Unit 3

Polynomial & Quadratic Functions

 

Algebra I 

Unit Length and Description:

 

35 days

 

In this unit, students build on their knowledge from unit 2, where they extended the laws of exponents to rational exponents. Students apply this new understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of equations involving quadratic expressions. In preparation for work with quadratic relationships students explore distinctions between rational and irrational numbers. They consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to the way in which extending the whole numbers to the negative numbers allows x+1 = 0 to have a solution.  (Mathematics Appendix A, p.23, 25)

 

·        Use properties of rational and irrational numbers.

·        Solve equations and inequalities in one variable.

·        Create equations that describe number or relationships.

·        Perform arithmetic operations on polynomials.

·        Write expressions in equivalent forms to solve problems.

·        Interpret the structure of expressions.

·        Interpret functions that arise in applications in terms of a context.

·        Analyze functions using different representations.

·        Build new functions from existing functions.

 

Standards:

 

A-APR.A.1   Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

 

A-APR.B.3   Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

o   Standards Clarification: In Algebra I, tasks are limited to quadratic and cubic polynomials, in which linear and quadratic factors are available. For example, find the zeros of (x – 2)(x2 – 9). 

 

A-CED.A.1   Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

  • Standards Clarification: In Algebra I, tasks are limited to linear, quadratic, or exponential equations with integer exponents.

 

A-CED.A.2   Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

 

A-REI.B.4   Solve quadratic equations in one variable.

  1. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
  2. Solve quadratic equations by inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

o   Standards Clarification: Tasks do not require students to write solutions for quadratic equations that have roots with nonzero imaginary parts. However, tasks can require that students recognize cases in which a quadratic equation has no real solutions.

 

A-REI.D.11 Explain why the 𝑥-coordinates of the points where the graphs of the equations 𝑦=𝑓(𝑥) and 𝑦=𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥)=𝑔(𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where (𝑥) and/or (𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

o   Standards Clarification: In Algebra I, tasks that assess conceptual understanding of the indicated concept may involve any of the function types mentioned in the standard except exponential and logarithmic functions. Finding the solutions approximately is limited to cases where f(x) and g(x) are polynomial functions.

 

A-SSE.A.1   Interpret expressions that represent a quantity in terms of its

context.

  1. Interpret parts of an expression, such as terms, factors, and coefficients.
  2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

 

A-SSE.A.2   Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

o   Standards Clarification: In Algebra I, tasks are limited to numerical expressions and polynomial expressions in one variable. Examples: Recognize that 532 – 472 is the difference of squares and see an opportunity to rewrite it in the easier-to-evaluate form (53 – 47)(53 + 47). See an opportunity to rewrite a2 + 9a + 14 as (a + 7)(a + 2). Can include the sum or difference of cubes (in one variable), and factoring by grouping. 

 

A-SSE.B.3   Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

  1. Factor a quadratic expression to reveal the zeros of the function it defines.
  2. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

 

F-BF.B.3     Identify the effect on the graph of replacing 𝑓(𝑥) by 𝑓(𝑥)+𝑘, 𝑘 𝑓(𝑥), 𝑓(𝑘𝑥), and 𝑓(𝑥+𝑘) for specific values of 𝑘 (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

o   Standards Clarification: In Algebra I, identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k (both positive and negative) is limited to linear and quadratic functions. Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. Tasks do not involve recognizing even and odd functions.

 

F-IF.B.4      For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

o   Standards Clarification: Tasks have a real-world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers.

 

F-IF.B.5      Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function (𝑛) gives the number of person-hours it takes to assemble 𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function.

 

F-IF.B.6      Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

o   Standards Clarification: Tasks have a real-world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers.

 

F-IF.C.7      Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

  1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
  2. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • Standards Clarification: Tasks have a real-world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions, radical functions, and absolute value functions), and exponential functions with domains in the integers.

 

F-IF.C.8      Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

  1. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

 

F-IF.C.9      Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

o   Standards Clarification: In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers.

 

N-RN.B.3    Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

 

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

A-APR.A.1

·       I can identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are closed under the operations of addition, subtraction, and multiplication

·       I can define “closure”

·       I can apply arithmetic operations of addition, subtraction, and multiplication to polynomials

A-APR.B.3

·       I can factor polynomials using any method.

·       I can sketch graphs of polynomials using zeroes and a sign chart.

A-CED.A.1

·       I can solve linear and exponential equations in one variable

·       I can solve inequalities in one variable

·       I can describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve

·       I can create equations (linear and exponential) and inequalities in one variable and use them to solve problems

·       I can create equations and inequalities in one variable to model real-world situations

·       I can compare and contrast problems that can be solved by different types of equations (linear and exponential)

A-CED.A.2

·       I can identify the quantities in a mathematical problem or real-world situation that should be represented by distinct variables and describe what quantities the variables represent

·       I can create at least two equations in two or more variables to represent relationships between quantities

·       I can justify which quantities in a mathematical problem or real-world situation are dependent and independent of one another and which operations represent those relationships

·       I can determine appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more variables

·       I can graph one or more created equation on a coordinate axes with appropriate labels and scales

A-REI.B.4

·       I can use the method of completing the square to transform any quadratic equation in x into an equation of the form (x-p)2 = q that has the same solutions

·       I can solve quadratic equations in one variable

·       I can derive the quadratic formula by completing the square on a quadratic equation in x

·       I can solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring

·       I can determine appropriate strategies (see first knowledge target listed) to solve problems involving quadratic equations, as appropriate to the initial form of the equation

·       I can recognize when the quadratic formula gives complex solutions

A-REI.D.11

·        I can recognize and use function notation to represent linear and exponential equations

·        I can recognize that if (x1 , y1) and (x2 , y2) share the same location in the coordinate plane that x1 = x2 and y1 = y2

·        I can recognize that f(x) = g(x) means that there may be particular inputs of f and g for which the outputs of f and g are equal

·        I can explain why the x-coordinates of the points where the graph of the equations y=f(x) and y=g(x) intersect are the solutions of the equations

f(x) = g(x). (Include cases where f(x) and/or g(x) are linear and exponential equations)

·        I can approximate/find the solution(s) using an appropriate method for example, using technology to graph the functions, make tables of values or find successive approximations (Include cases where f(x) and/or g(x) are linear and exponential equations)

A-SSE.A.1

·       I can, for expressions that represent a contextual quantity, define and recognize parts of an expression, such as terms, factors, and coefficients

·       I can, for expressions that represent a contextual quantity, interpret parts of an expression, such as terms, factors, and coefficients in terms of the context

·       I can, for expressions that represent a contextual quantity, interpret complicated expressions, in terms of the context, by viewing one or more of their parts as a single entity.

A-SSE.A.2

·       I can identify ways to rewrite expressions, such as difference of squares, factoring out a common monomial, regrouping, etc

·       I can identify ways to rewrite expressions based on the structure of the expression

·       I can use the structure of an expression to identify ways to rewrite it.

·       I can classify expression by structure and develop strategies to assist in classification

A-SSE.B.3

·       I can factor a quadratic expression to produce an equivalent form of the original expression

·       I can explain the connection between the factored form of a quadratic expression and the zeros of the function it defines

·       I can explain the properties of the quantity represented by the quadratic expression

·       I can choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression

·       I can complete the square on a quadratic expression to produce an equivalent form of an expression

·       I can explain the connection between the completed square form of a quadratic expression and the maximum or minimum value of the function it define

·       I can explain the properties of the quantity represented by the expression

·       I can choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression

F-BF.B.3

·        I can identify the effect a single transformation will have on the function (symbolic or graphic)

·        I can use technology to identify effects of single transformations on graphs of functions

·        I can graph a given function by replacing f(x) with f(x)+k, kf(x), f(kx), or f(x+k) for specific values of k (both positive and negative)

·        I can describe the differences and similarities between a parent function and the transformed function

·        I can find the value of k, given the graphs of a parent function, f(x), and the transformed function: f(x)+k, kf(x), f(kx), or f(x+k)

·        I can recognize even and odd functions from their graphs and from their equations

·        I can experiment with cases and illustrate an explanation of the effects on the graph using technology

F-IF.B.4

·        I can define and recognize the key features in tables and graphs of linear and exponential functions: intercepts; intervals where the function is increasing, decreasing, positive, or negative, and end behavior

·        I can identify whether the function is linear or exponential, given its table or graph

·        I can interpret key features of graphs and tables of function in the terms of the contextual quantities the function represents

·        I can sketch graphs showing key features of a function that models a relationship between two quantities from a given verbal description of the relationship

F-IF.B.5

·        I can, given the graph or a verbal/written description of a function, identify and describe the domain of the function

·        I can identify an appropriate domain based on the unit, quantity , and type of the function it describes

·        I can relate the domain of the function to its graph and, where applicable, to the quantitative relationship it describes

·        I can explain why a domain is appropriate for a given situation

F-IF.B.6

·        I can recognize slope as an average rate of change

·        I can calculate the average rate of change of a function (presented symbolically or as a table) over a specified interval

·        I can estimate the rate of change from a linear or exponential graph

·        I can interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval

F-IF.C.7

·       I can graph linear functions by hand in simple cases or using technology for more complicated cases and show/label intercepts of the graph

·       I can graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, by hand in simple cases or using technology for more complicated cases, and show/label key features of the graph

·       I can determine the difference between simple and complicated linear, quadratic, square root, cube root, and piecewise-defined functions, including step functions and absolute value functions and know when the use of technology is appropriate

·       I can compare and contrast the domain and range of absolute vale, step and piece-wise defined functions with linear, quadratic, and exponential

F-IF.C.8a

·       I can identify different forms of a quadratic expression

·       I can write functions in equivalent forms using the process of factoring

·       I can identify zeros, extreme values, and symmetry of the graph of a quadratic function

·       I can interpret different but equivalent forms of a function defined by an expression in terms of context

·       I can use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and intercept these in terms of a context

F-IF.C.9

·        I can identify types of functions based on verbal, numerical, algebraic, and graphical descriptions and state key properties (e.g. intercepts, growth rates, average rates of change, and end behaviors)

·        I can differentiate between exponential and linear functions using a variety of descriptors (graphically, verbally, numerically, and algebraically)

·        I can use a variety of function representations algebraically, graphically, numerically in tables, or by verbal descriptions) to compare and contrast properties of two functions

N-RN.B.3

·       I can find the sums and products of rational and irrational numbers

·       I can recognize that the sum of a rational number and an irrational number is irrational

·       I can recognize that the product of a nonzero rational number and irrational number is irrational

·       I can explain why rational numbers are closed under addition or multiplication

 

Enduring Understandings:

 

·        There are multiple algorithms for finding a mathematical solution and those algorithms are frequently associated with different contexts.

·        Quadratic functions, like linear and exponential, can be used to model real-life situations.

·        There is an important distinction between solving an equation and solving an applied problem modeled by an equation.

·        Algebraic representations can be used to generalize patterns in mathematics.

·        Relationships between quantities can be represented symbolically, numerically, graphically, and verbally in the exploration of real world situations.

·        The characteristics of radical functions and its representations are useful in solving real-world problems.

·        Families of functions exhibit properties and behaviors that can be recognized across representations.

·        Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.

·        Mathematical functions are relationships that assign each member of one set to a unique member of another set and the relationship is recognizable across representations.

 

Essential Questions:

 

·       How can patterns, relations, and functions be used as tools to best describe and help explain relationships between quantities?

·       How do parameters introduced in the context of the problem affect the symbolic, numeric and graphical representations of a quadratic function?

·       How are patterns of change related to the behavior of functions?

·       What characteristics of problems would determine how to model the situation and develop a problem solving strategy?

·       How do radical functions model real-world problems and their solutions?

·       How are expressions involving radicals and exponents related?

·       How can an equation, table, and graph be used to analyze the rate of change and other applicable information, related to a real-world problem and the representative function?

·       How can a given function be represented graphically, within a table, by an equation, and in the real-world?

·       What connections can be made between various functions and various representations of functions?