Algebra 2, Quarter 1, Unit 1.1 Quadratic Functions and ...cranstonmath.weebly.com/uploads/5/4/8/3/5483566/cranston-math-gr11... · Quadratic Functions and Their Graphs ... A-REI.11

Cranston Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin

1

Algebra 2, Quarter 1, Unit 1.1

Quadratic Functions and Their Graphs

Overview Number of instructional days: 6 (1 day = 45 minutes)

Content to be learned Mathematical practices to be integrated • Create a graph of a quadratic function from a

table of values.

• Create a graph of a quadratic function from an equation in two variables.

• Identify the intercept(s), vertex, and maximum or minimum values of a graph.

• Compare graphs of quadratic functions in terms of translations, reflections, and dilations by understanding the affects a, h, and k have on the graph of f(x) = a(x – h)2 + k.

• Explain why the intersection(s) of f(x) and g(x) is (are) a solution to the equation f(x) = g(x).

Use appropriate tools strategically.

• Use pencil and paper to sketch graphs of quadratics.

• Use graphing calculators or other technology to compare and analyze graphs of quadratics.

Look for and make use of structure.

• Look for patterns to determine translations, reflections, and dilations.

Essential questions • Can you identify key features of a quadratic

function from a graph?

• Can you generate a graph of a quadratic function given the equation or a table of values?

• Can you describe the transformations of f(x) = a(x – h)2 + k?

Algebra 2, Quarter 1, Unit 1.1 Quadratic Functions and Their Graphs (6 days)


2

Written Curriculum

Common Core State Standards for Mathematical Content

Creating Equations★ A-CED

Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple root functions]

A-CED.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.★

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

Reasoning with Equations and Inequalities A-REI

Represent and solve equations and inequalities graphically [Combine polynomial, rational, radical, absolute value, and exponential functions]

A-REI.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 f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

Interpreting Functions F-IF

Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models]

F-IF.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.★

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

Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, piecewise-defined]

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

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.



3

Building Functions F-BF

Build new functions from existing functions [Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types]

F-BF.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.

Common Core Standards for Mathematical Practice

5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Clarifying the Standards

Prior Learning

In grade 8, students used image formulas to analyze the transformations of geometric figures. For example, T(x, y) = (x + a, y + b), where a is the number of units moved horizontally and b is the number of units moved vertically. In grade 8 or Algebra I, students observed and compared graphs and tables of quadratic to linear functions.



4

Current Learning

Students graph quadratic functions from tables of values and equations in two variables. From a graph, they identify the intercept(s) and maximum or minimum value(s). Students extend their understanding of quadratic functions by applying translations, dilations, and reflections to the parent function f(x) = x2.

Future Learning

Students will apply their knowledge of translations, reflections, and dilations to graph various other classes of functions and identify key features of the respective function.

Additional Findings

Beyond Numeracy indicates that quadratics have real-world applications in engineering, physics, and elsewhere. It is more than just plugging in numbers. (p. 198)

Atlas of Science Literacy, Volume 1 states that students have difficulty translating graphical and algebraic representations, especially moving from a graph to an equation. (p. 114)


5


Solving Quadratic Equations with Real and Complex Solutions


Content to be learned Mathematical practices to be integrated • Define the imaginary number i.

• Solve a quadratic equation by factoring, completing the square, and using the quadratic formula.

• Recognize that the x-intercept(s) of a parabola are the real roots of a quadratic equation.

• Determine that parabolas with no x-intercepts have complex roots.

• Apply quadratic models to real-world scenarios.


• Analyze graphs of functions and solutions, with or without graphing technology.

• When making mathematical models, use technology to help visualize the results.

Attend to precision.

• Calculate accurately and efficiently by choosing the appropriate strategies.

• Express solutions with a degree of precision appropriate for the context of the problem.

Essential questions • How can you choose an appropriate strategy to

determine the roots/solutions to a quadratic equation?

• Can you appropriately apply factoring, completing the square, or the quadratic formula to determine the solutions/roots of a quadratic equation?

• What is the correlation between the solutions of a quadratic equation and its graph?

• Can you develop a mathematical model given a real-world problem and make predictions or find solutions based on the model?

Algebra 2, Quarter 1, Unit 1.2 Solving Quadratic Equations with Real and Complex Solutions (15 days)


6

Written Curriculum


The Complex Number System N-CN

Perform arithmetic operations with complex numbers.

N-CN.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.

Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients]

N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.


Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function]

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

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

F-IF.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.



Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.



7

6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.


Prior Learning

In Algebra 1, students developed skills necessary to complete the square, use the quadratic formula, and factor quadratic equations.

Current Learning

Students define I as i2 = –1. They build upon their understanding of roots of quadratic equations and extend possible solutions to include complex solutions. They use the techniques of factoring, completing the square, graphing, and the quadratic formula.

Future Learning

Later in Algebra 2, students will define the set of imaginary numbers as a subset of the complex numbers, including performing operations within this set of numbers. They will also determine the roots of higher order polynomial functions both in Algebra 2 and Precalculus.

Additional Findings

It may be necessary to point out to the students that in Algebra 1, parabolas with no x-intercepts have no real roots/solutions, not “no solution.” There is, in fact, a solution in the set of complex numbers.

It may be helpful to remind students that prime polynomials do not imply that there is no solution, and they must apply another strategy.



8


9


Operations on Polynomials and Complex Numbers


Content to be learned Mathematical practices to be integrated • Add, subtract, and multiply polynomial

expressions.

• Understand that polynomials are closed under addition, subtraction, and multiplication as are the set of integers.

• Divide polynomials with or without a remainder.

• Add, subtract, and multiply complex numbers.

• Compute in for any non-negative integer n.

• (+)Know and apply the Binomial Theorem to expand (x + y)n.

Reason abstractly and quantitatively.

• Attend to the meaning of quantities as well as how to compute them.

• Know and flexibly use properties of different operations.

Look for and make use of structure.

• Connect the operations under the set of real numbers to complex numbers and polynomials.

Essential questions • How do you perform the four basic arithmetic

operations using complex numbers? • How do you perform the four basic arithmetic

operations using polynomial expressions?

Algebra 2, Quarter 1, Unit 1.3 Operations on Polynomials and Complex Numbers (6 days)


10

Written Curriculum


Arithmetic with Polynomials and Rational Expressions A-APR

Perform arithmetic operations on polynomials [Beyond quadratic]

A-APR.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.

Use polynomial identities to solve problems

A-APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1 1 The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.

Rewrite rational expressions [Linear and quadratic denominators]

A-APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.


Perform arithmetic operations with complex numbers.

N-CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Seeing Structure in Expressions A-SSE

Interpret the structure of expressions [Polynomial and rational]

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

a. Interpret parts of an expression, such as terms, factors, and coefficients.


2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent



11

representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


Prior Learning

In Algebra 1, students developed skills performing operations on simple polynomial expressions. They only performed division of polynomials with a remainder of zero.

Current Learning

Students build mastery of the skills required to perform the operations on polynomial expressions, and they apply these skills to complex numbers. Students use division of polynomials to determine the quotient and remainder, including expressions with remainders not equal to zero.

Future Learning

Students will use these operations to solve polynomial equations in the next unit of study in Algebra 2 and again in Precalculus. The division of polynomials will lead to the remainder and factor theorems used to solve polynomial equations.

Additional Findings

After multiplying polynomial expressions, caution students to pay close attention to the operation indicated. For example, caution them not to multiply when they should be adding.



12


13


Graphing and Solving Polynomial Functions


Content to be learned Mathematical practices to be integrated • Determine the zeros of a polynomial equation

by factoring, using the factor theorem, and using the remainder theorem.

• Graph a polynomial function.

• Describe the end behavior, intervals of increasing/decreasing values, and relative extreme values of a polynomial function.

• Determine the zero(s) of a polynomial function using graphing technology when appropriate.

• Recognize even and odd functions both graphically and algebraically.

• Calculate the average rate of change of a function over a specific interval.

• (+) Show the Fundamental Theorem of Algebra is true for quadratic polynomials.

Model with mathematics.

• Make sense of situations using polynomial functions.

• Use the polynomial function to make predictions and analyze a situation.


• Use paper and pencil to find rational roots of simple polynomial functions.

• Use graphing technology to determine irrational roots of polynomial functions.

Essential questions • Where are the zeros of a polynomial function

located?

• How can you determine the possible rational zeros of a polynomial function?

• How can you use the remainder and factor theorems to determine rational zeros?

• What is the end behavior for any polynomial function?

• Where are the local extrema located on the graph of a polynomial?

Algebra 2, Quarter 1, Unit 1.4 Graphing and Solving Polynomial Functions (14 days)


14

Written Curriculum




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

Seeing Structure in Expressions A-SSE

Interpret the structure of expressions [Polynomial and rational]

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

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

A-SSE.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).

Arithmetic with Polynomials and Rational Expressions A-APR

Understand the relationship between zeros and factors of polynomials

A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A-APR.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.

Use polynomial identities to solve problems

A-APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.


Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients]

N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.



15


Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models]

F-IF.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.★


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

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Building Functions F-BF

Build new functions from existing functions [Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types]

F-BF.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.


4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.


Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their



16

limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.


Prior Learning

In Algebra 1, students performed operations with polynomial expressions and evaluated polynomials in function notation. Previously, in Algebra 2, students divided polynomial expressions.

Current Learning

Students reinforce and build mastery of division of polynomial expressions, a skill that is critical to performing the factor theorem. They use the factor and remainder theorems to determine zeros of polynomials of degree greater than two. Students further their understanding of critical graphical features.

Future Learning

Students will further develop these skills in work with rational functions and future advanced mathematical courses.

Additional Findings

None at this time.

Documents

Algebra 2, Quarter 1, Unit 1.1 Quadratic Functions and ...cranstonmath.weebly.com/uploads/5/4/8/3/5483566/cranston-math-gr11... · Quadratic Functions and Their Graphs ... A-REI.11