2B
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FAIM Assessment Form 2A
FAIM Form 2A
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FAIM Answer Key Form 1B
FAIM Answer Key 1B