# Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 1: Quadratic Functions

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1 Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 1: Quadratic Functions These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

4 STANDARDS ADDRESSED IN THIS UNIT KEY STANDARDS Interpret the structure of expressions MGSE9-1.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MGSE9-1.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients, in context. MGSE9-1.A.SSE.1b Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors. MGSE9-1.A.SSE. Use the structure of an expression to rewrite it in different equivalent forms. For example, see x 4 y 4 as (x ) - (y ), thus recognizing it as a difference of squares that can be factored as (x y ) (x + y ). Write expressions in equivalent forms to solve problems MGSE9-1.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. MGSE9-1.A.SSE.3a Factor any quadratic expression to reveal the zeros of the function defined by the expression. MGSE9-1.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression. Create equations that describe numbers or relationships MGSE9-1.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only). MGSE9-1.A.CED. Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase in two or more variables refers to formulas like the compound interest formula, in which A = P(1 + r/n) nt has multiple variables.) Mathematics : Quadratic Functions July 015 Page 4 of 19

5 MGSE9-1.A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. Examples: Rearrange Ohm s law V = IR to highlight resistance R; Rearrange area of a circle formula A = π r to highlight the radius r. Solve equations and inequalities in one variable MGSE9-1.A.REI.4 Solve quadratic equations in one variable. MGSE9-1.A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) = q that has the same solutions. Derive the quadratic formula from ax + bx + c = 0. MGSE9-1.A.REI.4b Solve quadratic equations by inspection (e.g., for x = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions). Interpret functions that arise in applications in terms of the context MGSE9-1.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. MGSE9-1.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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. MGSE9-1.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. Analyze functions using different representations MGSE9-1.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. MGSE9-1.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context). MGSE9-1.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MGSE9-1.F.IF.8a 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 Mathematics : Quadratic Functions July 015 Page 5 of 19

6 terms of a context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. MGSE9-1.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 function and an algebraic expression for another, say which has the larger maximum. Build a function that models a relationship between two quantities MGSE9-1.F.BF.1 Write a function that describes a relationship between two quantities. MGSE9-1.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with \$15 and earns \$ a day, the explicit expression x+15 can be described recursively (either in writing or verbally) as to find out how much money Jimmy will have tomorrow, you add \$ to his total today. J = J +, J = 15 n n 1 0 Build new functions from existing functions MGSE9-1.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. Construct and compare linear, quadratic, and exponential models and solve problems MGSE9-1.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. MGSE9-1.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. MGSE9-1.S.ID.6a Decide which type of function is most appropriate by observing graphed data, charted data, or by analysis of context to generate a viable (rough) function of best fit. Use this function to solve problems in context. Emphasize linear, quadratic and exponential models. Mathematics : Quadratic Functions July 015 Page 6 of 19

8 CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. 1. Use Function Notation. Put data into tables 3. Graph data from tables 4. Solve one variable linear equations 5. Determine domain of a problem situation 6. Solve for any variable in a multi-variable equation 7. Recognize slope of a linear function as a rate of change 8. Graph linear functions 9. Graph inequalities SELECTED TERMS AND SYMBOLS The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The website below is interactive and includes a math glossary. Definitions and activities for these and other terms can be found on the Intermath website. Links to external sites are particularly useful. Formulas and Definitions: Complete factorization over the integers: Writing a polynomial as a product of polynomials so that none of the factors is the number 1, there is at most one factor of degree zero, each polynomial factor has degree less than or equal to the degree of the product polynomial, each polynomial factor has all integer coefficients, and none of the factor polynomial can written as such a product. Completing the Square: Completing the Square is the process of converting a quadratic equation into a perfect square trinomial by adding or subtracting terms on both sides. Difference of Two Squares: a squared (multiplied by itself) number subtracted from another squared number. It refers to the identity a b = ( a+ b)( a b) in elementary algebra. Horizontal shift: A rigid transformation of a graph in a horizontal direction, either left or right. Mathematics : Quadratic Functions July 015 Page 8 of 19

13 What s the Pattern? (Spotlight Task) Georgia Department of Education Standards Addressed in this Task MGSE9-1.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MGSE9-1.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients, in context. MGSE9-1.A.SSE.1b Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors. MGSE9-1.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only). MGSE9-1.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. MGSE9-1.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MGSE9-1.F.IF.8a 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. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. MGSE9-1.F.BF.1 Write a function that describes a relationship between two quantities. MGSE9-1.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with \$15 and earns \$ a day, the explicit expression x+15 can be described recursively (either in writing or verbally) as to find out how much money Jimmy will have tomorrow, you add \$ to his total today. J = J +, J = 15 n n 1 0 Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them by requiring students to interpret and make meaning of a problem and find a logical starting point, and to monitor their progress and change their approach to solving the problem, if necessary.. Reason abstractly and quantitatively by requiring students to make sense of quantities and their relationships to one another in problem situations. Mathematics : Quadratic Functions July 015 Page 13 of 19

14 3. Construct viable arguments and critique the reasoning of others by engaging students on discussion of why they agree or disagree with responses, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 6. Attend to precision by requiring students to calculate efficiently and accurately; and to communicate precisely with others by using clear mathematical language to discuss their reasoning. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. 1. How many 1 x 1 squares are in each stage of this pattern? Stage Number of squares What might stage 5 of this pattern look like? How many 1 x 1 squares would be in stage 5? There will be 1 squares in stage 5 3. Write an expression that describes the number of 1 x 1 squares in stage n of the pattern. Justify your answer geometrically by referring to the pattern. See A, B, C, and D below. Mathematics : Quadratic Functions July 015 Page 14 of 19

15 Encourage students to write their expressions according to how they see the pattern, not to write any particular form of the expression. If students have trouble writing an expression symbolically, encourage them to verbally express how they found the number of squares in each stage and how would expect to find the next number of squares. 4. How much does the number of squares change from stage 1 to stage of the pattern? 5. How much does the number of squares change from stage to stage 3 of the pattern? 4 6. How much does the number of squares change from stage 3 to stage 4 of the pattern? 6 7. What do your answers to 4-6 tell you about the rate of change of the number of squares with respect to the stage number? Because the amount of change in the number of squares does not change by the same amount per each stage, the rate of change is not constant, but instead it is a variable rate of change. 8. You have previously worked with linear and exponential functions. Can this pattern be expressed as a linear or exponential function? Why or why not? No. A linear function would have a constant rate of change, and an exponential function would have a constant ratio between successive values. This has a changing rate of change that is different from what we have seen before. Note: This is the time to introduce the term quadratic and let students know that you will be studying a new type of function, quadratic functions, in this unit. It is not necessary to go into more depth at this point, as students will engage in more in-depth study in the next few tasks. As most groups finish, you should expect expressions of the following forms: A. n(n 1) +1 B. n n +1 Mathematics : Quadratic Functions July 015 Page 15 of 19

16 C. n (n 1) D. n + (n 1)(n 1) You should wrap up this task with a discussion in which different groups of students share their solutions, particularly to question 3. Invite students to share why these expressions are equivalent, and check to see if students know how to expand or simplify to demonstrate algebraically that the expressions are equivalent. If students do not yet know how to multiply binomials, this is a good opportunity to move to area models and discuss this skill. It is essential that students be able to multiply binomials prior to engaging in the rest of the activities of this unit. If students need additional instruction, a good resource is the use of Algebra Tiles. Many sites are available online with additional resources for this. Additionally, be sure to discuss question 8 and the differences between linear and exponential functions, which they studied in Coordinate Algebra, and quadratic functions. This task is adapted from one discussed on page 55 in the following book: Boaler, J. & Humphreys, C. (005). Connecting mathematical ideas: Middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann. Mathematics : Quadratic Functions July 015 Page 16 of 19

17 What s the Pattern? (Spotlight Task) Georgia Department of Education Standards Addressed in this Task MGSE9-1.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. MGSE9-1.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients, in context. MGSE9-1.A.SSE.1b Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors. MGSE9-1.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only). MGSE9-1.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. MGSE9-1.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MGSE9-1.F.IF.8a 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. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. MGSE9-1.F.BF.1 Write a function that describes a relationship between two quantities. MGSE9-1.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with \$15 and earns \$ a day, the explicit expression x+15 can be described recursively (either in writing or verbally) as to find out how much money Jimmy will have tomorrow, you add \$ to his total today. J = J +, J = 15 n n 1 0 Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them by requiring students to interpret and make meaning of a problem and find a logical starting point, and to monitor their progress and change their approach to solving the problem, if necessary.. Reason abstractly and quantitatively by requiring students to make sense of quantities and their relationships to one another in problem situations. Mathematics : Quadratic Functions July 015 Page 17 of 19

18 3. Construct viable arguments and critique the reasoning of others by engaging students on discussion of why they agree or disagree with responses, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 6. Attend to precision by requiring students to calculate efficiently and accurately; and to communicate precisely with others by using clear mathematical language to discuss their reasoning. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. 1. How many 1 x 1 squares are in each stage of this pattern?. What might stage 5 of this pattern look like? How many 1 x 1 squares would be in stage 5? 3. Write an expression that describes the number of 1 x 1 squares in stage n of the pattern. Justify your answer geometrically by referring to the pattern. 4. How much does the number of squares change from stage 1 to stage of the pattern? 5. How much does the number of squares change from stage to stage 3 of the pattern? 6. How much does the number of squares change from stage 3 to stage 4 of the pattern? Mathematics : Quadratic Functions July 015 Page 18 of 19

19 7. What do your answers to 4-6 tell you about the rate of change of the number of squares with respect to the stage number? 8. You have previously worked with linear and exponential functions. Can this pattern be expressed as a linear or exponential function? Why or why not? Mathematics : Quadratic Functions July 015 Page 19 of 19

21 Standards for Mathematical Practice This lesson uses all of the practices with emphasis on: 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. 8. Look for and express regularity in repeated reasoning by expecting students to understand broader applications and look for structure and general methods in similar situations. Mathematics : Quadratic Functions July 015 Page 1 of 19

23 MGSE9-1.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 function and an algebraic expression for another, say which has the larger maximum. Standards for Mathematical Practice This task uses all of the practices with emphasis on: 3. Construct viable arguments and critique the reasoning of others by engaging students on discussion of why they agree or disagree with responses, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. Mathematics : Quadratic Functions July 015 Page 3 of 19

24 Graphing Transformations Adapted from Marilyn Munford, Fayette County School System Standards Addressed in this Task MGSE9-1.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. Standards of Mathematical Practice 4. Model with mathematics by expecting students to apply the mathematics concepts they know in order to solve problems arising in everyday situations, and reflect on whether the results are sensible for the given scenario. 5. Use appropriate tools strategically by expecting students to consider available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a compass, a calculator, software, etc. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. Common Student Misconceptions 1. Students may believe that the graph of y=(x-4) 3 is the graph of y=x 3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by hand and on a graphing calculator to overcome this misconception.. Students often confuse the shift of a function with the stretch of a function. 3. Students may also believe that even and odd functions refer to the exponent of the variable, rather than the sketch of the graph and the behavior of the function. 4. Additionally, students may believe that all functions have inverses and need to see counter examples, as well as examples in which a non-invertible function can be made into an invertible function by restricting the domain. For example, f(x)=x has an inverse (f -1 (x)= xprovided that the domain is restricted to x 0. Teacher Notes: This problem set is provided for students to explore the transformations of the graph of f(x)=x. Depending on their comfort level with technology, some students may struggle using the graphing Mathematics : Quadratic Functions July 015 Page 4 of 19

25 calculator. The teacher should plan on circulating throughout the room to help students use the correct functions on the graphing calculator. For this problem set, students will learn more about the graphs of parabolas if they struggle a bit. Teachers should make sure that students actually make the graphs, because many students will be able to immediately write the equations with the given information. Materials Needed: Graphing Calculators, or, if available, computers with Geometer's Sketchpad or Geogebra. Working on computers rather than graphing calculators will allow students to see more clearly, and some functions are more user-friendly on the computer. You will graph various functions and make conjectures based on the patterns you observe from the original function y=x Follow the directions below and answer the questions that follow. Fill in the t-chart and sketch the parent graph y = x below. x y = x Now, for each set of problems below, describe what happened to the parent graph (y 1 = x ) to get the new functions. Equation Changes to parent graph. y 1 = x Students should observe the graph shifting up y = x +3 y 3 = x Conjecture: The graph of y = x + a will cause the parent graph to shift up. Equation Changes to parent graph. y 1 = x Students should observe the graph shifting down y = x -3 y 3 = x -7. Conjecture: The graph of y = x - a will cause the parent graph to shift down. Mathematics : Quadratic Functions July 015 Page 5 of 19

27 Based on your conjectures, write the equations for the following transformations to y=x. 10. Translated 6 units up 11. Translated units right y = x + 6 y = (x-) 1. Stretched vertically by a factor of Reflected over the x-axis, units left and down 5 units y = 3x y = -(x+) -5 More problems can be added as an extension. Mathematics : Quadratic Functions July 015 Page 7 of 19

28 Graphing Transformations Adapted from Marilyn Munford, Fayette County School System You will graph various functions and make conjectures based on the patterns you observe from the original function y=x Follow the directions below and answer the questions that follow. Fill in the t-chart and sketch the parent graph y = x below. x y = x Now, for each set of problems below, describe what happened to the graph (y 1 = x ) to get the new functions. Equation Changes to parent graph. y 1 = x y = x +3 y 3 = x Conjecture: The graph of y = x + a will cause the parent graph to. Equation Changes to parent graph. y 1 = x y = x -3 y 3 = x -7. Conjecture: The graph of y = x - a will cause the parent graph to. Equation Changes to parent graph. y 1 = x y = (x+3) y 3 = (x+7) 3. Conjecture: The graph of y = (x + a) will cause the parent graph to. Equation Changes to parent graph. y 1 = x y = (x-3) y 3 = (x-3) 4. Conjecture: The graph of y = (x - a) will cause the parent graph to. Mathematics : Quadratic Functions July 015 Page 8 of 19

29 Equation Changes to parent graph. y 1 = x y = -x y 3 = -3x 5. Conjecture: Multiplying the parent graph by a negative causes the parent graph to. For the following graphs, please use the descriptions vertical stretch (skinny) or vertical shrink (fat). Equation Changes to parent graph. y 1 = x y = 3x y 3 = 7x 6. Conjecture: Multiplying the parent graph by a number whose absolute value is greater than one causes the parent graph to. y 1 = x y = ½ x y 3 = ¼ x 7. Equation Changes to parent graph. Conjecture: Multiplying the parent graph by a number whose absolute value is between zero and one causes the parent graph to. Based on your conjectures above, sketch the graphs without using your graphing calculator. 8. y = (x+3) y = -x + 5 y y x x Now, go back and graph these on your graphing calculator and see if you were correct. Were you? Based on your conjectures, write the equations for the following transformations to y=x. 10. Translated 6 units up 11. Translated units right 1. Stretched vertically by a factor of Reflected over the x-axis, units left and down 5 units Mathematics : Quadratic Functions July 015 Page 9 of 19

30 Paula s Peaches Standards Addressed in this Task MGSE9-1.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. MGSE9-1.A.SSE.3a Factor any quadratic expression to reveal the zeros of the function defined by the expression. MGSE9-1.A.CED. Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase in two or more variables refers to formulas like the compound interest formula, in which A = P(1 + r/n) nt has multiple variables.) MGSE9-1.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. MGSE9-1.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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. MGSE9-1.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. MGSE9-1.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. MGSE9-1.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context). MGSE9-1.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MGSE9-1.F.IF.8a 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. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. Mathematics : Quadratic Functions July 015 Page 30 of 19

31 MGSE9-1.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 function and an algebraic expression for another, say which has the larger maximum. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them by requiring students to interpret and make meaning of a problem and find a logical starting point, and to monitor their progress and change their approach to solving the problem, if necessary.. Reason abstractly and quantitatively by requiring students to make sense of quantities and their relationships to one another in problem situations. 4. Model with mathematics by expecting students to apply the mathematics concepts they know in order to solve problems arising in everyday situations, and reflect on whether the results are sensible for the given scenario. 5. Use appropriate tools strategically by expecting students to consider available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a compass, a calculator, software, etc. 6. Attend to precision by requiring students to calculate efficiently and accurately; and to communicate precisely with others by using clear mathematical language to discuss their reasoning. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. 8. Look for and express regularity in repeated reasoning by expecting students to understand broader applications and look for structure and general methods in similar situations. Common Student Misconceptions 1. Some students may believe that factoring and completing the square are isolated techniques within a unit of quadratic equations. Teachers should help students to see the value of these skills in the context of solving higher degree equations and examining different families of functions.. Students may think that the minimum (the vertex) of the graph of y=(x+5) is shifted to the right of the minimum (the vertex) of the graph y=x due to the addition sign. Students should explore expels both analytically and graphically to overcome this misconception. 3. Some students may believe that the minimum of the graph of a quadratic function always occur at the y-intercept. Mathematics : Quadratic Functions July 015 Page 31 of 19

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