8 HMH Unit 5: Quadratic Functions

Before: Throughout Middle school students learn the building blocks of mathematical modeling. They observe the behavior of functions or relations in tables, graphs, or . Students have also done extensive work linear functions.

During: In 8th Grade Algebra students bring together their work on , Functions, and Statistics. For example, students use the concept of volume to explore to explore functions (8. F). For example, students can represent the volume of a cylinder as a of its height and recognize that the perpendicular height of a cylinder is proportional to the volume of a cylinder. They also recognize that the area of the of the cylinder is the constant of proportionality. Students understand quadratic relationships • classifying • evaluating expressions • adding, subtracting, and multiplying polynomials • factoring polynomials, Students will learn about • transforming quadratic functions • maxima, minima, zeros of functions, and axes of symmetry for quadratic equations • comparing linear, exponential, and quadratic models • other functions and their transformations, Students will connect • quadratic functions and functions • quadratic graphs and square root graphs • modeling with quadratic functions and modeling with other functions.

After: Functions presented as expressions can model many important phenomena. Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with a constant term of zero describe proportional relationships.

A graphing utility or a computer algebra system can be used to experiment with properties of these functions and their graphs and to build computational models of functions, including recursively defined functions.

DRAFT Algebra 8

UNIT 5: Quadratic Functions and Modeling Unit Planner TIME: 6 weeks UNIT NARRATIVE: An important nonlinear function category is quadratics. Understanding characteristics of quadratic functions and connections between various representations should be developed in this unit. When examining the table form of a , the change in the rate of change distinguishes it from a linear relationship. In particular, looking at the second rate of change or difference is where a constant value occurs. The symmetry of the function values can be found in the table. The graphical form shows common characteristics of quadratic functions including maximum or minimum values, symmetric shapes (parabolas), location of the y-intercept, and the ability to determine roots of the function. Quadratic functions can be written in a variety of formats: polynomial form f (x) = ax2 + bx + c, factored form f (x) = a (x - p) (x - q), and vertex form f (x) = a (x - h)2 + k. The impact of changing the parameters a, b, c, h, k, p, and q should be explored and understood. Connections should be made between each explicit form and its graph and table. Real- world situations that can be modeled by quadratic functions include projectile motion, television dish antennas, and revenue and profit models in business. In this unit students synthesize and generalize what they have learned about a variety of function families. They explore the effects of transformations on Graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always hav ethe same effect regardless of the type of the underlying functions. They identify appropriate types of functions to model a situation, they adjust parameters t oimprove the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit.

Textbook Correlations: Additional Resources HMH Go Math Unit 5: Modules 17, 18, and 19 Personal Math Trainer, Math on the Spot, Formative Assessment Lessons, and Real Player Activator videos ESSENTIAL QUESTIONS: ACADEMIC VOCABULARY: function, , symmetry, 1. How do quadratic functions relate to their graphs? quadratic function, parabola, vertex of a parabola, maximum value, 2. How does the graph of f(x) = (x - h)2 + k change as the constants h and k are minimum value, , axis of symmetry, absolute value, changed? domain, integer, parent function, range, greatest integer function, 3. How does the graph of f(x) = ax2 change as the constant a is changed? piecewise function, step function, inverse (of a function), parameter, 4. How can you obtain the graph of g(x) = a(x - h)2 + k from the graph of g(x) = x2? reflection, translation, cube root function, square root function, linear, 5. How are the characteristics of quadratic functions related to the key features of quadratic, exponential, function model their graphs? 6. How can you use the graph of a quadratic function to solve a quadratic ? 7. How can you solve a system of equations when one equation is linear and the other is quadratic? 8. How can you decide which function type to use when modeling? 9. How do piecewise functions differ from other types of functions? 10. What are the characteristics of an absolute value function? 11. How do the square and cube root function families relate to their graphs?

DRAFT Algebra 8

CLUSTER HEADING & STANDARDS: MATHEMATICAL PRACTICE: Understand and apply the Pythagorean Theorem. All mathematical practice standards are addressed in every unit. 8. G.6 Explain a proof of the Pythagorean Theorem and its converse. MP 1 Make sense of problems and persevere when solving them. 8. G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right MP 2 Reason quantitatively and abstractly. triangles in real-world and mathematical problems in two and three dimensions. MP 3 Construct viable arguments and critique the reasoning of others. 8. G.8 Apply the Pythagorean Theorem to find the distance between two points in a MP 4 Model with . coordinate system. MP 5 Use appropriate tools strategically. MP 6 Attend to precision. Interpret functions that arise in applications in terms of the context. MP 7 Look for and make use of structure. F.IF.4 For a function that models a relationship between two quantities, interpret MP 8 Look for and express regularity in repeating reasoning. 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.*

Analyze functions using different representations. 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.* • F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. • F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Construct and compare linear, quadratic, and exponential models and solve problems. F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Solve equations and inequalities in one variable. DRAFT Algebra 8 A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve systems of equations. A.REI.7 Solve a simple system consisting of a linear equation and a in two variables algebraically and graphically. For example, find the points of intersection between a and a .

Represent and solve equations and inequalities graphically. 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.

Build new functions from existing functions. 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.

Learning Outcomes: Apply Pythagorean Theorem in solving real-world problems dealing with two- and three-dimensional shapes. Solve basic mathematical Pythagorean Theorem problems and its converse to find missing lengths of sides of triangles in two and three-dimensions. Determine the differences between simple and complicated linear, exponential and quadratic functions and know when the use of technology is appropriate. 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. Interpret different yet equivalent forms of a function defined by an expression in terms of a context. Given the expression of a quadratic function, interpret zeros, extreme values, and symmetry of the graph in terms of a real-world context. Write a quadratic function defined by an expression in different but equivalent forms to reveal and explain different properties of the function and determine which form of the quadratic is the most appropriate for showing zeros and symmetry of a graph in terms of a real-world context. 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. Use a variety of function representations (algebraic, graphical, numerical in tables, or by verbal descriptions) to compare and contrast properties of two functions. Write a DRAFT Algebra 8 function that describes a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context. Describe the differences and similarities between a parent function and the transformed function. Find the value of k, given the graphs of a parent function, f(x), and the transformed function: f(x) + k, k f(x), f (kx), or f(x + k). Recognize even and odd functions from their graphs and equations. Experiment with cases and illustrate an explanation of the effects on the graph, using technology. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. End of the Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Balancing equations using Hands-On Algebra. Virtual Technology of real-world situations. ELD Literacy Standards Small instruction Scaffolding 8th grade Common Core regarding Use of Rational expressions Graphic organizers One on one peer support one step equations and inverse properties. Multiple step with multiple simplifications. Highlighting : “cloze” activities Smaller size quantities Skill base lessons regarding properties and Multi standard processes SIOP strategies Talks integers. Talk Moves Real-world visuals Visual aids: Allow students to take lines Presentations Group collaboration world situations. Number Talks Number talks Scaffolding of basic concepts regarding Visual charts using T-Tables prime . Visual aids: Allow students to take real world Begin with basic monomials and situations. progress. Scaffolding of basic concepts regarding prime Revisit 7th, 8th, Common Core factorization. Standards in regard to factors and Begin with basic monomials and progress. primes. Revisit 7th, 8th, Common Core Standards in Allow conceptual understanding using regard to factors and primes. the area model of multiplication. Allow conceptual understanding using the Develop lessons with various area model of multiplication. polynomial processes prior to Develop lessons with various polynomial complex . processes prior to complex numbers.

DRAFT