Function Concepts



Algebra 2 StandardsFunction ConceptsI can…determine if a given relation is a function (F.IF.1; A1.3.A)This is an Algebra 1 standard under both Common Core and WA 2008. It is included here for the purposes of review.determine the domain and range of a function (F.IF.1; A1.3.A)This is an Algebra 1 standard under both Common Core and WA 2008. It is included here for the purposes of review.use function notation to evaluate functions for inputs in their domains and interpret statements that use function notation in a context (F.IF.2; A1.2.B; A1.3.C)This is an Algebra 1 standard under both Common Core and WA 2008. It is included here for the purposes of review.relate the domain of a function to its graph and the quantitative relationship it describes (F.IF.5; A1.3.A)For example: A theatre seats 150 people and tickets are sold for $10. Write a function relating the revenue, r, to the number of people in attendance, n. State the domain and range of the function.This standard should be embedded within all tasks throughout the year.estimate, calculate, and interpret the average rate of change of a function over a specified interval (F.IF.6)An emphasis should be placed on using the average rate of change in the context of solving real-world problems. This standard should be embedded within all tasks throughout the bine standard function types using arithmetic operations (F.BF.1b; A2.5.A)The goal is for students to write functions to model situations in a context and use these functions to solve problems. This standard should be embedded within all tasks throughout the year.For example: Model the temperature of a cooling body by adding a constant function to a decaying exponential. Model the paying down of a loan by subtracting a linear function representing monthly payments from an exponential function representing interest.determine if a given graph or algebraic expression represents an even or odd function (F.BF.3)find the inverse of a function (F.BF.4a)Students should be able to identify appropriate domain restrictions to ensure that the inverse is defined both mathematically and for a given problem pare properties of two functions represented, including those in different ways (F.IF.9)Students should be able to compare possible representations and identify strengths and weaknesses of each in a given context. This standard should be embedded within all tasks throughout the year.Representations include: algebraically, graphically, numerically in tables, or verbal descriptions.Properties include (but are not limited to): zeroes, end behavior, asymptotes, maxima, minima, periodicity, and whether the function is even or odd.Graphing FunctionsI can…recognize and use the following parent functions: constant, linear, absolute value, quadratic, cubic, square root, exponential, logarithmic, reciprocal, sine, cosine, and tangent.graph cubic, step, absolute value, and piece-wise defined functions (F.IF.7b; A2.5.D)For A2.5.D, students should be able to graph cubic functions. The remaining functions are required of F.IF.7b, which is an Algebra 1 standard under Common Core. It is included here for the purposes of review.interpret and sketch key features of graphs, tables, and functions, including (but not limited to): intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; whether the function is even or odd; and periodicity (F.IF.4)An emphasis should be placed on using these key features to solve problems in a context. This standard should be embedded within all tasks throughout the year.identify the effect on the graph of a function of replacing f(x) by fx+ k, k f(x), f(kx), and f(x+k) for specific values of k; and find the value of k given a graph (F.BF.3; A2.5.A)Linear FunctionsI can…create and solve linear equations to represent relationships between quantities (A.CED.1–3; A1.4.A)Students should be able to find the equation of a line when given: two points on the line; a point on the line and the slope of the line; a table of values; and a context in which points on the line can be determined.This is an Algebra 1 standard under both Common Core and WA 2008. It is included here for the purposes of review.apply the slope criteria for parallel and perpendicular lines (G.GPE.5; G.2.A; G.4.A)This is a Geometry standard under both Common Core and WA 2008. It is included here for the purposes of review.Quadratic FunctionsI can…create quadratic equations and inequalities to represent relationships between quantities (A.CED.1–3; A2.1.A, C; A2.3.C)Students should be able to write the equation represented by a word problem or other context. In-class work should emphasize interpreting and using these equations within the given context.solve quadratic equations (A.CED.1, 4; F.IF.8; .7; A2.1.C; A2.3.C)Students should choose an appropriate method for solving a given quadratic equation, and be able to justify their choice. Solution methods that students should be comfortable with include: the quadratic equation, factoring, completing the square, graphing, and computer algebra systems.Students should be able to solve quadratic equations with real or complex solutions.move between standard form, vertex form, and factored form of a quadratic equation (F.IF.8; A2.3.A)The goal is for students to select an appropriate representation for their needs when solving a given problem. This standard should be embedded within all tasks throughout the unit.Students should also be able to select an appropriate form to determine: zeroes, extreme values, and symmetry.determine the number and nature of the roots of a quadratic equation (F.IF.8; A2.3.B)Students should be able to recognize and interpret the discriminant.solve quadratic inequalities (A.CED.1; A2.3.C)graph quadratic functions and inequalities (F.IF.7a)Polynomial FunctionsI can…factor polynomials and prove polynomial identities (A.APR4; A.SSE.2; A1.2.E)This is an Algebra 1 standard under WA 2008.Students should know and recognize the following identities:x+y2=x2+2xy+y2x-y2=x2-2xy+y2x+yx-y=x2-y2x+ax+b=x2+a+bx+abx+y3=x3-3x2y+3xy2-y3x-y3=x3-3x2y+3xy2-y3x3+y3=x+yx2-xy+y2x3-y3=(x-y)(x2+xy+y2)add, subtract, multiply, and divide polynomials (A.APR.1, 6; A1.2.F)This is an Algebra 2 standard under Common Core. It is an Algebra 1 standard under WA 2008.identify the zeroes of a polynomial (A.APR.2; F.IF.8)Students should know and be able to apply the Remainder and Rational Root Theorems.Students should be able to use the zeroes to sketch a rough graph of the function defined by the polynomial.identify the end behavior of a polynomial (A.APR.3; F.IF.7c)Students should be able to use the end behavior to sketch a rough graph of the function defined by the polynomial.construct a rough graph of the function defined by a polynomial (A.APR.3; F.IF.7c)Students should know and be able to apply the Remainder and Rational Root Theorems.create and solve polynomial equations to represent relationships between quantities (A.CED.1–3)(+) know and apply the Binomial Theorem for expanding a binomial raised to a power (A.APR.5; A2.6.D)This standard is required of all students in Algebra 2 under WA 2008.(+) use the Fundamental Theorem of Algebra to number and characterize the roots of a polynomial (.9; A2.3.B)This standard is required of all students in Algebra 2 under WA 2008.Students should be able to demonstrate the truth of the theorem for quadratic expressions.Radical FunctionsI can…create radical equations in one variable to represent relationships between quantities (A.CED.1–3; A2.1.A)solve radical equations in one variable and identify extraneous solutions (A.REI.2; A2.5.B)Students should be able to explain how extraneous solutions arise, in addition to being able to identify them.For A2.5.B, functions should be of the form fx=ax-c+d.graph square root and cube root functions (F.IF.7b; A2.5.B)Rational FunctionsI can…rewrite simple rational expressions in different forms (A.APR.6)Students should be able to use polynomial division to write a(x)/b(x) as q(x)+r(x)/b(x).Students should be able to use this to state properties of the polynomial and its graph; for example, to find a horizontal asymptote.(+) add, subtract, multiply, and divide rational expressions (A.APR.7; A2.2.C)This standard is required of all students in Algebra 2 under WA 2008.create rational equations in one variable to represent relationships between quantities (A.CED.1–3; A2.1.A)solve rational equations in one variable and identify extraneous solutions (A.REI.2; A2.1.E; A2.5.C)For A2.1.E, students should be able to solve problems of the form f(x)=a/x+b, f(x)=a/x2+b, and f(x)=a/(bx+c).For A.REI.2, students should be able to solve problems with rational expressions on both sides of the equality and that require factoring in any or all positions. For example: Solve3(x-4)(x+2)=1x-4 .(+) construct a rough graph of a rational function (F.IF.7d)Student graphs should accurately show zeroes, asymptotes, and end behavior.Exponential and Logarithmic FunctionsI can…evaluate logarithms (F.LE.4; A2.4.A)Students should derive and be able to use the change of base formula.Students should derive and be able to use the formulas logba?logab=1.Students should be able to use the properties of logarithms, and their relationships to exponentials, to evaluate some logarithms by hand. Students should know when it is appropriate to evaluate by hand, and when to use technology.simplify exponential and logarithmic expressions using the properties of exponents and logarithms (F.LE.4; A2.4.A)Students should be able to move between exponential and logarithmic form as necessary to solve a given problem.create exponential and logarithmic equations to represent relationships between quantities (A.CED.1–3; F.IF.8; F.LE.4; A2.1.A)Students should be able to write the equation represented by a word problem or other context. In-class work should emphasize interpreting and using these equations within the given context.solve exponential and logarithmic equations (A.CED.1, 4; F.LE.4; A2.1.D; A2.4.C)Students should be able to solve equations with variable coefficients.For F.LE.4, exponentials should be of the form abct=d, with a, b, c, d∈R and b∈2, 10, egraph exponential and logarithmic functions, showing intercepts and end behavior (F.IF.7e; A2.4.B)Systems of EquationsI can…find approximate solutions to fx=gx when f(x) and/or g(x) are: linear, quadratic, polynomial, rational, absolute value, exponential, and/or logarithmic (A.REI.11; A2.1.B)Students should be able to find solutions by: graphing the functions by hand, using technology to graph the functions, making tables of values, or finding successive approximations.An emphasis should be placed on students understanding why their solutions methods are valid. Students should understand 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 fx=g(x).create and solve systems of equations and inequalities, including systems of three equations and three unknowns (A.CED.3; A2.1.B; A2.7.A)This standard should focus on exact, algebraic solutions to systems of equations. The systems should consist of classes of equations from throughout the Algebra 2 curriculum.For A2.7.A, students should be able to solve systems of three equations and three unknowns.TrigonometryI can…convert between radian and degree measure of an angle (F.TF.1)Students should know the definition of one radian as the angle subtended at the center of a circle by an arc that is equal in length to the length of the radius of the circle.use trigonometric functions to solve problems involving any real number (F.TF.2)For example: Line l is the line between the origin and the point P(9, 4). If θ is the angle between l and the x-axis, find tan(θ).use trigonometric functions to write and solve equations that model periodic phenomena with specific amplitude, frequency, and midline (A.CED.1–3; F.TF.5; A2.1.A)In-class work should emphasize interpreting and using these equations within the given context. Students should be able to use these equations to solve problems.graph trigonometric functions (F.IF.7e)Student graphs should accurately show period, midline, and amplitude.use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find sinθ,cos(θ), or tan(θ) given sinθ,cos(θ), or tan(θ) and the quadrant of the angle (F.TF.8)Students should be able to prove the identity using a reference triangle and the definitions of sine, cosine, and tangent.(+) use the addition and subtraction formulas for sine, cosine, and tangent to solve problems (F.TF.9)Complex NumbersI can…know that -1=i and write the square root of a negative number in terms of i. (.1)Students should be able to simplify square roots involving negative numbers, including rationalizing the denominator.add, subtract and multiply complex numbers (.2)(+) write polynomial identities using complex numbers (.8)Students should be able to show a chain of reasoning that leads from an expression with all real-valued coefficients to another expression with complex numbers.For example: Show that x2+4=(x+2i)(x-2i).ProbabilityI can…solve problems and find probabilities using combinations and permutations (A2.1.F; A2.6.C)For example: The company Ali works for allows her to invest in her choice of 10 different mutual funds, 6 of which grew by 5% over the last year. Ali randomly selected 4 of the 10 funds in which to invest. What is the probability that 3 of Ali’s funds grew by 5%?use the Fundamental Counting Principle to calculate probabilities of compound events (A2.6.A)determine whether two events A and B, within a finite sample space, are dependent or independent (S.CP.2; A2.6.B)This is a Geometry standard under Common Core. It is an Algebra 2 standard under WA 2008.find the conditional probability of A given B (S.CP.3; A2.6.B)This is a Geometry standard under Common Core. It is an Algebra 2 standard under WA 2008.use the Binomial Theorem to solve problems involving probability (A.APR.5; A2.6.D)A.APR.5 is marked as content for fourth-year courses. A2.6.D is required of all students in Algebra 2 under WA 2008.(+) use probability concepts to analyze decisions and strategies (S.MD.6–7)The goal is for students to analyze situations and strategies to determine if they are mathematically fair. It would also make sense to discuss this while evaluating statistical claims and studies.For example: Why is a random number generator a fair way to choose the winner of a raffle? Given a company’s testing procedures for defective products, decide whether the process is fair and likely to lead to valid conclusions.StatisticsI can…understand statistics as a process for making inferences about population parameters based on a random sample from that population (S.IC.1)The goal is for students to understand that statistics involves extrapolation from a sample to the whole population. This understanding should be embedded within all tasks throughout the unit.evaluate a report based on data (S.IC.6)The goal is for students to think critically about claims they see in the media. This standard should be embedded within all tasks throughout the unit.Students might look at: data collection methods; potential biases; the inferences made by the researchers; sampling techniques; scales on graphs; outliers; randomization; and which statistical tests were used.recognize the purposes of, and differences among, sample surveys, experiments, and observational studies (S.IC.3)Students should design their own surveys, experiments, and studies and be able to argue for their choices, including the effects of randomization on their results.Students should analyze various sampling methods and identify the strengths and weaknesses of each type for a given scenario.determine if a specific model is consistent with results from a given data-generating process (S.IC.2)The goal is for students to evaluate the effectiveness of a given model and the implications of a result based on the chosen model. This standard should be embedded within all tasks throughout the unit.For example: A model says that the probability of a coin landing on heads is 0.5; would a result of 5/20/50 tails in a row cause you to question the model?determine if a bivariate data set can be better modeled with a linear, quadratic, or exponential function, and use the model to make predictions (S.ID.6; A2.6.E)This is an Algebra 1 standard under Common Core. It is an Algebra 2 standard under WA 2008.find and interpret the mean and standard deviation of a data set (S.ID.4; A2.6.F)determine if a normal distribution is appropriate for a given data set (S.ID.4; A2.6.F)fit a data set to the normal distribution and use it to estimate population percentages and area under the normal curve (S.ID.4; A2.6.F)Students should be able to apply the Empirical Rule to estimate population percentages.use data from a sample survey to estimate a population mean or proportion and calculate a margin of error and confidence interval (S.IC.4; A2.6.G)As part of this standard, students should design, execute, and interpret their own survey.use data from a randomized experiment to compare two treatments and use simulations to determine if differences between parameters are significant (S.IC.5)Students might do this using a t-test.Sequences and SeriesI can…find the terms and partial sums of an arithmetic series (A2.7.B)find the terms, partial sums, and infinite sum of a geometric series (A.SSE.4; A2.7.B)Students should derive the formula for the sum of a geometric series and use this formula to solve problems in context.Miscellaneous Mathematical KnowledgeI can…know and use correct vocabulary to decompose, understand, and refer to expressions (A.SSE.1)This standard should be embedded within all tasks throughout the year.use the structure of an expression to identify ways to rewrite it (A.SSE.2; F.IF.8)This standard includes the factoring that students will do throughout the course.Students should be able to reason about quantities based on structural considerations. For example: If Q>P then Q+P>2P.classify numbers as real, complex, irrational, rational, natural, whole, and/or integer (A2.2.A)Students should be able to identify in which number systems a given equation can be answered. For example: x/7=π does not have any solutions in the rational number system, but does in the irrationals.write inequalities in interval, set-builder, and inequality notation.create and solve proportions (7.RP.2; 7.2.B)This is a 7th grade standard under both Common Core and WA 2008. It is included here for the purposes of review.use the properties of exponents to rewrite, simplify, and evaluate exponential expressions (N.RN.2; A2.2.B)This is an Algebra 1 standard under Common Core. It is an Algebra 2 standard under WA 2008.For N.RN.2, students should be comfortable moving between an expression with rational exponents and an equivalent expression with radicals.Students should know the rules for multiplying and dividing two exponential expressions with common bases, and for raising a power to a power.simplify square root expressions, including rationalizing the denominator, and add, subtract, multiply, and divide square roots (N.RN.2; A1.2.C)This is an Algebra 1 standard under both Common Core and WA 2008. It is included here for the purposes of review. ................
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