# Math Curriculum Guide Portales Municipal Schools Pre-Calculus

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1 Math Curriculum Guide Portales Municipal Schools Pre-Calculus Revised Spring 2016

2 PRECALCULUS COMMON CORE STANDARDS 1st Nine Weeks Extend the properties of exponents to rational exponents N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Perform arithmetic operations on polynomials CMS APR.8 Factor polynomials of various types (e.g., difference of squares, perfect square trinomials, sum and difference of cubes). Rewrite rational expressions A-APR.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Create equations that describe numbers or relationships. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Understand solving equations as a process of reasoning and explain the reasoning A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable A-REI.4 Solve quadratic equations in one variable. 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)2 = q that has the same solutions. Derive the quadratic formula from this form. A-REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), 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". Understand the concept of a function and use function notation F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If "f" is a function and "x" is an element of its domain, then f(x) denotes the output of "f" corresponding to the input "x". The graph of "f" is the graph of the equation y = f(x). F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context 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.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. 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. (linear) F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (step/piecewise) Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).

3 PRECALCULUS COMMON CORE STANDARDS 2nd Nine Weeks 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. N-CN.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Use complex numbers in polynomial identities and equations. N-CN.7 Solve quadratic equations with real coefficients that have complex solutions. N-CN.8 (+) Extend polynomial identities to the complex numbers. N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Rewrite rational expressions 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. Create equations that describe numbers or relationships A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. A-REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Interpret functions that arise in applications in terms of the context F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 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. (quadratic) F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (square root/cube root) F-IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F-IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Build a function that models a relationship between two quantities F-BF.1 Write a function that describes a relationship between two quantities. F-BF.1c (+) Compose functions. Build new functions from existing functions F-BF.4 Find inverse functions. F-BF.4a 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. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).

4 PRECALCULUS COMMON CORE STANDARDS 3rd Nine Weeks Analyze functions using different representations CMS IF-7f Graph trigonometric functions, showing period, midline, and amplitude. Extend the domain of trigonometric functions using the unit circle. F-TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π x, π+x, and 2π x in terms of their values for "x", where "x" is any real number. F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F-TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%). 4th Nine Weeks Model periodic phenomena with trigonometric functions F-TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. Prove and apply trigonometric identities F-TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Apply trigonometry to general triangles G-SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).

5 Grade: PreCalculus Revision Date: May 2016 Page # : 01 Domain: RN- Real Number System Extend the properties of exponents to rational exponents. N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. MP.3 Construct viable arguments and critique the reasoning of others. - Chapter P, Section 2 Understand that the denominator of the rational exponent is the root index and the numerator is the exponent of the radicand. For example, 5¹/² = 5 Extend the properties of exponents to justify that (5¹/² )² = 5. - Chapter P Midchapter Test p. 67, Questions N- RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - Chapter P, Section 3 Students should be able to use the properties of exponents to rewrite expressions involving radicals as expressions using rational exponents. - Chapter P Test p. 133, Questions 6-9

6 Grade: PreCalculus Revision Date: May 2016 Page # : 02 Domain: CN- Complex Number System Perform arithmetic operations with complex numbers. (+) For advanced course: calculus, advanced statistics, etc. N-CN.1 Know there is a complex number is such that i 2 = 1, and every complex number has the form a + bi with a and b real. MP.6 Attend to precision. - Chapter 2, Section 1 Web Resources orglib.com (select High School and Domain) Every number is a complex number or the form a + bi, where a and b are elements of the Real Numbers and bi is an element of the Pure Imaginary Numbers. Students should know the sets and subsets of the Complex Number System. The identity, i= -1, is not only used to identify non-real solutions for particular functions, but is also be used to find the identity, i 2 = -1, which is used to simplify expressions. Example: Is it possible to simplify using the Real Number System? Justify your reasoning. - Chapter 2 Midchapter Test p. 339, Questions 4, 32, 33 N-CN.2 Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - Chapter 2, Section 1 Web Resources orglib.com (select High School and Domain) When adding, subtracting, or multiplying Complex Numbers and i 2 remains in the expression, use the identity i 2 = -1 and the commutative, associative, and distributive properties to simplify the expression further. MP.8 Look for and express regularity in repeated reasoning. - Chapter 2 Midchapter Test p. 339, Questions 1-7

7 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: CN- Complex Number System Use complex numbers in polynomial identities and equations. N-CN.7 Solve quadratic equations with real coefficients that have complex solutions. MP.1 Make sense of problems and persevere in solving them. - Chapter 2, Section 2 Web Resources orglib.com (select High School and Domain) Extend strategies for solving quadratics such as; taking the square root and applying the quadratic formula, to find solutions of the form, a + bi, for quadratic equations. This extension is made when the identity i= -1 is used to simplify radicals having negative numbers under the radical. - Chapter 2 Midchapter Test p. 339, Questions N-CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i). - Chapter 2, Section 5 Web Resources orglib.com (select High School and Domain) Use polynomial identities to write equivalent expressions in the form of complex numbers - Chapter 2 Midchapter Test p. 339, Questions 12-26

8 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: CN- Complex Number System Use complex numbers in polynomial identities and equations. N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning. - Chapter 2, Section 5 Web Resources orglib.com (select High School and Domain) Understand The Fundamental Theorem of Algebra, which says that the number of complex solutions to a polynomial equation is the same as the degree of the polynomial. Show that this is true for a quadratic polynomial. Examples: If a function has a degree of 5, how many solutions would that function have? Why? Solve the general form of a quadratic equation for x. What do the solutions show in relation to the Fundamental Theorem of Algebra for quadratic equations? - Chapter 2 Test p. 386, Questions 13-17

9 Grade: PreCalculus Revision Date: May 2016 Page # : 05 Domain: APR- Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials CMS Factor polynomials of various types (e.g., difference of squares, perfect square trinomials, sum and difference of cubes). - Chapter P, Sections 4-5 Web Resources orglib.com (select High School and Domain) - Chapter P Midchapter Test p. 68, Questions MP.8 Look for and express regularity in repeated reasoning.

10 Rewrite rational expressions Common Core State Standards-Mathematics Grade: PreCalculus Revision Date: May 2016 Page # : 06 Domain: APR- Arithmetic with Polynomials and Rational Expressions (+) For advanced course: calculus, advanced statistics, etc. 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. MP.8 Look for and express regularity in repeated reasoning. - Chapter 2, Section 4 Rewrite rational expressions,a(x)/b(x), in the form q(x)+r(x) /b(x) by using factoring, long division, or synthetic division. Use a computer algebra system for complicated examples to assist with building a broader conceptual understanding. *****See explanation to change formula to correct format. - Chapter 2 Midchapter Test p. 339, Questions A-APR.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. MP.8 Look for and express regularity in repeated reasoning. - Chapter P, Section 6 Web Resources orglib.com (select High School and Domain) Simplify rational expressions by adding, subtracting, multiplying, or dividing. Understand that rational expressions are closed under addition, subtraction, multiplication, and division (by a nonzero expression). - Chapter P Test p. 133, Questions 14-18

11 Grade: PreCalculus Revision Date: May 2016 Page # : 07 Domain: CED- Creating Equations Create equations that describe numbers or relationships Modeling Standard A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. - Chapter 7, Sections 1-2 Given a contextual situation, write equations in two variables that represent the relationship that exists between the quantities. Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations arising from the functions they have studied. Example: The height of a ball t seconds after it is kicked vertically depends upon the initial height and velocity of the ball and on the downward pull of gravity. Suppose the ball leaves the kicker s foot at an initial height of 0.7 m with initial upward velocity of 22m/sec. Write an algebraic equation relating flight time t in seconds and height h in meters for this punt. - Chapter 7 Test p. 802, Questions 1-5, A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. MP.1 Make sense of problems and persevere in solving them. - Chapter P, Sections 7-8 Solve multi-variable formulas or literal equations, for a specific variable. Explicitly connect this to the process of solving equations using inverse operations. - Chapter P Test p. 133, Questions MP.4 Model with mathematics.

12 Grade: PreCalculus Revision Date: May 2016 Page # : 08 Domain: REI- Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. - Chapter P, Section 7 - Chapter P Test p. 133, Questions 35-40, 43

13 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: REI- Reasoning with Equations and Inequalities Solve equations and inequalities in one variable A-REI.4 Solve quadratic equations in one variable. - Chapter P, Section 7 - Chapter P Test p. 99, Questions p. 133, Questions MP.8 Look for and express regularity in repeated reasoning. 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)2 = q that has the same solutions. Derive the quadratic formula from this form. MP.8 Look for and express regularity in repeated reasoning. - Chapter P, Section 7 Transform a quadratic equation written in standard form to an equation in vertex form (x - p)² = q by completing the square. Derive the quadratic formula by completing the square on the standard form of a quadratic equation. - Chapter P Test p. 99, Questions p. 133, Questions 36-38

14 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: REI- Reasoning with Equations and Inequalities Solve equations and inequalities in one variable A-REI.4b Solve quadratic equations by inspection (e. g., for x2 = 49), 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. MP.8 Look for and express regularity in repeated reasoning. - Chapter P, Section 7 Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. Understand why taking the square root of both sides of an equation yields two solutions. Use the quadratic formula to solve any quadratic equation, recognizing the formula produces all complex solutions. Write the solutions in the form a ± bi, where a and b are real numbers. Explain how complex solutions affect the graph of a quadratic equation. - Chapter P Test p. 99, Questions 61-66

15 Grade: PreCalculus Revision Date: May 2016 Page # : 11 Domain: REI- Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically Modeling Standard 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. MP.4 Model with mathematics. MP.6 Attend to precision. - Chapter 7, Section 4 Construct an argument to demonstrate understanding that the solution to every equation can be found by treating each side of the equation as separate functions that are set equal to each other, f(x) = g(x). Allow y 1 =f (x)and y 2 = g(x) and find their intersection(s). The x-coordinate of the point of intersection is the value at which these two functions are equivalent, therefore the solution(s) to the original equation. Students should understand that this can be treated as a system of equations and should also include the use of technology to justify their argument using graphs, tables of values, or successive approximations. - Chapter 7 Test p. 803, Questions 1-5, 17 A-REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. MP.4 Model with mathematics. - Chapter 7, Section 5 By graphing a two variable inequality, students understand that the solutions to this inequality are all the ordered pairs located on a portion or side of the coordinate plane that, when substituted into the inequality, make the equation true. Students should be able to graph the inequality, specifying whether the points on the boundary line are also solutions by using a dotted or solid line. Using a variety of methods, which include selecting and substituting test points into the inequality, students should be able to determine which portion or side of the graph contains the ordered pairs that are the solutions to the original inequality. - p. 787, Questions 1-12, 27-32, 65

16 Grade: PreCalculus Revision Date: May 2016 Page # : 12 Domain: IF- Interpreting Functions Understand the concept of a function and use function notation F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). MP.6 Attend to precision. - Chapter 1, Section 2 The domain of a function is the set of all x-values (input), called the independent variable. The range of a function is the set of all y-values (output), and is dependent on a particular x-value, thus called the dependent variable. The idea of a function should be developed through an understanding that each input has exactly one output. If a specific rule can be written which models the relation between the input and one unique output, then it is a function. When this relationship is established, the variable y becomes f(x), meaning y=f(x). For example in the equation y=3x+5, for every x-value, there exist one and only one y-value, therefore you may rewrite the equation using function notation, f(x)=3x+5. - Chapter 1 Test p. 275, Questions 1-3 F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MP.6 Attend to precision. - Chapter 1, Section 3 Students should continue to use function notation throughout high school mathematics, understanding f(input) = output, f (x)=y. Students should be comfortable finding output given input (i.e. f(3) =?) and finding inputs given outputs (f(x) = 10) and describe their meanings in the context in which they are used. - Chapter 1 Test p. 275, Questions 1-3

17 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: IF- Interpreting Functions Interpret functions that arise in applications in terms of the context Modeling Standard 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. MP.4 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning. - Chapter 1, Section 3 Given a function, identify key features in graphs and tables including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Given the key features of a function, sketch the graph. This standard should be revisited with every function your class is studying. Students should be able to move fluidly between graphs, tables, words, and symbols and understand the connections between the different representations. For example, when given a table and graph of a function that models a real-life situation, explain how the table relates to the graph and vise versa. - Chapter 1 Test p. 275, Questions 1-3 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. MP.4 Model with mathematics. MP.6 Attend to precision. - Chapter 1, Section 10 Given a function, determine its domain. Describe the connections between the domain and the graph of the function. Know that the domain taken out of context is a theoretical domain and that the practical domain of a function is found based on a contextual situation given, and is the input values that make sense to the constraints of the problem context. - Chapter 1 Test p. 276, Questions 27, 35

18 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: IF- Interpreting Functions Interpret functions that arise in applications in terms of the context Modeling Standard 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. MP.4 Model with mathematics. - Chapter 1, Sections 4-5 Students should be able to describe patterns of changes from tables and/or graphs of linear and exponential functions. Sample vocabulary may include, increasing/decreasing at a constant rate or increasing or decreasing at an increasing or decreasing rate. Students should be comfortable in their understanding of rates of change to apply their knowledge linear and non-linear graphical display. - p. 189, Questions p. 200, Questions 13-18

19 Grade: PreCalculus Revision Date: May 2016 Page # : 15 Domain: IF- Interpreting Functions Analyze functions using different representations Modeling Standard (+) For advanced course: calculus, advanced statistics, etc. 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. F-IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. MP.6 Attend to precision. - Chapter 1, Sections 2, 5, 6 - Chapter 2, Sections 2-6 This standard should be seen as related to F-IF.4 with the key difference being students can create graphs, by hand and using technology, from the symbolic function in this standard. Example: The all-star kicker kicks a field goal for the team and the path of the ball is modeled by f (x) = 4.9t 2 +20t Find the realistic maximum and minimum values for the path of the ball and describe what each means in the context of this problem. - Chapter 1 Test p. 276, Questions (linear) - Chapter 2 Midchapter Test p. 339, Questions 8-11 (quadratic) and (polynomial). - Chapter 2 Review p (square root) and (cube root) - Chapter 2 Review p (step/piecewise) F-IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. CMS Graph trigonometric functions, showing period, midline, and amplitude. - Chapter 1, Sections 2, 5, 6 - Chapter 2, Sections Chapter 4, Section 5 - Chapter 2 Test p. 386, Questions (rational) - Chapter 4 Test p. 583, Questions (trigonometric)

20 Grade: PreCalculus Revision Date: May 2016 Page # : 16 Domain: BF- Building Functions Build a function that models a relationship between two quantities Modeling Standard (+) For advanced course: calculus, advanced statistics, etc. F-BF.1 Write a function that describes a relationship between two quantities. MP.1 Make sense of problems and persevere in solving them. - Chapter 1, Sections 7-8 Compose functions, which means to evaluate a function when the value used to evaluate is another function; i.e., f(g(x)). F-BF.1c (+) Compose functions. For example, if T (y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning. - p. 229, Questions 1-30, 49-64

21 Grade: PreCalculus Revision Date: May 2016 Page # : 17 Domain: BF- Building Functions Build new functions from existing functions (+) For advanced course: calculus, advanced statistics, etc. F-BF.4 Find inverse functions. F-BF.4a 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. For example, f(x) =2 x 3 or f(x) = (x+1)/(x 1) for x 1. MP.4 Model with mathematics. - Chapter 1, Section 8 Solve a function for the dependent variable and write the inverse of a function by interchanging the values of the dependent and independent variables. Note: In certain situations, the inverse of a function is not always a function, which is why many people say, find the inverse, if it exists. Every function has an inverse; however, that inverse may not be a function. Connect this concept to the vertical and horizontal line tests as well as the concept of having a one-toone correspondence between the values in the domain and range. Example: At Cosmo Creamery the sundae is \$3.50 plus an additional \$.30 per topping. Write a function f(x), to model the cost of a Sundae with x toppings. b. Find f(5) and explain its meeting in this context. If deluxe Sundae costs \$5.60, how many toppings does the deluxe Sundae include? Explain your reasoning. Using function notation, write a rule that determines the number of toppings given any cost. What is the relationship that exists between f(x) and the new equation? Is the new equation a function? Use the equation from part (d) to determine how many toppings Johnny ordered if his sundae cost \$ p. 240, Questions 1-28

22 Grade: PreCalculus Revision Date: May 2016 Page # : 18 Domain: TF- Trigonometric Functions Extend the domain of trigonometric functions using the unit circle (+) For advanced course: calculus, advanced statistics, etc. F-TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π x, π+x, and 2π x in terms of their values for x, where x is any real number. MP.6 Attend to precision. - Chapter 4, Sections 1-4 Use 30º-60º-90º and 45º-45º-90º triangles to determine the values of sine, cosine, and tangent for values of π/3, π/4, and π/6. - Chapter 4 Midchapter Test p. 514, Questions 1-30 F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. MP.3 Construct viable arguments and critique the reasoning of others. - Chapter 4, Sections 4-6 Use the unit circle and periodicity to find values of sine, cosine, and tangent for any value of θ, such as π +θ, 2π + θ, where is a real number. Use the values of the trigonometric functions derived from the unit circle to explain how trigonometric functions repeat themselves. Use the unit circle to explain that f(x) is an even function if f(-x) = f(x), for all x, and an odd function if f(-x) = -f(x). Also know that an even function is symmetric about the y-axis. - p. 533, Questions p. 547, Questions 45-48

23 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: TF- Trigonometric Functions Model periodic phenomena with trigonometric functions Modeling Standard (+) For advanced course: calculus, advanced statistics, etc. F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. MP.4 Model with mathematics. - Chapter 4, Section 5 Use sine and cosine to model periodic phenomena such as the ocean s tide or the rotation of a Ferris wheel. Given the amplitude; frequency; and midline in situations or graphs, determine a trigonometric function used to model the situation. - p. 533, Questions F-TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. - Chapter 4, Section 6 Know that the inverse for a trigonometric function can be found by restricting the domain of the function so it is always increasing or decreasing. - p. 547, Questions 1-40

24 Grade: PreCalculus Revision Date: May 2016 Page # : Domain: TF- Trigonometric Functions Model periodic phenomena with trigonometric functions Modeling Standard (+) For advanced course: calculus, advanced statistics, etc. F-TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. - Chapter 4, Section 5 Use the inverse of trigonometric functions to solve equations that arise in real-world contexts. Use technology to evaluate the solutions to the inverse trigonometric functions, and interpret their meaning in terms of the context. - Chapter 5 Test p. 642, Questions 12-18

25 Grade: PreCalculus Revision Date: May 2016 Page # : 21 Domain: TF- Trigonometric Functions Prove and apply trigonometric identities (+) For advanced course: calculus, advanced statistics, etc. F-TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. MP.3 Construct viable arguments and critique the reasoning of others. MP.8 Look for and express regularity in repeated reasoning. - Chapter 5, Sections 1-2 Use the unit circle to prove the Pythagorean identity sin²(θ) + cos²(θ) = 1. Given the value of the sin(θ) or cos(θ), use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to calculate other trigonometric ratios. - p. 603, Questions 9-12, F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. MP.3 Construct viable arguments and critique the reasoning of others. - Chapter 5, Sections 2, 5 Prove the addition and subtraction formulas sin(ά±β), cos(ά ±β), and tan(ά±β). Use the addition and subtraction formulas to determine exact trigonometric values such as sin(75º) or cos(π/12). - p. 603, Questions 45-46, 57-64

26 Grade: PreCalculus Revision Date: May 2016 Page # : 22 Domain: SRT- Similarity, Right Triangles, and Trigonometry Apply trigonometry to general triangles (+) For advanced course: calculus, advanced statistics, etc. G-SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. - Chapter 6, Sections 1-2 A surveyor standing at point C is measuring the length of a property boundary between two points located at A and B. Explain what measurements he is able to collect using his transit. Create a plan for this surveyor to find the length of the boundary between A and B. How does the surveyor use the law of sines and/or cosines in this problem? Will your process develop a reliable answer? Why or why not? A B C - p. 652, Questions 1-16, 33-38, p. 661, Questions 1-30

27 Common Core State Standards-Mathematic