The study of linear phenomena, begun in middle school and expanded in Algebra I, serves as a starting point for this course. The introduction of complex numbers opens the door to the understanding and solution of all quadratic equations and their related functions. Power, root, polynomial, and rational functions; expressions; and equations increase student experience with non-linear behavior and its representation. In this course, students are asked to compare and contrast the properties of all of these different algebraic forms. They are expected to relate changes in the algebraic structure of each function to transformations of its graphical representation and are expected to recognize and solve problems that can be modeled using this range of functions. An optional enrichment unit on operations on functions introduces function composition and permits the definition of logarithms. This course can be enhanced even further by including another optional enrichment unit in the area of statistical studies and models, addressing such topics as transforming data and identifying a class of functions from among those studied that can be used to model data.

Throughout Algebra II, technology is an important tool for visualization and for deepening understanding of function relationships and transformations. Data utilities are essential for the optional enrichment unit that addresses data transformations. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of Algebra II to those encountered in earlier mathematics courses as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for this Algebra II course. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Systems of Linear Equations and Inequalities
• Extending the Number System
• Quadratic Functions, Equations, and Inequalities
• Polynomial Functions, Expressions, and Equations
• Radical and Rational Functions and Equations
• Function Prototypes and Transformations
• Function Operations and Logarithms [OPTIONAL ENRICHMENT UNIT]
• Statistical Studies and Models

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this Algebra II course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for Algebra II.

• Real Numbers and their Applications
• Rational Exponents and Radicals
• Linear Functions, Equations, and Inequalities
• Systems of Linear Equations
• Simple Exponential Relationships
• Quadratic Functions and Equations with Real Zeros/Roots

A. Systems of Linear Equations and Inequalities

Students who have completed a rigorous Algebra I course should be very familiar with solving systems of linear equations in two variables. While linear behavior was addressed in Geometry as well, this Algebra II unit provides an opportunity to reinforce that knowledge by extending it to systems of inequalities in two variables and to systems of equations in three variables. It also serves to reinforce understanding of linear behavior in general and extend it to lines in space.

Successful students will:

A1 Solve systems of two or more linear inequalities in two variables and graph the solution set.

Example: The set of points (x, y) that satisfy all three inequalities
5x - y ≥ 3, 3x + y ≤ 10, and 4x - 3y ≤ 6 is a triangle; the solution to the system of inequalities x + y + z ≥ -10, 2x − y + 3z ≤ 20, 8x − 2y + z ≥ −3 is the intersection of three half-planes whose points satisfy each inequality separately.

A2 Solve systems of linear equations in three variables using algebraic procedures; describe the possible arrangements of their graphs.

1. Relate the possible arrangements of the graphs of three linear equations in three variables to the number of solutions of the corresponding system of equations.

A3 Recognize and solve problems that can be modeled using a system of linear equations or inequalities; interpret the solution(s) in terms of the context of the problem.

Examples: Break-even problems, such as those comparing costs of two services; optimization problems that can be approached through linear programming.

B. Extending the Number System

While all linear equations and inequalities and systems of linear equations and inequalities can be solved within the real number system, solution of some higher degree equations requires an extension of the real numbers to the complex number system. Students should understand that each successive number system from the natural numbers to the integers, rational numbers, real numbers, and complex numbers is contained in, or embedded in, the succeeding system. However, students should also understand that for complex numbers, a new form of number (a + bi, for a and b real numbers) requiring the use of two real numbers must be defined and that different operations of addition and multiplication must be defined in order to accommodate this new form. The complex number system maintains some, but not all, of the properties of the real number system.

Successful students will:

B1 Identify expressions of the form a + bi as complex numbers.

The imaginary unit, sometimes represented as , is a solution to the equation x² = −1.

1. Explain why every real number is a complex number.

   Every real number, a, is a complex number because it can be expressed as a + 0i.

2. Express the square root of a negative number in the form bi or 0 + bi, where b is real.

   Just as with square roots of positive numbers, there are two square roots for negative numbers; in , 2i is taken to be the principal square root based on both the Cartesian and trigonometric representations of complex numbers.

   Examples: Determine the principal square root for each of the following: ;

B2 Compute with complex numbers. [OPTIONAL ENRICHMENT UNIT]

1. Add, subtract, and multiply complex numbers using the rules of arithmetic.
2. Use conjugates to divide complex numbers.

   Example:

   This process can also be applied to the division of irrational numbers involving square roots such as .

C. Quadratic Functions, Equations, and Inequalities

In Algebra I, work was limited to quadratic equations and functions for which there were real roots or zeros. Now that students have the system of complex numbers in which to work, they can apply the algebraic and graphical techniques learned earlier to all quadratic equations and inequalities. They should be able to predict the nature of the roots of a quadratic equation from its discriminant, identify characteristics of the solutions and coefficients of a quadratic equation in two variables from its graph, and relate the factored form of a quadratic to its solutions even when those solutions are not real. The usefulness of quadratic functions in expressing important real-world relationships should be reinforced with students now able to interpret complex as well as real solutions when they arise.

Successful students will:

C1 Solve quadratic equations over the complex numbers.

1. Use the quadratic formula or completing the square to solve any quadratic equation in one variable and write it as a product of linear factors.
2. Use the quadratic formula to show that the x-coordinate (abscissa) of the vertex of the corresponding parabola is halfway between the roots of the equation.
3. Use the discriminant D = b² - 4ac to determine the nature of the roots of the equation ax² + bx + c = 0.
4. Identify quadratic functions that do not have real zeros by the behavior of their graphs.

   A quadratic function that does not cross the horizontal axis has no real zeros.

5. Show that complex roots of a quadratic equation having real coefficients occur in conjugate pairs; show that multiplying factors related to conjugate pairs results in a quadratic equation having real coefficients.

   Example: The complex numbers are the roots of the equation = x² - 6x +14 = 0 whose coefficients are real.

C2 Manipulate simple quadratic equations or functions to extract information.

1. Describe the effect that changes in the the leading coefficient or constant term of f(x) = ax² + bx + c have on the shape, position, and characteristics of the graph of f(x); identify the relationship of such transformations to the type of solutions of the equation.

   Examples: If a and c have opposite signs, then the roots of the quadratic must be real and have opposite signs; varying c varies the y-intercept of the graph of the parabola; if a is positive, the parabola opens up, if a is negative, it opens down; as |a| increases, the graph of the parabola is stretched vertically, i.e., it looks narrower.

2. Use completing the square to determine the center and radius of a circle.

C3 Solve and graph quadratic inequalities in one or two variables.

Example: Solve (x - 5)(x + 1) > 0 and relate the solution to the graph of (x - 5)(x + 1) > y.

D. Polynomial Functions, Expressions, and Equations

To extend the graphing of quadratic functions, power functions form a natural bridge into the study of polynomial functions and expressions. The development of facility with algebraic expressions involving polynomials, including the development of the binomial expansion theorem and its connections to probability and combinatorics, is included here, along with understanding about how leading coefficients and constant terms of a polynomial contribute to characteristics of its graph.

Successful students will:

D1 Analyze power functions and identify their key characteristics.

In this course, a power function is any function defined over the real numbers of the form f(x) = a^p where p is a rational number. Power functions include positive integer power functions such as f(x) = -3⁴, root functions such as , and reciprocal functions such as f(x) = kx^-4

1. Recognize that the inverse proportional function f(x) = k/x (f(x) = kxⁿ for n = -1) and the direct proportional function f(x) = kx (f(x) = kxⁿ for n = 1) are special cases of power functions.
2. Distinguish between odd and even power functions.

   Examples: When the exponent of a power function is a positive integer, then even power functions have either a minimum or maximum value, while odd power functions have neither; even power functions have reflective symmetry over the y-axis, while odd power functions demonstrate rotational symmetry about the origin.

3. Transform the algebraic expression of power functions using properties of exponents and roots.

   Example: can be more easily identified as a root function once it is rewritten as .

4. Explain and illustrate the effect that a change in a parameter has on a power function (a change in a or n for f(x) = axⁿ).

D2 Analyze polynomial functions and identify their key characteristics.

1. Know that polynomial functions of degree n have the general form f(x) = axⁿ + bx^n-1 + ... + px² + qx + r for n an integer, n ≥ 0 and a ≠ 0.

   The degree of the polynomial function is the largest power of its terms for which the coefficient is non-zero.

2. Know that a power function with an exponent that is a positive integer is a particular type of polynomial function, called a monomial, whose graph contains the origin.
3. Distinguish among polynomial functions of low degree, i.e., constant functions, linear functions, quadratic functions, or cubic functions.
4. Explain why every polynomial function of odd degree has at least one zero; identify any assumptions that contribute to your argument.

   At this level, students are expected to recognize that this result requires that polynomials are connected functions without “holes.” They are not expected to give a formal proof of this result.

5. Communicate understanding of the concept of the multiplicity of a root of a polynomial equation and its relationship to the graph of the related polynomial function.

   If a root, r₁, of a polynomial function has multiplicity 3, (x - r₁)³ is a factor of the polynomial. The graph of the polynomial touches the horizontal axis at r1 but does not change sign (does not cross the axis) if the multiplicity of r₁ is even; it changes sign (crosses over the axis) if the multiplicity is odd.

D3 Use key characteristics to identify the graphs of simple polynomial functions.

Simple polynomial functions include constant functions, linear functions, quadratic functions or cubic functions such as f(x) = x³, f(x) = x³ - a, or f(x) = x(x - a)(x + b).

1. Decide if a given graph or table of values suggests a simple polynomial function.
2. Distinguish between the graphs of simple polynomial functions.
3. Where possible, determine the domain, range, intercepts, and end behavior of polynomial functions.

   The end behavior of a graph refers to the trend of the values of the function as x approaches ±∞. It should be noted that it is not always possible to determine exact x-intercepts. Graphing utilities are excellent vehicles for providing indications of end behavior and approximations for intercepts.

D4 Recognize and solve problems that can be modeled using power or polynomial functions; interpret the solution(s) in terms of the context of the problem.

1. Use power or polynomial functions to represent quantities arising from numeric or geometric contexts such as length, area, and volume.

   Examples: The number of diagonals of a polygon as a function of the number of sides; the areas of simple plane figures as functions of their linear dimensions; the surface areas of simple three-dimensional solids as functions of their linear dimensions; the sum of the first n integers as a function of n.

2. Solve simple polynomial equations and use technology to approximate solutions for more complex polynomial equations.

D5 Perform operations on polynomial expressions.

1. Add, subtract, multiply, and factor polynomials.
2. Divide one polynomial by a lower-degree polynomial.
3. Know and use the binomial expansion theorem.
4. Relate the expansion of (a + b)ⁿ to the possible outcomes of a binomial experiment and the nth row of Pascal's triangle.

E. Radical and Rational Functions and Equations

Radical equations and functions are a natural outgrowth of power functions. Simple rational equations and functions build on the reciprocal functions of the form f(x) = k/x studied earlier and are directly related to linear and quadratic polynomials and their reciprocals. Algebraic facility with radical and rational forms should be seen as extending earlier work with numeric fractions in upper elementary and middle school.

Successful students will:

E1 Use factoring to reduce rational expressions that consist of the quotient of two simple polynomials.

E2 Perform operations on simple rational expressions.

Simple rational expressions are those whose denominators are linear or quadratic polynomial expressions.

1. Add subtract, multiply, and divide rational expressions having monomial or binomial denominators.
2. Rewrite complex fractions composed of simple rational expressions as a simple fraction in lowest terms.

   Example:

E3 Solve simple rational and radical equations in one variable.

1. Use algebraic, numerical, graphical, and/or technological means to solve rational equations.
2. Use algebraic, numerical, graphical, and/or technological means to solve equations involving a radical.
3. Know which operations on an equation produce an equation with the same solutions and which may produce an equation with fewer or more solutions (lost or extraneous roots) and adjust solution methods accordingly.

E4 Graph simple rational and radical functions in two variables.

1. Graph simple rational functions in two variables; identify the domain, range, intercepts, zeros, and asymptotes of the graph.
2. Graph simple radical functions in two variables; identify the domain, range, intercepts and zeros, of the graph.
3. Relate the algebraic properties of a rational or radical function to the geometric properties of its graph.

   Examples: The graph of has vertical asymptotes at x = 1 and x = -1, while the graph of has a vertical asymptote at x = -2 but a hole at (2, ¼); the graph of is the same as the graph of translated five units to the left and 2 units down.

F. Function Prototypes and Transformations

Having now added power, polynomial, rational, and radical functions to the classes, or families, of functions studied earlier—linear, direct proportional, reciprocal, exponential, and quadratic—students should begin to understand how functions behave as a class of mathematical objects. They should be able to identify or create prototypes for most function families—determining a prototype for polynomials presents difficulties—and should be able to identify or describe the effect of certain transformations on each function, seeing that a specific type of transformation affects each class of functions in a similar way.

Successful students will:

F1 Analyze exponential functions and relate key characteristics in their algebraic and graphical representations.

1. Describe key characteristics of the graphs of exponential functions and relate these to the coefficients in the general form f(x) = ab^x + c for b > 0, b ≠ 1.

   Examples: Know that, if b > 1, exponential functions are increasing and that they approach a lower limit if a > 0 and an upper limit if a < 0 as x decreases; know that, if 0 < b < 1, exponential functions are decreasing and that they approach a lower limit if a > 0 and an upper limit if a < 0 as x increases.

F2 Distinguish between the graphs of simple exponential and power functions by their key characteristics.

Be aware that it can be very difficult to distinguish graphs of these various types of functions over small regions or particular subsets of their domains. Sometimes the context of an underlying situation can suggest a likely type of function model.

1. Decide whether a given exponential or power function is suggested by the graph, table of values, or underlying context of a problem.
2. Distinguish between the graphs of exponential growth functions and those representing exponential decay.
3. Distinguish among the graphs of power functions having positive integral exponents, negative integral exponents, and exponents that are positive unit fractions (f(x) = x^1/n, n ≥ 0).
4. Identify and explain the symmetry of an even or odd power function.
5. Where possible, determine the domain, range, intercepts, asymptotes, and end behavior of exponential and power functions.

   Range is not always possible to determine with precision. End behavior refers to the trend of the graph as x approaches ±∞.

F3 Distinguish among linear, exponential, power, polynomial, and rational expressions, equations, and functions by their symbolic form.

1. Use the position of the variable in an expression to determine the classification of the expression.

   Examples: f(x) = 3^x is an exponential function because the variable is in the exponent, while f(x) = x³ has the variable in the position of a base and is a power function; f(x) = x³ - 5 is a polynomial function but not a power function because of the added constant.

2. Identify or determine a prototypical representation for each family of functions.

   Examples: f(x) = x² is a prototype for quadratic functions; g(x) = 1/x is a prototypical reciprocal function.

F4 Explain, illustrate, and identify the effect of simple coordinate transformations on the graphs of power, polynomial and exponential functions; compare these to transformations occasioned by changes in parameters in linear, direct proportional and reciprocal functions.

1. Interpret the graph of y = f(x – a) as the graph of y = f(x) shifted |a| units to the right (a > 0) or the left (a < 0).
2. Interpret the graph of y = f(x) + a as the graph of y = f(x) shifted |a| units up (a > 0) or down (a < 0).
3. Interpret the graph of y = f(ax) as the graph of y = f(x) expanded horizontally by a factor of 1/|a| if 0 < |a| < 1 or compressed horizontally by a factor of |a| if |a| > 1 and reflected over the y-axis if a < 0.
4. Interpret the graph of y = af(x) as the graph of y = f(x) compressed vertically by a factor of 1/|a| if 0 < |a| < 1 or expanded vertically by a factor of |a| if |a| > 1 and reflected over the x-axis if a < 0.

G. Functions Operations and Logarithms [OPTIONAL ENRICHMENT UNIT]

If time permits, the basic families of functions can be extended through a unit that introduces logarithmic functions. The unit begins by formalizing operations on functions including function composition. While addition, subtraction, multiplication, and division of functions follow directly from earlier work, the new operation, function composition, permits the definition of a new class of functions. Logarithms are defined as the inverses of exponential functions. This offers an opportunity to revisit and reinforce the understanding of exponential functions originally introduced in Algebra I. The introduction of logarithms permits new solution strategies for exponential equations, encouraging expanded applications including those involving continuous growth or decay.

Successful students will:

G1 Perform operations on simple functions; identify any necessary restrictions on the domain.

1. Determine the sum, difference, product, quotient, and composition of simple functions.
2. Analyze the transformations of a function from its graph, formula, or verbal description.

   Example: The graph of f(x) = –3x² + 4 is a vertical dilation by a factor of 3 of the prototype f(x) = x² followed by a reflection over the x-axis and a translation 4 units up. The resulting vertex of the parabola (0, 4) reflects these transformations and is evident when f(x) = –3x² + 4 is compared to the vertex form of a parabola f(x) = a(x – h)² + k.

G2 Analyze characteristics of inverse functions.

1. Identify and explain the relationships among the identity function, composition of functions, and the inverse of a function, along with implications for the domain.
2. Identify the conditions under which the inverse of a function is a function.
3. Determine whether two given functions are inverses of each other.
4. Explain why the graph of a function and its inverse are reflections of one another over the line y = x.

G3 Determine the inverse of linear and simple non-linear functions, including any necessary restrictions on the domain.

1. Determine the inverse of a simple polynomial or simple rational function.
2. Identify a logarithmic function as the inverse of an exponential function.

   If x^y = z, x > 0, x ≠ 1, and z > 0, then y is the logarithm to the base x of z. The logarithm y=log_xz is one of three equivalent forms of expressing the relation x^y = z (the other being ).

3. Determine the inverse of a given exponential or logarithmic function.

   Example: If 5^a = b, then log₅(b) = a.

G4 Apply properties of logarithms to solve equations and problems and to prove theorems.

1. Know and use the definition of logarithm of a number and its relation to exponents.

   Examples: log₂32 = log₂2⁵ = 5; if x = log₁₀3, then 10^x = 3 and vice versa.

2. Prove basic properties of logarithms using properties of exponents (or the inverse exponential function).
3. Use properties of logarithms to manipulate logarithmic expressions in order to extract information.
4. Use logarithms to express and solve problems.

   Example: Explain why the number of digits in the binary representation of a decimal number N is approximately the logarithm to base 2 of N.

5. Solve logarithmic equations; use logarithms to solve exponential equations.

   Examples: log(x - 3) + log(x - 1) = 0.1; 5^x = 8.

H. Statistical Studies and Models

Now that students have deepened their experience with many types of functions and with the effect of transformations on them, they would benefit from engaging in a project that requires collecting and analyzing data. This unit provides students with the opportunities they will need to understand the differences among the major types of statistical studies. For the purposes of applying what they have learned about functions, a project that generates bivariate data would be effective. Included here is an optional enrichment unit that provides students with the opportunity to make meaningful connections between functions, modeling, and data analysis.

Successful students will:

H1 Describe the nature and purpose of sample surveys, experiments, and observational studies, relating each to the types of research questions they are best suited to address.

1. Identify specific research questions that can be addressed by different techniques for collecting data.
2. Critique various methods of data collection used in analyzing real-world problems, such as a clinical trial in medicine, an opinion poll or a report on the effect of smoking on health.
3. Explain why observational studies generally do not lead to good estimates of population characteristics or cause-and-effect conclusions regarding treatments.

H2 Plan and conduct sample surveys, observational studies, and experiments.

1. Recognize and explain the rationale for using randomness in research designs; distinguish between random sampling from a population in sample surveys and random assignment of treatments to experimental units in an experiment.

   Random sampling is how items are selected from a population so that the sample data can be used to estimate characteristics of the population; random assignment is how treatments are assigned to experimental units so that comparisons among the treatment groups can allow cause-and-effect conclusions to be made.

2. Use simulations to analyze and interpret key concepts of statistical inference.

   Key concepts of statistical inference include margin of error and how it relates to the design of a study and to sample size; confidence interval and how it relates to the margin of error; and p-value and how it relates to the interpretation of results from a randomized experiment.

H3 Identify an appropriate family of functions as a model for real data. [OPTIONAL ENRICHMENT UNIT]

1. Analyze and compare key characteristics of different families of functions; identify prototypical functions as potential models for given data.
2. Apply transformations of data for the purpose of “linearizing” a scatter plot that exhibits curvature.

   Examples: Apply squaring, square root, and reciprocal and logarithmic functions to input data, output data, or both; evaluate which transformation produces the strongest linear trend.

3. Use and interpret a residual plot of the relationship among the standard deviation, correlation coefficients and slope of the line to evaluate the goodness of fit of a regression line to transformed data.
4. Estimate the rate of exponential growth or decay by fitting a regression model to appropriate data transformed by logarithms.
5. Estimate the exponent in a power model by fitting a regression model to appropriate data transformed by logarithms.
6. Analyze how linear transformations of data affect measures of center and spread, the slope of a regression line and the correlation coefficient.

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

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Jump to:

A. Systems of Linear Equations and Inequalities

B. Extending the Number System

C. Quadratic Functions, Equations and Inequalities

D. Polynomial Functions, Expressions and Equations

E. Radical and Rational Functions and Equations

F. Function Prototypes and Transformations

G. Functions Operations and Logarithms [OPTIONAL ENRICHMENT UNIT]

H. Statistical Studies and Models

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