Secondary Mathematics Benchmarks Progressions, Grades 7–12: Algebra (A)

The language of algebra provides the means to express and illuminate mathematical relationships. Multiple representations—verbal, symbolic, numeric, and graphic—are used to describe change, to express the interaction of forces, and to describe and compare patterns. Algebra allows its users to generate new knowledge by drawing broad, rigorous generalizations from specific examples. Every mathematical strand makes extensive use of algebra to symbolize, to clarify, and to communicate its concepts and content. Learning algebra is an important step in a student’s cognitive mathematical development. It opens the door to organized abstract thinking, supplies a tool for logical reasoning, and helps us to model and understand the quantitative relationships so vital in today’s world.

A.A.1 Variables and expressions

a. Interpret and compare the different uses of variables and describe patterns, properties of numbers, formulas, and equations using variables.

• Compare the different uses of variables.

  Examples: When a + b = b + a is used to state the commutative property for addition, the variables a and b represent all real numbers; the variable a in the equation 3a – 7 = 8 is a temporary placeholder for the one number, 5, that will make the equation true; the symbols C and r refer to specific attributes of a circle in the formula C = 2πr; the variable m in the slope-intercept form of the line, y = mx + b serves as a parameter describing the slope of the line.

• Express patterns, properties, formulas, and equations using and defining variables appropriately for each case.

b. Analyze and identify characteristics of algebraic expressions.

• Analyze expressions to identify when an expression is the sum of two or more simpler expressions (called terms) or the product of two or more simpler expressions (called factors).
• Identify single-variable expressions as linear or non-linear.

c. Evaluate, interpret, and construct simple algebraic expressions.

• Evaluate a variety of algebraic expressions at specified values of their variables.

  Algebraic expressions to be evaluated include polynomial and rational expressions as well as those involving radicals and absolute value.

  Example: Evaluate 3x² − 2y³ + for x = 6, y = -3, and w = 81.

• Write linear and quadratic expressions representing quantities arising from geometric and real–world contexts.

  Examples: Area of a rectangle of length l and width w; area of a circle of radius r; cost of buying 5 apples at price p and 7 oranges at price q.

• Analyze the structure of an algebraic expression and identify the resulting characteristics.

  Example: -5(u² + 4) is a product of two factors, the second of which is always positive because it is the sum of a square and a positive number; since the first factor is negative, the algebraic expression is negative for all values of u.

d. Identify and transform expressions into equivalent expressions.

Two algebraic expressions are equivalent if they yield the same result for every value of the variables in them.

• Use commutative, associative, and distributive properties of number operations to transform simple expressions into equivalent forms in order to collect like terms or to reveal or emphasize a particular characteristic.

  Examples: Add, subtract, and multiply linear expressions, such as

  (2x + 5) + (3 – 2x) = 2x + 5 + 3 – 2x = 8,

  (2x + 5) – (3 – 2x) = 2x + 5 – 3 + 2x = 2 + 4x, or

  –5(3 – 2x) = –15 + 10x.

  Transform simple nonlinear expressions, such as

  (3p)(5q) = 15pq or n(n + 1) = n² + n.

• Rewrite linear expressions in the form ax + b for constants a and b.
• Choose different but equivalent expressions for the same quantity that are useful in different contexts.

  Example: p + 0.07p shows the breakdown of the cost of an item into the price p and the tax of 7%, whereas (1.07)p is a useful equivalent form for calculating the total cost.

Tasks related to this benchmark: Common Differences, Counting Cubes, Out of the Swimming Pool

e. Determine whether two algebraic expressions are equivalent.

• Demonstrate equivalence through algebraic transformations.
• Show that expressions are not equivalent by evaluating them at the same value(s) to get different results.
• Show that certain expressions are equivalent by checking at a small number of different values (e.g., two linear expressions are equivalent if they yield equal results at two distinct values of the variable), and identify the special circumstances under which this may be true.

  Great care must be taken to demonstrate that, in general, a finite number of instances is not sufficient to demonstrate equivalence.

• Set each expression equal to y, consider all ordered pairs of these newly constructed equations, and know that if the graph of all ordered pairs that satisfy one equation is identical to the graph of all ordered pairs that satisfy the other, then the expressions are equivalent.

Task related to this benchmark: Out of the Swimming Pool

f. Apply the properties of exponents to transform variable expressions involving integral exponents.

• Know and apply the laws of exponents.

  Examples: a^p · a^q = a^p+q; - x^-2; 9^x = 3^2x; = 32.

• Factor out common factors with exponents.

  Factoring transforms an expression that was written as a sum or difference into one that is written as a product.

  Examples: 6v⁷ + 12v⁵ – 8v³ = 2v³ (3v⁴ + 6v² – 4); 3x(x + 1)² – 2(x + 1)² = (x + 1)²(3x – 2).

  Chunking is a term often used to describe treating an expression, such as the x + 1 above, as a single entity.

g. Interpret rational exponents; translate between rational exponents and notation involving integral powers and roots.

Examples: ; ; .

A.A.2 Functions

a. Determine whether a relationship is or is not a function.

In general, a function is a rule that assigns a single element of one set—the output set—to each element of another set—the input set. The set of all possible inputs is called the domain of the function, while the set of all outputs is called the range.

• Identify the independent (input) and dependent (output) quantities/variables of a function.

b. Represent and interpret functions using graphs, tables, words, and symbols.

• Make tables of inputs x and outputs f(x) for a variety of rules that take numbers as inputs and produce numbers as outputs.

  The notation f(x) or P(t) represents the number that the function f or P assigns to the input x or t.

• Define functions algebraically, e.g., g(x) = 3 + 2(x – x²).

  When functions are defined by algebraic expressions, these expressions are sometimes called formulas. Not every function can be defined by means of an algebraic expression. Many are stated using algorithms or verbal descriptions. Spreadsheet software packages offer an abundant source of function rules.

• Create the graph of a function f by plotting the ordered pairs (x, f(x)) in the coordinate plane.
• Analyze and describe the behavior of a variety of simple functions using tables, graphs, and algebraic expressions.

  Examples: f(x) = 3x + 1; f(x) = 3x; f(x) = 3x² + 1; f(x) = 2x³; f(x) = 2^x; f(x) = 3/x.

  To understand the breadth of the function concept, it is important for students to work with a variety of examples.

• Construct and interpret functions that describe simple problem situations using expressions, graphs, tables, and verbal descriptions and move flexibly among these multiple representations.

  Caution should be taken when using tables, as they only indicate the value of the function at a finite number of points and could arise from many different functions.

Task related to this benchmark: Counting Cubes, Function Transformations, Out of the Swimming Pool, Talk Is Cheap, You're Toast, Dude!

A.A.3 Linear functions

a. Analyze and identify linear functions of one variable.

A function exhibiting a constant rate of change is called a linear function. A constant rate of change means that for any pair of inputs x₁ and x₂, the ratio of the corresponding change in value f(x₂) – f(x₁) to the change in input x₂ – x₁ is constant (i.e., it does not depend on the inputs).

• Explain why any function defined by a linear algebraic expression has a constant rate of change.

  Examples: f(x) = 2x; f(x) = 5 – 3x; f(side of square) = perimeter of square.

• Explain why the graph of a linear function defined for all real numbers is a straight line, and identify its constant rate of change and create the graph.
• Explain why a vertical line is not the graph of a function.
• Determine whether the rate of change of a specific function is constant; use this to distinguish between linear and nonlinear functions.

Tasks related to this benchmark: Match That Function, Talk Is Cheap

b. Know the definitions of x- and y-intercepts, know how to find them, and use them to solve problems.

An x-intercept is the value of x where f(x) = 0. A y-intercept is the value of f(0).

c. Know the definition of slope, calculate it, and use slope to solve problems.

The slope of a linear function is its constant rate of change.

• Know that a line with positive slope tilts from lower left to upper right, whereas a line with a negative slope tilts from upper left to lower right.
• Know that a line with slope equal to zero is horizontal, while the slope of a vertical line is undefined.

d. Express a linear function in several different forms for different purposes.

• Recognize that in the form f(x) = mx + b, m is the slope, or constant rate of change of the graph of f, that b is the y-intercept, and that in many applications of linear functions, b defines the initial state of a situation; express a function in this form when this information is given or needed.
• Recognize that in the form f(x) = m(x – x₀) + y₀, the graph of f(x) passes through the point (x₀, y₀); express a function in this form when this information is given or needed.

e. Recognize contexts in which linear models are appropriate; determine and interpret linear models that describe linear phenomena.

Common examples of linear phenomena include distance traveled over time for objects traveling at constant speed; shipping costs under constant incremental cost per pound; conversion of measurement units (e.g., pounds to kilograms or degrees Celsius to degrees Fahrenheit); cost of gas in relation to gallons used; the height and weight of a stack of identical chairs.

• Identify situations that are linear and those that are not linear and justify the categorization based on whether the rate of change is constant or varies.
• Express a linear situation in terms of a linear function f(x) = mx + b and interpret the slope (m) and the y-intercept (b) in terms of the original linear context.

Task related to this benchmark: Talk Is Cheap

A.A.4 Proportional functions

a. Recognize, graph, and use direct proportional relationships.

A proportion is composed of two pairs of real numbers, (a, b) and (c, d), with at least one member of each pair non-zero, such that both pairs represent the same ratio. A linear function in which f(0) = 0 represents a direct proportional relationship. The function f(x) = kx, where k is constant, describes a direct proportional relationship.

• Show that the graph of a direct proportional relationship is a line that passes through the origin (0, 0) whose slope is the constant of proportionality.
• Compare and contrast the graphs of x = k, y = k, and y = kx, where k is a constant.
• If f(x) is a linear function, show that g(x) = f(x) – f(0) represents a direct proportional relationship.

  In this case, g(0) = 0, so g(x) = kx. The graph of f(x) = mx + b is the graph of the direct proportional relationship g(x) = mx shifted up (or down) by b units. Since the graph of g(x) is a straight line, so is the graph of f(x).

b. Recognize, graph, and use reciprocal relationships.

A function of the form f(x) = k/x where k is constant describes a reciprocal relationship. The term “inversely proportional” is sometimes used to identify such relationships, however, this term can be very confusing since the word "inverse" is also used in the term "inverse function" (the function y = f^-1(x) with the property that f f^-1(x) = f^-1 f(x) = x, which describes the identity function).

• Analyze the graph of f(x) = k/x and identify its key characteristics.

  The graph of f(x) = k/x is not a straight line and does not cross either the x– or the y–axis (i.e., there is no value of x for which f(x) = 0, nor is there any value for f(x) if x = 0). It is a curve consisting of two disconnected branches, called a hyperbola.

• Recognize quantities that are inversely proportional and express their relationship symbolically.

  Example: The relationship between lengths of the base and side of a rectangle with fixed area.

c. Distinguish among and apply linear, direct proportional, and reciprocal relationships.

• Identify whether a table, graph, formula, or context suggests a linear, direct proportional, or reciprocal relationship.
• Create graphs of linear, direct proportional, and reciprocal functions by hand and using technology.
• Identify practical situations that can be represented by linear, direct, or inversely proportional relationships; analyze and use the characteristics of these relationships to answer questions about the situation.

d. Explain and illustrate the effect of varying the parameters m and b in the family of linear functions and varying the parameter k in the families of directly proportional and reciprocal functions.

A.A.5 Equations and identities

a. Distinguish among an equation, an expression, and a function.

An equation is a statement of equality between algebraic expressions or functions.

Example: If f(x) = 3x + 2 and g(x) = 5x – 8, the statement f(x) = g(x) is an equation in one variable.

• Know that solving an equation means finding all its solutions.

  A solution of an equation (in one variable) is a value of the variable that makes the equation true. Because the solutions of an equation are often not known (or at least not apparent from the form of the equation), a variable in an equation is often called an unknown.

• Predict the number of solutions that should be expected for various simple equations and identities.
• Explain why solutions to the equation f(x) = g(x) are the x-values (abscissas) of the set of points in the intersection of the graphs of the functions f(x) and g(x).
• Recognize that f(x) = 0 is a special case of the equation f(x) = g(x) and solve the equation f(x) = 0 by finding all values of x for which f(x) = 0.

  The solutions to the equation f(x) = 0 are called roots of the equation or zeros of the function. They are the values of x where the graph of the function f crosses the x-axis. In the special case where f(x) equals 0 for all values of x, f(x) =0 represents a constant function where all elements of the domain are zeros of the function.

  Example: The graph of the linear function f(x) = 2x – 4 crosses the x-axis at x = 2. Hence, 2 is a root of the equation f(x) = 0, since f(2) = 2(2) – 4 = 0.

  Beware of the confusion inherent in two apparently different meanings of the word "root": The root of an equation (e.g., 3x² – 4x + 1 = 0) and the root of a number (e.g., ). Although different, these uses do arise from a common source: the root of a number such as is a root (or a solution) of an associated equation, namely x² – 5 = 0.

• Interpret the notation for the equation y = f(x) as a function for which each specific input, x, has a specific y-value as its output.

  In this representation, y stands for the output f(x) of the function f and corresponds to the y–axis on an x–y coordinate grid.

  Example: The two-variable equation y = 3x + 8 corresponds to the single-variable linear function f(x) = 3x + 8.

b. Solve linear and simple nonlinear equations involving several variables for one variable in terms of the others.

Example: Solve A = πr²h for h or for r.

c. Interpret identities as a special type of equation and identify their key characteristics.

An identity is an equation for which all values of the variables are solutions. Although an identity is a special type of equation, there is a difference in practice between the methods for solving equations that have a small number of solutions and methods for proving identities. For example, (x + 2)² = x² + 4x + 4 is an identity which can be proved by using the distributive property, whereas (x + 2)² = x² + 3x + 4 is an equation that can be solved by collecting all terms on one side.

• Use identities to transform expressions.

d. Make regular fluent use of basic algebraic identities such as (a + b)² = a² + 2ab + b²; (a – b)² = a² – 2ab + b²; and (a + b)(a – b) = a² – b².

• Use the distributive law to derive each of these formulas.

  Examples: (a + b)(a – b) = (a + b)a – (a + b)b = (a² + ab) – (ab + b²) = a² + ab – ab – b² = a² – b²; applying this to specific numbers, 37 · 43 = (40 – 3)(40 + 3) = 1,600 – 9 = 1,591.

• Use geometric constructs to illustrate these formulas.

  Example: Use a partitioned square or tiles to provide a geometric representation of (a + b)² = a² + 2ab + b².

e. Create, interpret, and apply mathematical models to solve problems arising from contextual situations.

Mathematical modeling consists of recognizing and clarifying mathematical structures that are embedded in other contexts, formulating a problem in mathematical terms, using mathematical strategies to reach a solution, and interpreting the solution in the context of the original problem.

• Distinguish relevant from irrelevant information, identify missing information, and find what is needed or make appropriate estimates.
• Apply problem solving heuristics to practical problems: Represent and analyze the situation using symbols, graphs, tables, or diagrams; assess special cases; consider analogous situations; evaluate progress; check the reasonableness of results; and devise independent ways of verifying results.

A.A.6 Linear equations and inequalities

a. Solve linear equations in one variable algebraically.

An equation of the form f(x) = g(x) is linear if the function f(x) – g(x) is linear. Combining terms makes each linear equation in a single variable equivalent to an equation in the standard form ax + b = 0.

• Solve equations using the facts that equals added to equals are equal and that equals multiplied by equals are equal. More formally, if A = B and C = D, then A + C = B + D and AC = BD.

  Together with the ordinary laws of arithmetic (commutative, associative, distributive), these principles justify the steps used to transform linear equations into equivalent equations in standard form and then solve them.

• Using the fact that a linear expression ax + b is formed using the operations of multiplication by a constant followed by addition, solve an equation ax + b = 0 by reversing these steps.
• Be alert to anomalies caused by dividing by 0 (which is undefined), or by multiplying both sides by 0 (which will produce equality even when things were originally unequal).

  Example: Multiplying both sides of an equation by x – 1 is appropriate only when x ≠ 1.

b. Solve and graph the solution of linear inequalities in one variable.

A solution to a linear inequality in one variable consists of all points on the number line whose coordinates satisfy the inequality.

• Graph a linear inequality in one variable and explain why the graph is always a half-line (open or closed).
• Know that the solution set of a linear inequality in one variable is infinite, and contrast this with the solution set of a linear equation in one variable.

  It is also possible that some contextual situations may limit the reasonable solutions of a linear inequality to a finite number.

• Explain why, when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality is reversed, but that when all other basic operations involving non-zero numbers are applied to both sides, the direction of the inequality is preserved.

c. Identify the relationship between linear functions of one variable and linear equations in two variables.

• Translate fluently between the linear function of one variable f(x) = mx + b and the related linear equation in two variables y = mx + b.
• Rewrite a linear equation in two variables in any of three forms: ax + by = c, ax + by + c = 0, or y = mx + b; select a form depending upon how the equation is to be used.
• Know that the graph of a linear equation in two variables consists of all points (x, y) in the coordinate plane that satisfy the equation and explain why, when x can be any real number, such graphs are straight lines.

d. Use graphs to help solve linear equations in one variable.

• Explain why the solution to an equation in standard (or polynomial) form (ax + b = 0) will be the point at which the graph of f(x) = ax + b crosses the x-axis.
• Identify the solution of an equation that is in the form f(x) = g(x) and relate the solution to the x-value (abscissa) of the point at which the graphs of the functions f(x) and g(x) intersect.

  Example: To solve the linear equation 3x + 1 = x + 5, graph f(x) = 3x + 1 and g(x) = x + 5. The graphs of f(x) and g(x) intersect at the point (2, 7); thus the solution to the linear equation is x = 2. Alternatively, the linear equation 3x + 1 = x + 5 is equivalent to 2x – 4 = 0. This yields a single linear function h(x) = 2x – 4 whose graph crosses the x-axis at x = 2.

Task related to this benchmark: Cycling Situations

e. Represent any straight line in the coordinate plane by a linear equation in two variables.

• Represent any line in its standard form ax + by = c whether or not the line is the graph of a function.

  Vertical lines have the equation x = k (or 1x + 0y = k) with undefined slope and do not represent functions.

• Know that pairs of non-vertical lines have the same slope if and only if they are parallel (or the same line) and slopes that are negative reciprocals if and only if they are perpendicular; apply these relationships to analyze and represent equations.

f. Solve and graph the solution of a linear inequality in two variables.

• Know what it means to be a solution of a linear inequality in two variables, represent solutions algebraically and graphically, and provide examples of ordered pairs that lie in the solution set.
• Graph a linear inequality in two variables and explain why the graph is always a half-plane (open or closed).

  In analogy with the vocabulary of equations, the collection of all points (x, y) that satisfy the linear inequality ax + by < c is called the graph of the inequality. These points lie entirely in one of the half-planes determined by the graph of the equation ax + by = c.

g. Recognize and solve problems that can be modeled using linear equations or inequalities in one or two variables; interpret the solution(s) in terms of the context of the problem.

Common problems are those that involve time/rate/distance, percentage increase or decrease, ratio and proportion.

• Represent linear relationships using tables, graphs, verbal statements, and symbolic forms; translate among these forms to extract information about the relationship.

Task related to this benchmark: Cycling Situations

h. Solve equations and inequalities involving the absolute value of a linear expression in one variable.

A.B.1 Quadratic functions

a. Identify quadratic functions expressed in multiple forms; identify the specific information each form clarifies.

• Express a quadratic function as a polynomial, f(x) = ax² + bx + c, where a, b, and c are constants with a ≠ 0, and identify its graph as a parabola that opens up when a > 0 and down when a < 0; relate c to where the graph of the function crosses the y-axis.
• Express a quadratic function in factored form, f(x) = (x – r)(x – s), when r and s are integers; relate the factors to the solutions of the equation (x – r)(x – s) = 0 (x = r and x = s) and to the points ((r, 0) and (s, 0)) where the graph of the function crosses the x-axis.

Task related to this benchmark: Match That Function

b. Graph quadratic functions and use the graph to help locate zeros.

A zero of a quadratic function f(x) = ax² + bx + c is a value of x for which f(x) = 0.

• Sketch graphs of quadratic functions using both graphing calculators and tables of values.
• Estimate the real zeros of a quadratic function from its graph.
• 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.

c. Recognize contexts in which quadratic models are appropriate; determine and interpret quadratic models that describe quadratic phenomena.

Examples: The relationship between length of the side of a square and its area; the relationship between time and distance traveled for a falling object.

Task related to this benchmark: Match That Function, Season Pass

A.B.2 Simple quadratic equations

a. Solve quadratic equations that can be easily transformed into the form (x - a)(x - b) = 0 or (x + a)² = b, for a and b integers.

b. Estimate the roots of a quadratic equation from the graph of the corresponding function.

c. Solve simple quadratic equations that arise in the context of practical problems and interpret their solutions in terms of the context.

Examples: Determine the height of an object above the ground t seconds after it has been thrown upward at an initial velocity of v₀ feet per second from a platform d feet above the ground; find the area of a rectangle with perimeter 120 in terms of the length, L, of one side.

A.B.3 Systems of linear equations and inequalities

a. Solve systems of linear equations in two variables using algebraic procedures.

A system of simultaneous linear equations in two variables consists of two or more different linear equations in two variables. A solution to such a system is the set of ordered pairs of values (x₀, y₀) that makes all of the equations true.

• Determine whether a system of two linear equations has one solution, no solutions, or infinitely many solutions, and know that these are the only possibilities.

Task related to this benchmark: Cycling Situations, Season Pass

b. Use graphs to help solve systems of simultaneous linear equations in two variables.

• Use the graph of a system of equations in two variables to suggest solution(s).

  Since the solution is a set of ordered pairs that satisfy the equations, it follows that these ordered pairs must lie on the graph of each of the equations in the system; the point(s) of intersection of the graphs is (are) the solution(s) to the system of equations.

  Example: To solve the system 3x + 5y = 11; 7x – 9y = 5, first graph each of the two equations. It appears that the two graphs intersect at point x₀ = 2, y₀ = 1. Substitution of these values in both equations establishes that (2, 1) is indeed a solution of both equations and the actual point of intersection.

• Represent the graphs of a system of two linear equations as two intersecting lines when there is one solution, parallel lines when there is no solution, and the same line when there are infinitely many solutions.

Task related to this benchmark: Cycling Situations

c. 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 intersection of three half-planes whose points satisfies each inequality separately.

d. Solve systems of simultaneous linear equations in three variables using algebraic procedures.

A system of simultaneous linear equations in three variables consists of three or more different linear equations in three variables. A solution to such a system is the set of ordered triples of values (x₀, y₀, z₀) that makes all of the equations true.

e. Describe the possible arrangements of the graphs of three linear equations in three variables and relate these to the number of solutions of the corresponding system of equations.

f. 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.

A.C.1 Elementary functions

a. Identify key characteristics of absolute value, step, and other piecewise-linear functions and graph them.

• Interpret the algebraic representation of a piecewise-linear function; graph it over the appropriate domain.
• Write an algebraic representation for a given piecewise-linear function.
• Determine vertex, slope of each branch, intercepts, and end behavior of an absolute value graph.
• Recognize and solve problems that can be modeled using absolute value, step, and other piecewise-linear functions.

  Examples: Postage rates, cellular telephone charges, tax rates.

Task related to this benchmark: Function Transformations

b. Graph and analyze exponential functions and identify their key characteristics.

• Know that exponential functions have the general form f(x) = ab² + c for b > 0, b ≠ 1; identify the general shape of the graph and its lower or upper limit (asymptote).
• Explain and illustrate the effect that a change in a parameter has on an exponential function (a change in a, b, or c for f(x) = ab^x + c).

Tasks related to this benchmark: Equal Salaries for Equal Work?, Match That Function

c. Analyze power functions and identify their key characteristics.

Power functions include positive integer power functions such as f(x) = –3x⁴, root functions such as , and reciprocal functions such as f(x) = kx^–4.

• 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.
• 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.

Task related to this benchmark: Match That Function

d. 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 .

• 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).

e. Distinguish among 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.

• Decide whether a given exponential or power function is suggested by the graph, table of values, or underlying context of a problem.
• Distinguish between the graphs of exponential growth functions and those representing exponential decay.
• Distinguish among the graphs of power functions having positive integral exponents, negative integral exponents, and exponents that are positive unit fractions (f(x) = , n ≥ 0).
• Identify and explain the symmetry of an even or odd power function.
• 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.

Task related to this benchmark: Match That Function

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

• Use exponential functions to represent growth functions, such as f(x) = an^x (a > 0 and n > 1), and decay functions, such as f(x) = an^-x (a > 0 and n > 1).

  Exponential functions model situations where change is proportional to quantity (e.g., compound interest, population grown, radioactive decay).

• Use power functions to represent quantities arising from geometric contexts such as length, area, and volume.

  Examples: The relationships between the radius and area of a circle, between the radius and volume of a sphere, and between the volumes of simple three-dimensional solids and their linear dimensions.

• Use the laws of exponents to determine exact solutions for problems involving exponential or power functions where possible; otherwise approximate the solutions graphically or numerically.

Tasks related to this benchmark: Bighorn Sheep, Match That Function

g. Explain, illustrate, and identify the effect of simple coordinate transformations on the graph of a function.

• 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).
• 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).
• Interpret the graph of y = f(ax) as the graph of y = f(x) expanded horizontally by a factor of if 0 < |a| < 1 or compressed horizontally by a factor of |a| if |a| > 1 and reflected over the y-axis if a < 0.
• Interpret the graph of y = af(x) as the graph of y = f(x) compressed vertically by a factor of if 0 < |a| < 1 or expanded vertically by a factor of |a| if |a| > 1 and reflected over the x-axis if a < 0.

Task related to this benchmark: Function Transformations

A.C.2 Polynomial functions

a. Transform quadratic functions and interpret their graphical forms.

• Write a quadratic function in polynomial or standard form, f(x) = ax² + bx + c, to identify the y-intercept of the function’s parabolic graph or the x-coordinate of its vertex, .
• Write a quadratic function in factored form, f(x) = a(x – b)(x – c), to identify the zeros of the function’s parabolic graph.
• Write a quadratic function in vertex form, f(x) = a(x – h)² + k, to identify the vertex and axis of symmetry of the function.
• Describe the effect that changes in 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).

  Examples: If a and c have opposite signs, then the zeros of the quadratic function 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.

• Determine domain and range, intercepts, axis of symmetry, and maximum or minimum for quadratic functions whose intercepts and vertices are real.

b. Analyze polynomial functions and identify their key characteristics.

• 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.

• 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.
• Distinguish among polynomial functions of low degree, i.e., constant functions, linear functions, quadratic functions, or cubic functions.
• Explain why every polynomial function of odd degree has at least one zero.
• 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 zero, 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.

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

Simple polynomial functions include constant functions, linear functions, quadratic functions such as f(x) = ax² + b or g(x) = (x – a)(x + b), or cubic functions such as f(x) = x³, f(x) = x³ – a or f(x) = x(x – a)(x + b).

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

  It is not always possible to determine exact horizontal intercepts.

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

• Use 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.

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

Task related to this benchmark: Season Pass

A.C.3 Polynomial and rational expressions and equations

a. Solve and graph quadratic equations having real solutions using a variety of methods.

• Solve quadratic equations having real solutions by factoring, by completing the square, and by using the quadratic formula.
• Use a calculator to approximate the roots of a quadratic equation and as an aid in graphing.
• Select and explain a method of solution (e.g., exact vs. approximate) that is effective and appropriate to a given problem.

b. Relate the coefficients a, b, and c of the quadratic equation ax² + bx + c = 0 to its roots.

• Describe how the discriminant D = b² – 4ac indicates the nature of the roots of the equation ax² + bx + c = 0.

  The roots are real and distinct if D > 0, real and equal if D = 0, and not real if D < 0.

• Use the quadratic formula to prove that the abscissa of the vertex of the corresponding parabola is halfway between the roots of the equation.

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

• Use the position of the variable in a term to determine the classification of the related expression, equation, or function.

  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.

d. Perform operations on polynomial expressions.

In general, a polynomial expression is any expression equivalent to one of the form axⁿ + bx^n–1 + . . . + px² + qx + r for n an integer, n ≥ 0 and a ≠ 0. A polynomial expression is a sum of monomials.

• Add, subtract, multiply, and factor polynomials.
• Divide one polynomial by a lower-degree polynomial.

e. Know and use the binomial expansion theorem.

• Relate the expansion of (a + b)ⁿ to the possible outcomes of a binomial experiment and the nth row of Pascal’s triangle.

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

g. Perform operations on simple rational expressions.

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

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

  Example: .

A.D.1 General quadratic equations and inequalities

a. Solve and graph quadratic equations having complex solutions.

• Use the quadratic formula to solve any quadratic equation and write it as a product of linear factors.
• Use the discriminant D = b² – 4ac to determine the nature of the roots of the equation ax² + bx + c = 0.
• Show that complex roots of a quadratic equation having real coefficients occur in conjugate pairs.

b. 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.

Task related to this benchmark: Leo's Painting

c. Manipulate simple quadratic equations to extract information.

Example: Use the completing the square method to determine the center and radius of a circle from its equation given in general form.

A.D.2 Rational and radical equations and functions

a. Solve simple rational and radical equations in one variable.

• Use algebraic, numerical, graphical, and/or technological means to solve rational equations.
• Use algebraic, numerical, graphical, and/or technological means to solve equations involving a radical.
• 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.

Task related to this benchmark: You're Toast, Dude!

b. Graph simple rational and radical functions in two variables.

• Graph rational functions in two variables; identify the domain, range, intercepts, zeros, and asymptotes of the graph.
• Graph simple radical functions in two variables; identify the domain, range, intercepts, and zeros of the graph.
• 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.

Task related to this benchmark: You're Toast, Dude!

A.E.1 Trigonometric functions

a. Relate sine and cosine functions to a central angle of the unit circle.

• Interpret the sine, cosine, and tangent functions corresponding to a central angle of the unit circle in terms of horizontal and vertical sides of right triangles based on that central angle.

b. Define and graph trigonometric functions over the real numbers.

• Use the unit circle to extend the domain of the sine and cosine function to the set of real numbers.
• Explain and use radian measures for angles; convert between radian and degree measure.
• Create, interpret, and identify key characteristics (period, amplitude, vertical shift, phase shift) of basic trigonometric graphs (sine, cosine, tangent).
• Identify the zeros of trigonometric functions that have a vertical shift of 0.
• Describe the effect that changes in each of the coefficients of f(x) = A sin B(x - C) + D have on the position and key characteristics of the graph of f(x).

c. Analyze periodic functions and identify their key characteristics.

Periodic functions are used to describe cyclic behaviors.

• Recognize periodic phenomena.

  Examples: The height above the pavement of a point on the tread of a truck tire; the length of time from sunrise to sunset over a 10–year period.

• Identify the length of a cycle in situations exhibiting periodic behavior and use it to make predictions.
• Use basic properties of frequency and amplitude to solve problems.
• Recognize that the graphs of trigonometric functions represent periodic behavior.

d. Recognize and solve problems that can be modeled using equations and inequalities involving trigonometric functions.

Examples: Circular motion as in approximate orbits, water wheels, or Ferris wheels; periodic behavior as in sound waves, tides, or minutes of daylight.

• Demonstrate graphically the relation between the sine function and common examples of harmonic motion.

e. Derive and use basic trigonometric identities.

• Derive the basic Pythagorean identities for sine and cosine, for tangent and secant, and for cotangent and cosecant.
• Know and use the angle addition formulas for sine, cosine, and tangent.
• Derive and use formulas for sine, cosine, and tangent of double angles.

f. Solve geometric problems using the sine and cosine functions.

• Know and use the Law of Sines and the Law of Cosines to solve problems involving triangles.
• Know and use the area formula Area = ab sin(C) to determine the area of triangle ABC.

A.E.2 Matrices and linear equations

a. Know and use matrix notation for rows, columns, and entries of cells.

b. Compute the determinant of a 2x2 or 3x3 matrix.

c. Know and perform addition, subtraction, and scalar multiplication of matrices.

• Recognize that matrix addition is associative and commutative and explain why that is the case.
• Distinguish between multiplication of a matrix by a scalar (a number or variable representing a number) and the multiplication of two matrices.

d. Know and perform matrix multiplication.

• Describe the characteristics of matrices that can be multiplied and those that cannot.
• Utilize knowledge of the algorithm for matrix multiplication. Compute the product by hand for 2 x 2 or 3 x 3 matrices and use technology for matrices of larger dimension.
• Know that matrix multiplication is not commutative and provide examples of square matrices A and B such that AB ≠ BA.
• Know and apply the associative property of matrix multiplication.

  The associative property of matrix multiplication states that if there are three matrices, A, B, and C such that AB and BC are defined, then (AB)C and A(BC) are defined and (AB)C = A(BC).

e. Relate vector and matrix operations to transformations in the coordinate plane.

• Interpret vector and matrix addition as translation in the coordinate plane.Examples: If vector v = (x, y), then v + (–1, 6) represents the translation of the plane 1 unit to the left and 6 units up; is a way of representing the translation of the point A(3, 1) to the left 1 unit and up 6 units; combining , and describes the translation of the triangle A(3,1), B(–2,0), C(4,5) to the left 1 unit and up 6 units to the new triangle A’(2,7), B’(–3,6), C’(3,11), which is congruent to triangle ABC.
• Interpret matrix multiplication as reflection about the axes or rotation about the origin in the coordinate plane.

  Examples: represents the rotation of the point A(3, 1) 180° about the origin; represents the reflection of the point A(3, 1) over the y-axis.

f. Apply the concept of inverse to matrix multiplication.

• Know the definition and properties of the identity matrix.
• Find the inverse of a 2 x 2 matrix if the inverse exists.
• Use row reduction to find inverses of 3 x 3 matrices when the inverses exist.
• Use the inverse of a matrix, when one exists, to solve a matrix equation.

g. Write and solve systems of 2 x 2 and 3 x 3 linear equations in matrix form.

• Switch between equation notation and matrix notation for linear systems.
• Solve linear systems by row reduction.
• Solve linear systems using the inverse matrix.

  If the inverse of a matrix associated with a linear system involving the same number of distinct equations as variables does not exist, then it is not always possible to solve the system; when it is possible, there will be infinitely many solutions.

A.E.3 Operations on functions

a. Compare and contrast properties of different types (families) of functions.

These types (families) include algebraic (linear, quadratic, polynomial, rational), piecewise (absolute value, step, piecewise-linear), and transcendental (trigonometric, exponential, logarithmic).

• Determine key characteristics of a function (e.g., domain, range, zeros, symmetries, asymptotes, end behavior) from its context or from its symbolic or graphical form.

  Some details (e.g., range) may require technological assistance to determine.

• Recognize and solve problems that can be modeled with various types (families) of functions.

b. Analyze the transformations of a function from its graph, formula, or verbal description.

• Select a prototypical representation for each family of functions.

  Examples: f(x) = x² is a prototype for quadratic functions; g(x) = sin(x) is a prototypical trigonometric function.

• Analyze a graph to identify properties that provide useful information about a given context.
• Identify changes in the graph of a function related to various transformations (vertical/horizontal translations, reflections over the x- or y-axis, dilation/contraction) and relate them to changes in the function’s algebraic representation.

  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.

c. Compute the sum, difference, product, and quotient of two functions.

d. Determine the composition of simple functions, including any necessary restrictions on the domain.

• Know the relationships among the identity function, composition of functions, and the inverse of a function, along with implications for the domain.

A.E.4 Inverse functions

a. Analyze characteristics of inverse functions.

• Identify the conditions under which the inverse of a function is a function.
• Determine whether two given functions are inverses of each other.
• Explain why the graph of a function and its inverse are reflections of one another over the line y = x.

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

• Determine the inverse of a simple polynomial or simple rational function.
• 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 x = ).

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

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

• 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.

• Use properties of logarithms to manipulate logarithmic expressions in order to extract information.
• 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.

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

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

• Prove basic properties of logarithms using properties of exponents (or the inverse exponential function).

A.E.5 Relations

a. Know the definition of a relation and distinguish non-function relations from functions.

Example: x² + y² = 1 defines a relation, but not a function.

b. Determine whether a function has the characteristics of reflexivity, symmetry, and transitivity; know that relations exhibiting these characteristics are members of a special class of relations called equivalence relations.

c. Explain how some geometric concepts, such as equality, parallelism, and similarity, can be defined as equivalence relations.

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