PreCalculus Curriculum Timeline GRADING PERIOD 1 Topic 1: Learning Goal A Learning Goal B Learning Goal C No. of days: 5-30 Learning Goal D Learning Goal E Learning Goal F Topic : Polynomial, Power, and Rational Functions Learning Goal A Learning Goal B Learning Goal C Learning Goal D No. of days: 40-45 (Continued in nd 9-Weeks) GRADING PERIOD Topic : Polynomial, Power, and Rational Functions Learning Goal A Learning Goal B Learning Goal C Learning Goal D No. of days: 40-45 (Continued from 1st 9-Weeks) Topic 3: Exponential, Logarithmic, and Logistic Functions Learning Goal A Learning Goal B Learning Goal C Learning Goal D No. of days: 5-30 (Continued in 3 rd 9-Weeks) GRADING PERIOD 3 Topic 3: Exponential, Logarithmic, and Logistic Functions Learning Goal A Learning Goal B Learning Goal C Learning Goal D No. of days: 5-30 (Continued from nd 9-Weeks) Topic 4: Trigonometry and Trigonometric Functions Learning Goal A Learning Goal F Learning Goal B Learning Goal G Learning Goal C Learning Goal H Learning Goal D Learning Goal I Learning Goal E Learning Goal J No. of days: 14-16 (Continued in 4 th 9-Weeks) GRADING PERIOD 4 Topic 4: Trigonometry and Trigonometric Functions Topic 5: Noncartesian Representations Learning Goal A Learning Goal F Learning Goal B Learning Goal G Learning Goal C Learning Goal H Learning Goal D Learning Goal I Learning Goal E Learning Goal J No. of days: 14-16 (Continued from 3 rd 9-Weeks) Learning Goal A Learning Goal B Learning Goal C Learning Goal D No. of days: 10-15 June 005
PreCalculus 1 st Nine-Weeks Scope and Sequence Topic 1: (5-30 days) A) Identifies properties of functions by investigating intercepts, zeros, domain, range, horizontal and vertical asymptotes, and local and global behavior and uses functions to model problems B) Identifies the characteristics of the following families of functions: polynomials of degree one, two and three, reciprocal, square root, exponential, logarithmic, sine, cosine, absolute value, greatest integer and logistic C) Performs operations with functions, including sum, difference, product, quotient, and composition and transformations. D) Represents the inverse of a function symbolically and graphically as a reflection about the line y=x. E) Identifies families of functions with graphs that have reflectional symmetry about the y-axis, x-axis, or y=x. F) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and make predictions. Topic : Polynomial, Power, and Rational Functions (40 45 days) (Continued in nd Nine-Weeks) A) Determines the characteristics of the polynomial functions of any degree, general shape, number of real and nonreal (real and nonreal), domain and range, and end behavior, and finds real and nonreal zeros. B) Identifies power functions and direct and inverse variation. C) Describes and compares the characteristics of rational functions; e.g., general shape, number of zeros (real and nonreal), domain and range, asymptotic behavior, and end behavior. D) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and make predictions.
COLUMBUS PUBLIC SCHOOLS MATHEMATICS CURRICULUM GUIDE SUBJECT STATE STANDARDS 4 and 5 TIME RANGE GRADING PreCalculus Patterns, Functions, and Algebra 5-30 days PERIOD Data Analysis and Probability 1 MATHEMATICAL TOPIC 1 CPS LEARNING GOALS A) Identifies properties of functions by investigating intercepts, zeros, domain, range, horizontal and vertical asymptotes, and local and global behavior and uses functions to model problems B) Identifies the characteristics of the following families of functions: polynomials of degree one, two and three, reciprocal, square root, exponential, logarithmic, sine, cosine, absolute value, greatest integer and logistic. C) Performs operations with functions, including sum, difference, product, quotient, and composition and transformations. D) Represents the inverse of a function symbolically and graphically as a reflection about the line y = x. E) Identifies families of functions with graphs that have reflectional symmetry about the y-axis, x-axis, or y = x. F) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and make predictions. COURSE LEVEL INDICATORS Course Level Indicators (i.e., How does a student demonstrate mastery?): Identifies the intervals on which a function is increasing or decreasing. Math A:11-A:04 Uses interval notation to describe domain and range and the solution to inequalities. Math MP:11/1-H Identifies removable, jump, and infinite discontinuities. Math A:11-A:03 Identifies boundedness of a function and intervals on which a function is bounded. Math A:11-A:03 Uses limit notation to describe asymptotic and end behaviors. Math A:1-A:07 Solves equations, inequalities, and systems of equations and inequalities graphically and algebraically. Math A:11-A:03 Identifies points of discontinuity and the intervals over which a function is continuous. Math A:11-A:03 Models real world data with functions. Math A:11-A:03, Math MP:11-D:11, and Math D:11-A:04 Determines points of discontinuity and intervals on which a function is continuous. Math A:11-A:03 Sketches graphs of basic functions and their transformations without technology. Math A:11-A:03 Connects geometric transformations on the graph to changes of parameters in an equation. Math A:11-A:03 Finds the composition of two or more functions. Math A:11-A:03 Page 1 of 73 Columbus Public Schools 7/0/05
Writes a given function as the composition of simple functions. Math A:11-A:03 Determines the equation of an inverse relation. Math A:11-A:06 Uses the horizontal line test to determine if a relation is one-to-one. Math A:11-A:03 Explains the concept of even and odd functions and uses algebraic tests to determine symmetry. Math A:11-A:05 Previous Level: Describes the behavior of functions involving absolute value. Math A:11-A:05 Determines the domain and range of a function. Math A:09-E:01 Defines function and uses function notation. Math A:10-E:01 Explains the concept of inverse relationships and reflections about the line y=x. Math A:11-A:06 Uses the vertical line test to determine if a relation is a function. Math A:10-B:01 Uses technology to find the Least Squares Regression Line, the regression coefficient, and the correlation coefficient for bivariate data with a linear trend, and interpret each of these statistics in the context of the problem situation. Math D:11-B:05 Next Level: Analyzes functions by investigating rates of change. Math A:1-A:10 Page of 73 Columbus Public Schools 7/0/05
The description from the state, for the Patterns, Functions, and Algebra Standard says: Students use patterns, relations, and functions to model, represent, and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs, and equations. The grade-band benchmark from the state, for this topic in the grade band 11 1 is: A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local and global behavior. The description from the state, for the Data Analysis and Probability Standard says: Students pose questions and collect, organize, represent, interpret, and analyze data to answer those questions. Students develop and evaluate inferences, predictions, and arguments that are abased on data. The grade-band benchmark from the state, for this topic in the grade band 11 1 is: A. Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators. The description from the state, for the Mathematical Processes Standard says: Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The grade-band benchmarks from the state, for this topic in the grade band 11 1 are: D. Select and use various types of reasoning and methods of proof. H. Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations. J. Apply mathematical modeling to workplace and consumer situations including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation. Page 3 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS - A Given the graph below, which is the correct description of the discontinuity? A. The discontinuity is removable and is an infinite discontinuity. B. The discontinuity is nonremovable and is an infinite discontinuity. C. The discontinuity is nonremovable and is a jump discontinuity. D. The discontinuity is removable and is a jump discontinuity. Given y = (5 x) 3 + 3x 10, which statement is true? A. The function is always decreasing B. The function is always increasing. C. The domain of the function is (-, 5]. D. The domain of the function is (-6.5, 5). Page 4 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A Answers/Rubrics Low Complexity Given the graph below, which is the correct description of the discontinuity? A. The discontinuity is removable and is an infinite discontinuity. B. The discontinuity is nonremovable and is an infinite discontinuity. C. The discontinuity is nonremovable and is a jump discontinuity. D. The discontinuity is removable and is a jump discontinuity. Answer: C Moderate Complexity Given y = (5 x) 3 + 3x 10, which statement is true? A. The function is always decreasing B. The function is always increasing. C. The domain of the function is (-, 5]. D. The domain of the function is (-6.5, 5). Answer: C Page 5 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A Given the graph below, which statement correctly describes the behavior of the function? A. lim x f (x) = B. lim x f (x) = 0 C. lim f (x) = x 1 D. lim x 1 f (x) = 0 x + The graph of the function f( x) = contains a discontinuity. Use a graphing calculator to x 1 sketch the graph and identify the type of discontinuity. Show algebraically how you verify the discontinuity. Page 6 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A Answers/Rubrics High Complexity Given the graph below, which statement correctly describes the behavior of the function? A. lim x f (x) = B. lim x f (x) = 0 C. lim f (x) = x 1 D. lim x 1 f (x) = 0 Short Answer/Extended Response Answer: B x + The graph of the function f( x) = contains a discontinuity. Use a graphing calculator to x 1 sketch the graph and identify the type of discontinuity. Show algebraically how you verify the discontinuity. This is an infinite discontinuity at x = 1. It occurs because the denominator x - 1 = 0 at x = 1. A point response correctly identifies the infinite discontinuity and its position. A 1 point response identifies the infinite discontinuity. A 0 point response show no mathematical understanding. Page 7 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A Use a graphing calculator to graph f (x) = x +. Choose the statement which is false. 3x 5 A. The value of the function at x = 3 is 5. B. lim f (x) = 0. x C. The range is the set of real numbers. D. lim f (x) = 0. x Use the function y = 3x, x 0 Which statement is true? x + 1, x > 0 A. The value of the function at x = 3 is 5. B. The relation is continuous. C. The value of the function at x = -1 is -3. D. The relation is a function. Page 8 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A Answers/Rubrics Low Complexity Use a graphing calculator to graph f (x) = x +. Choose the statement which is false. 3x 5 A. The value of the function at x= 3 is 5. B. lim x f (x) = 0 C. The range is the set of real numbers. D. lim f (x) = 0 x Moderate Complexity Answer: A Use the function y = 3x, x 0 Which statement is true? x + 1, x > 0 A. The value of the function at x = 3 is 5. B. The relation is continuous. C. The value of the function at x = -1 is -3. D. The relation is a function. Answer: D Page 9 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A The base of an isosceles triangle is half as long as the two equal sides. Which of the following gives the area of the triangle (A) as a function of the length of the base (b). A. A = 15b 4 B. A = 3b C. A = 5b 4 D. A = 3b x Graph the function f( x) = x 4 asymptotes, domain and range. using a graphing calculator. Identify the x-intercepts, Page 10 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS A Answers/Rubrics High Complexity The base of an isosceles triangle is half as long as the two equal sides. Which of the following gives the area of the triangle (A) as a function of the length of the base (b)? A. A = 15b 4 B. A = 3b 4 C. A = 5b 4 D. A = 3b Short Answer/Extended Response Answer: A x Graph the function f( x) = x 4 asymptotes, domain and range. Answer: using a graphing calculator. Identify the x-intercepts, 10 8 6 4-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8 - -4-6 -8-10 The zeros are 1 and -1. The vertical asymptotes are x = and x = -, the domain is (, ) (,) (, ), and the range is (-, ) (, ). A -point response correctly identifies the intercepts, asymptotes, domain and range. A 1-point response correctly identifies of the 4 required answers: the x-intercepts, the asymptotes, the domain, and the range. A 0-point response shows no mathematical understanding. Page 11 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS B Which is the graph of y = sin x on (-π, π)? A. B. C. D. Which function(s) have domain (-, )? A. y = cos x B. y = ln x C. y = 1/x D. all of the above E. none of the above Page 1 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS B Answers/Rubrics Low Complexity Which is the graph of y= sin x on (-π, π)? A. B. C. D. Answer: B Moderate Complexity Which function(s) have domain (-, )? A. y = cos x B. y = ln x C. y = 1/x D. all of the above E. none of the above Answer: A Page 13 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS B Which letter represents a group of functions that are all bounded below? A. f(x) = ln x, f(x) = x, f ( x) = x 1 B. f(x) = sin x, f( x), x 1 C. f(x) = sin x, f(x) = ln x = f ( x) = x + e x D. f(x) = cos x, f ( x) = e, 1 f( x) = 1 + e x Sketch the graph of the piecewise function this represent? Why? x if x 0 f( x) =. What basic function does x if x 0 Page 14 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS B Answers/Rubrics High Complexity Which letter represents a group of functions that are all bounded below? A. f(x ) = ln x, f(x) = x, f ( x) = x B. f(x) = sin x, f( x) 1 1 =, f ( x) = x + x e C. f(x) = sin x, f(x) = ln x D. f ( x) x = x, f(x) = cos x, f ( x) = e, 1 f( x) = 1 + e x Answer: D Short Answer/Extended Response Sketch the graph of the piecewise function this represent? Why? x if x 0 f( x) =. What basic function does x if x 0 Answer: The piecewise function is the same as the absolute value function because the definition of absolute value says for x 0, the absolute value of x= x and for x 0, the absolute value of x = -x. A -point response identifies the absolute value function and supports the answer. A 1-point response identifies the absolute value function without support. A 0-point response shows no mathematical understanding. Page 15 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS C Which represents (f + g) (x) if f(x) = x and gx ( ) = x+ 1 for x 0? A. (f + g) (x) = B. (f + g) (x) = C. (f + g) (x) = x x + x 1 + x + 1 + x + 1 D. (f + g) (x) = x + 1+ x What is the domain of f ( x) gx ( ) if f(x) = x and gx ( ) = x+ 1? A. x 0 B. All real numbers C. x 1 D. x -1 Page 16 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS C Answers/Rubrics Low Complexity Which represents (f + g) (x) if f(x) = x and, gx ( ) = x+ 1 for x 0? A. (f + g) (x) = x +1 B. (f + g) (x) = x + x + 1 C. (f + g) (x) = x + x + 1 D. (f + g) (x) = x + 1+ x Answer: B Moderate Complexity What is the domain of f ( x) gx ( ) if f(x) = x and gx ( ) = x+ 1? A. x 0 B. All real numbers C. x 1 D. x -1 Answer: D Page 17 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS C Which represents (f g) (x) if f(x) = x and gx ( ) = x+ 1? A. (f g) (x) = x + 1 B. (f g) (x) = x + 1 C. (f g) (x) = x + x + 1 D. (f g) (x) = x + x Given f(x) = x - 1 and gx ( ) = x, find (f g) (x) and (g f ) (x) and the domain of each. Page 18 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS Answers/Rubrics C High Complexity Which represents (f g) (x) if f(x) = x and gx ( ) = x+ 1? A. (f g) (x) = x + 1 B. (f g) (x) = x + 1 C. (f g) (x) = D. (f g) (x) = x x + x + 1 + x Answer: B Short Answer/Extended Response Given f(x) = x - 1 and gx ( ) = x, find (f g) (x) and (g f ) (x) and the domain of each. Answer: (f g) (x )= x-1 and the domain is [0, ) (g f) (x )= x 1 and the domain is (-,-1) (1, ). A point response correctly identifies the compositions and their domains. A 1-point response correctly identifies the compositions. A 0- point response shows no mathematical understanding of the topic. Page 19 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS D Given the graph below. which is the graph of the inverse? 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 A. B. C. D. 10 8 6 4 10 8 6 4 10 8 6 4 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 -10-8 -6-4 - 4 6 8 10 - -4-6 -8-10 -10-8 -6-4 - 4 6 8 10 - -4-6 -8-10 -10-8 -6-4 - 4 6 8 10 - -4-6 -8-10 Which is the inverse of f (x) = x 3 5? A. f 1 3 (x) = x 5 B. f 1 3 (x) = x 5 C. f 1 3 (x) = x + 5 D. f 1 3 (x) = x + 5 Page 0 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS D Answers/Rubrics Low Complexity Given the graph below, which is the graph of the inverse? 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 A. B. C. D. 10 8 6 4 10 8 6 4 10 8 6 4 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 -10-8 -6-4 - 4 6 8 10 - -4-6 -8-10 -10-8 -6-4 - 4 6 8 10 - -4-6 -8-10 -10-8 -6-4 - 4 6 8 10 - -4-6 -8-10 Answer: B Moderate Complexity Which is the inverse of f (x) = x 3 5? A. f 1 3 (x) = x 5 B. f 1 3 (x) = x 5 C. f 1 3 (x) = x + 5 D. f 1 3 (x) = x + 5 Answer: D Page 1 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS D Which function has an inverse that is a function? A. f(x) = x 3 B. f(x ) = ln x C. f(x) = x D. All of the above. E. None of the above. Give the equation of a function that is one-to-one and a function that is not one-to-one. Explain your choices both algebraically and graphically. Page of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS D Answers/Rubrics High Complexity Which function has an inverse that is a function? A. f(x) = x 3 B. f(x) = ln x C. f(x) = x D. All of the above. E. None of the above. Short Answer/Extended Response Answer: D Give the equation of a function that is one-to-one and a function that is not one-to-one. Explain your choices both algebraically and graphically. Sample Answer: y = x 3 is one-to-one. When you interchange x and y and solve for y, there is only one solution. The graph passes both the vertical and horizontal line tests. y = x is not one-to-one. When you interchange x and y and solve for y, there are two solutions. The graph fails the horizontal line test. A -point solution includes both a one-to-one function and a function which is not one-toone and supports the answer both graphically and algebraically. A 1-point solution includes both a one-to-one function and a function which is not one-toone and supports the answer EITHER graphically OR algebraically. A 0-point solution demonstrates no mathematical understanding. Page 3 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS D Given the graphs below, what symmetries are exhibited? I II III A. I: x-axis, II: y-axis, III: origin B. I: y-axis, II: x-axis, III: origin C. I: y-axis, II: origin, III: x-axis D. I: origin, II: y-axis, III: x-axis A graph of a relation which is symmetric about the x-axis contains the points (, 3), (-5, 1), and (-4, -6). Which other points must also be on the graph? A. (,-3), (5,1), and (-4, 6) B. (, -3), (-5, -1), and (-4, 6) C. (-, 3), (5,1), and (4, -6) D. (-, 3), (-5, -1), and (4, -6) Page 4 of 73 Columbus Public Schools 7/0/05
58PRACTICE ASSESSMENT ITEMS Answers/Rubrics D Low Complexity Given the graphs below, what symmetries are exhibited? I II III A. I: x-axis, II: y-axis, III: origin B. I: y-axis, II: x-axis, III: origin C. I: y-axis, II: origin, III: x-axis D. I: origin, II: y-axis, III: x-axis Moderate Complexity Answer: C A graph of a relation which is symmetric about the x-axis contains the points (, 3), (-5, 1), and (-4, -6). Which other points must also be on the graph? A. (, -3), (5, 1), and (-4, 6) B. (, -3), (-5, -1), and (-4, 6) C. (-, 3), (5, 1), and (4, -6) D. (-, 3), (-5, -1), and (4, -6) Answer: B Page 5 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS D Given a function f such that f ( x) = f (x), which statement is true? A. The function is symmetric about the x-axis. B. The function is symmetric about the y-axis. C. The function is symmetric about the origin. D. The function does not necessarily exhibit symmetry. Use your calculator to graph the function f (x) = x 9. Describe the symmetry and verify x 4 the symmetry algebraically. Page 6 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS E Answers/Rubrics High Complexity Given a function f such that f ( x) = f (x), which statement is true? A. The function is symmetric about the x-axis. B. The function is symmetric about the y-axis. C. The function is symmetric about the origin. D. The function does not necessarily exhibit symmetry. Answer: C Short Answer/Extended Response Use your calculator to graph the function f (x) = x 9. Describe the symmetry and verify x 4 the symmetry algebraically. ( x) 9 4 x = ( x) f( ) x 9 = x 4 = f( x) Because f ( x) = f( x), the function is symmetric about the y-axis. 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 A -point response includes a correct graph, identifies the function as symmetric about the y-axis, and provides algebraic support. A 1-point response includes a correct graph and identifies the function as symmetric about the y-axis. A 0-point response demonstrates no mathematical understanding. Page 7 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS E Given the scatterplot below, which of the following types of regression is likely to give the most accurate model? A. exponential B. sinusoidal C. quadratic D. quartic The table shows the population of a certain city in various years. Year 1981 1985 1989 1993 1997 Population (hundreds of thousands) 3. 4.1 5.7 9.6 14.1 Using x as the number of years which have elapsed since 1980 and y as the population of the city in hundreds of thousands, which exponential function models the data? A. y = 1.8506(0.5109) x B. y =.017(1.4647) x C. y = 1.705(0.53750) x D. y =.6797(1.1001) x Page 8 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS E Answers/Rubrics Low Complexity Given the scatterplot below, which of the following types of regression is likely to give the most accurate model? A. exponential B. sinusoidal C. quadratic D. quartic Moderate Complexity The table shows the population of a certain city in various years. Year 1981 1985 1989 1993 1997 Population (hundreds of thousands) 3. 4.1 5.7 9.6 14.1 Using x as the number of years which have elapsed since 1980 and y as the population of the city in hundreds of thousands, which exponential function models the data? A. y = 1.8506(0.5109) x B. y =.017(1.4647) x C. y = 1.705(0.53750) x D. y =.6797(1.1001) x Answer: B Answer: D Page 9 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS E The table below shows several examples of saturated vapor pressure and the associated relative humidity. Saturated Vapor Pressure (in millibars) 1.6 17.65 5.01 34.94 48.1 65.43 Relative Humidity % 100 69.5 49 35.1 5.5 18.7 Find an equation that models the data and use that equation to find the relative humidity if the saturated vapor pressure is 41.3 millibars. A. 7.6% B. 9.7% C. 31.6% D. 33.7% The table below shows the U.S. per capita income for the years 1990-000. Year 1990 1991 199 1999 1994 1995 Amount (in $) 19,614 0,16 1,105 1,736,593 3,571 Year 1996 1997 1998 1999 000???? Amount (in $) 4,660 5,876 7,317 8,534 30,069 45,000 Using the year 1990 as t = 0, make a scatterplot of the data and find the most appropriate regression model using a graphing calculator. Sketch the scatterplot and the graph of the regression equation. Use the model to predict the year in which the per capita income will exceed $45,000. Page 30 of 73 Columbus Public Schools 7/0/05
PRACTICE ASSESSMENT ITEMS E Answers/Rubrics High Complexity The table below shows several examples of saturated vapor pressure and the associated relative humidity. Saturated Vapor Pressure (in millibars) 1.6 17.65 5.01 34.94 48.1 65.43 Relative Humidity % 100 69.5 49 35.1 5.5 18.7 Find an equation that models the data and use that equation to find the relative humidity if the saturated vapor pressure is 41.3 millibars. A. 7.6% B. 9.7% C. 31.6% D. 33.7% Answer: B Short Answer/Extended Response The table below shows the U.S. per capita income for the years 1990-000. Year 1990 1991 199 1993 1994 1995 Amount (in $) 19,614 0,16 1,105 1,736,593 3,571 Year 1996 1997 1998 1999 000???? Amount (in $) 4,660 5,876 7,317 8,534 30,069 45,000 Using the year 1990 as t =0, make a scatterplot of the data and find the most appropriate regression model using a graphing calculator. Sketch the scatterplot and the graph of the regression equation. Use the model to predict the year in which the per capita income will exceed $45,000. Answer: The regression equation is y = 49.1x + 553.5x+ 1963.3. The model predicts that the per capita income will exceed $45,000 in 005. A -point response includes a correct, labeled scatterplot with the graph of regression equation, the regression equation, and a solution (either graphical or algebraic) and an answer in terms of the year. ) $ ( e m o c n I Income ($) 50000 40000 30000 0000 (15.5007, 40000.) 5 10 15 0 Year A 1-point response includes a correct scatterplot and graph and solution but does not express the answer in terms of the year. A 0-point response demonstrates no mathematical understanding. Page 31 of 73 Columbus Public Schools 7/0/05
Teacher Introduction This topic provides the foundation for the rest of the course. Many of the learning goals were introduced in Algebra II at a basic level. In this topic, students are required to draw upon their previous work and apply it to new situations. Students often encounter difficulty with this. In previous courses, the topics are fairly narrowly focused, and students do not need to draw upon concepts from outside that topic. In PreCalculus, it is essential that they bring their previous knowledge to bear on this general study of functions. In addition, they need to be prepared to apply the concepts learned in this topic to the rest of the PreCalculus course as they study families of functions in greater depth. This course consists mostly of the study of functions. The learning goals in this topic are essential to the remainder of the course and cannot be rushed. This guide has been created to be used in conjunction with the text, and pages that are indicated in the resources are essential for the implementation of the curriculum. The strategies used in this topic involve several different learning goals. They are not intended to be completed in one day or even on consecutive days. It is essential that the class come together to discuss the different parts frequently. The grouping of the subjects in the learning goals does not necessarily indicate the order in which they should be taught, and the nature of the topic requires that the learning goals be integrated. The pacing guide and correlations demonstrate this. This topic will probably require four to five weeks. Page 3 of 73 Columbus Public Schools 7/0/05
TEACHING STRATEGIES/ACTIVITIES Vocabulary: mathematical model, domain, range, function, removable discontinuity, jump discontinuity, infinite discontinuity, continuity, increasing, decreasing, constant, lower bound, upper bound, boundedness, local extrema, absolute extrema, odd function, even function, asymptote, identity function, squaring function, cubing function, reciprocal function, square root function, exponential function, natural logarithm function, sine function, cosine function, absolute value function, greatest function, logistic function, symmetry, piecewise function, composition, inverse, relation, implicit, inverse, transformation, translation, rigid transformation, reflection, stretch, shrink, regression, correlation coefficient, quadratic, end behavior, zeros. Core: Learning Goal A: Identifies properties of functions by investigating intercepts, zeros, domain, range, horizontal and vertical asymptotes, and local and global behavior and uses functions to model problems. 1. Emphasize that the correct use of vocabulary is essential. Students must understand how zeros, x-intercepts, and factors are related and use the correct word to describe each. With rational functions, look at specific examples and then generalize.. Do the activity The Bathtub (included in this Curriculum Guide). 3. Do the activity Piecewise Functions Step by Step (included in this Curriculum Guide). Learning Goal B: Identifies the characteristics of the following families of functions: polynomials of degree one, two and three, reciprocal, square root, exponential, logarithmic, sine, cosine, absolute value, greatest integer and logistic 1. Begin the activity, "Stacks of Cups" (included in this Curriculum Guide). This activity includes many of the ideas included in this topic including the idea of slope as a rate of change, and the greatest integer function. It also introduces the idea of inverse functions that will be studied and points out the difference between the domain and the range of a function and the situation that it models. This activity will be used throughout Topic One. Learning Goal C: Performs operations with functions, including sum, difference, product, quotient, and composition and transformations. 1. Do the activity Transformations (included in this Curriculum Guide.). This activity reinforces the basic functions Learning Goal D Represents the inverse of a function symbolically and graphically a reflection about the line y = x. 1. Introduce inverse functions by using the Teacher Notes (included in this Curriculum Guide.) Emphasize that the inverse of a function may not be a function. Students should investigate functions that are and are not one-to-one and restricting the domain and range of functions. Complete Stacks of Cups (included in this Curriculum Guide and started in Learning Goals B). Page 33 of 73 Columbus Public Schools 7/0/05
Learning Goal E: Identifies families of functions with graphs that have reflection symmetry about the y-axis, x-axis, or y = x. 1. Introduce symmetry by using the activity Introduction to Symmetry (included in this Curriculum Guide). This is not actually an activity, but is really a graphic organizer for their notes. Learning Goal F: Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and make predictions. 1. Do the activity The Mile Run (included in this Curriculum Guide.). This activity should take several days while the class is continuing to study from the textbook. Parts of this should be worked in groups in the classroom and parts should be assigned to be completed as homework. Reteach: 1. Review linear equations, x-intercepts, y-intercepts.. Review quadratic equations, factoring, use of the quadratic formula. 3. Complete the activity Concepts in Graphical Analysis (included in this Curriculum Guide.) Page 34 of 73 Columbus Public Schools 7/0/05
RESOURCES Learning Goal A: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (004): pp. 81-100, 131-141 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (004): Resource Manual pp. 13-14, 19-1 Learning Goal B: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (004): pp. 101-111 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (004): Resource Manual pp.15-16 Learning Goal C: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (004): pp. 1-18 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (004): Resource Manual pp. 17-18 Learning Goal D: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (004): pp. 93-95 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (004): Resource Manual pp.13-14 Learning Goal E: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (004): pp. 63-80. 15-156 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (004): Resource Manual pp.11-1, 1- Page 35 of 73 Columbus Public Schools 7/0/05
Introduction to Inverse Teacher Notes D The graphing calculator helps students to understand the idea of inverse, the difference between an inverse relation and function, and one-to-one functions. Start with the graphs of y= x 3 in Y1 (with the heavy style) and y = x in Y. From the home screen, go to the DRAW menu and choose #8: DrawInv. (Locate Y1 by VARS, YVARS, 1: Function, 1:Y1.) The resulting graph is a drawing, not a graph. You cannot trace on it or access the points on the table. However, you can discuss the idea of reversing ordered pairs and the reflection about the line y = x. Repeat with a graph of y = x in (with the heavy style) and the line y = x. It is obvious that by just reversing the ordered pairs, the result is not necessarily a function, i.e. not all functions are one-to-one. This points out the necessity for restricting the domain of the function in order to create an inverse function. To restrict the domain of y = x to values of x that are greater than or equal to zero, use the TEST ( nd MATH) menu as shown below in Y4. See the Graphing Calculator Resource Manual (included in this Curriculum Guide) for a further discussion of the use of the TEST menu. By restricting the domain and drawing the inverse, the result is one-to-one. Students should discuss how to find the function that represents the inverse (Y) and then test their conjectures. You can then find the other branches, using Y3 and Y4. Introducing the line y = x may make this more obvious. Page 36 of 73 Columbus Public Schools 7/0/05
The Bathtub A Name Below is a graph of the change in water level in a bathtub. At time t = 0, there is some unknown amount of water in the tub. The graph tracks how the water level changes over time. Change in water level (in inches) Time (in minutes) 1. Identify time intervals over which the water level is increasing. a. When does the bather enter the bathtub? How can you tell? b. Other than that, when is the water level increasing the fastest?. Identify time intervals over which the water level is decreasing. a. When does the bather exit the bathtub? How can you tell? b. Other than that, when is the water level decreasing the fastest? 3. Where are the x-intercepts of the graph? What do they mean in terms of the problem situation? 4. What is happening to the water depth as the time nears 16 minutes? How far below the initial water level is the bottom of the tub? 5. What was the initial water level? What was the greatest depth of the water in the tub? Page 37 of 73 Columbus Public Schools 7/0/05
The Bathtub A Answer Key Below is a graph of the change in water level in a bathtub. At time t = 0, there is some unknown amount of water in the tub. The graph tracks how the water level changes over time. Change in water level (in inches) Time (in minutes) 1. Identify time intervals over which the water level is increasing. (0, 4) a. When does the bather enter the bathtub? How can you tell? After 4 minutes; the water level increases instantaneously b. Other than that, when is the water level increasing the fastest? Between and 4 minutes. Identify time intervals over which the water level is decreasing. Between 10 and 16 minutes. (10, 16) a. When does the bather exit the bathtub? How can you tell? After 1 minutes; the water level decreases instantaneously b. Other than that, when is the water level decreasing the fastest? Between 1 and 13 minutes 3. Where are the x-intercepts of the graph? What do they mean in terms of the problem situation.? At 0 and 1 minutes. This is when the water level is the same as the initial value. 4. What is happening to the water depth as the time nears 16 minutes? How far below the initial water level is the bottom of the tub? The level is getting close to zero. About 3 inches. 5. What was the initial water level? What was the greatest depth of the water in the tub? About 3 inches. About 8 inches. Page 38 of 73 Columbus Public Schools 7/0/05
Name Piecewise Functions Step by Step A To graph the piecewise function: 1 x if x - f ( x ) 3 if - x x 4 if x 1 1 1. Show the breaking points on the number-line below, and indicate for which x we will be graphing which function: -10-8 -6-4 - 0 4 6 8 10. Graph the first piece of the function. Then, cross out the part you re not using. 3. Graph the second piece of the function. Then, cross out the part you re not using. Page 39 of 73 Columbus Public Schools 7/0/05
A 4. Graph the third piece of the function. Then, cross out the part you re not using. 5. Now put all three pieces together in the following grid. Page 40 of 73 Columbus Public Schools 7/0/05
1 x if x< - To graph the piecewise function f ( x) = 3 if - x 1 on your graphing calculator, you x + 4 if x > 1 must tell it which part to cross out. You do this with the test menu. To enter access the inequality symbols in the TEST menu ( nd MATH.) The inequality symbols are Boolean operators that are assigned the value 0 when the expression is false and the value 1 when the expression is true. The / leaves the equation unchanged when for the correct interval and divides by zero when the expression is false. Try these with and without your calculator. 3 if x 1 6. f (x) = if x > 1 A x if x < 1 7. f (x) = x if 1 x < 1 if x 1 x 3 if x 8. f (x) = x if < x < 1 x + 4 if x 1 Page 41 of 73 Columbus Public Schools 7/0/05
To graph the piecewise function: 1 x if x - Piecewise Functions Step by Step Answer Key f ( x) 3 if - x 1 x 4 if x 1 A 1. Show the breaking points on the number-line below, and indicate for which x we will be graphing which function: 1-x 3-10 -8-6 -4-0 4 6 8 10 x 4. Graph the first piece of the function. Then, cross out the part you re not using. 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 3. Graph the second piece of the function. Then, cross out the part you re not using. 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 Page 4 of 73 Columbus Public Schools 7/0/05
A 4. Graph the third piece of the function. Then, cross out the part you re not using. 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 5. Now put all three pieces together in the following grid. 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 Page 43 of 73 Columbus Public Schools 7/0/05
A 1 x if x< - To graph the piecewise function f ( x) = 3 if - x 1 on your graphing calculator, you x + 4 if x > 1 must tell it which part to cross out. You do this with the test menu. To enter access the inequality symbols in the TEST menu ( nd MATH.) The inequality symbols are Boolean operators that are assigned the value 0 when the expression is false and the value 1 when the expression is true. The / leaves the equation unchanged when for the correct interval and divides by zero when the expression is false. Try these with and without your calculator. 3 if x 1 6. f (x) = if x > 1 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 x if x < 1 7. f (x) = x if 1 x < 1 if x 1 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 8. x 3 if x f x x if x x+ 4 if x 1 ( ) = < < 1 10 8 6 4-10 -8-6 -4-4 6 8 10 - -4-6 -8-10 Page 44 of 73 Columbus Public Schools 7/0/05
Stack of Cups B Name In this activity, you will investigate the relationship between the number of cups and the height of the stack. 1. Using a sample of cups, all of the same size, complete the following chart. Use the data you collect to look for patterns that might help you determine the relationship between the height of the stack and the number of cups in that stack. Number of cups Height of stack (cm). Make a scatterplot of the data, with the number of cups on the x-axis and the height of the stack on the y-axis. 3. Use your graph to predict the height of a stack of 16 cups: and 5cups: _. Circle the points on your graph that you used for your predictions. Page 45 of 73 Columbus Public Schools 7/0/05
B 4. Define a function, f, such that f(n) gives the height of a stack, h, in terms of the number of cups in that stack, n. When you have your result, call in your teacher to verify your results. Why is the y-intercept of your linear function NOT the same as the height of one cup? What are the domain and range of the function that you wrote? How must you restrict these to be the same as the domain and range of the function defined by the set of points (n,h)? 5. Sketch the graph of f on the grid with your scatterplot. Use this information for #6-9. Another team used a different design of cup and found that the equation that modeled the height in centimeters as a function of the number of cups to be S(n)=.5n+1.5. 6. For this team: a) What is the height of one cup? b) What is change in height per cup? c) If you increased the stack by cups, how much would the height of the stack increase? d) If you increased the stack by 0 cups, how much would the height of the stack increase? e) In general, if you add k cups to an existing stack, how much will the height of the stack increase? 7. Suppose that another student claims that doubling the number of cups in the stack doubles the height of the stack. Explain why this statement is incorrect. Support your argument with examples. Page 46 of 73 Columbus Public Schools 7/0/05
8. B a) Using the cups from #6, if you increased the height of the carton by 5 cm, how many more cups could you fit in? b) If you increased the height of the carton by 6.4 cm, how many more cups could you fit in the carton? Remember that you should not have a "part" of a cup. c) In general, if the height of the carton were increased by d centimeters, how many more cups could you fit in the carton? 9. How many cups could you fit in a carton of height 36 cm? 50 cm? 10. The function S(n)=.5n+1.5 expresses the height of a stack of cups in terms of the number of cups in the stack. Now write a function g(h) that expresses the number of cups in a stack in terms of the height, h, of the stack. 11. The slope of a line represents a rate of change. What is the slope of S? In the language of rate of change, this means that the height increases.5 cm for each increase of 1 cup. What is the slope of g? Express this as a rate of change. S and g are inverses; that is, their ordered pairs are the reverse of each other. What is the relationship of their slopes? Use the concept of rate of change to explain why this is so. Page 47 of 73 Columbus Public Schools 7/0/05
Stack of Cups Answer Key B In this activity, you will investigate the relationship between the number of cups and the height of the stack. 1. Using a sample of cups, all of the same size, complete the following chart. Use the data you collect to look for patterns that might help you determine the relationship between the height of the stack and the number of cups in that stack. Answer will vary. Number of cups Height of stack (cm). Make a scatterplot of the data, with the number of cups on the x- axis and the height of the stack on the y-axis. Answer will vary. 3. Use your graph to predict the height of a stack of 16 cups: and 5cups:. Circle the points on your graph that you used for your predictions. Answer will vary. Page 48 of 73 Columbus Public Schools 7/0/05
B 4. Define a function, f, such that f(n) gives the height of a stack, h, in terms of the number of cups in that stack, n. When you have your result, call in your teacher to verify your results. Why is the y-intercept of your linear function NOT the same as the height of one cup? Answer will vary. What are the domain and range of the function that you wrote? How must you restrict these to be the same as the domain and range of the function defined by the set of points (n,h)? Answer will vary. 5. Sketch the graph of f on the grid with your statplot. Use this information for #6-9: Another team used a different design of cup and found that the equation that modeled the height in centimeters as a function of the number of cups to be S(n)=.5n+1.5. 6. a) What is the height of one cup? 13 cm b) What is change in height per cup?.5 cm c) If you increased the stack by cups, how much would the height of the stack increase? 1 cm d) If you increased the stack by 0 cups, how much would the height of the stack increase? 10 cm e) In general, if you add k cups to an existing stack, how much will the height of the stack increase?.5k 7. Suppose that another student claims that doubling the number of cups in the stack doubles the height of the stack. Explain why this statement is incorrect. Support your argument with examples. If we have 10 cups: s(10) =17.5 If we double this to 0 cups: s(0) =.5.5 is not double 17.5 Page 49 of 73 Columbus Public Schools 7/0/05
8. B d) Using the cups from #6, if you increased the height of the carton by 5 cm, how many more cups could you fit in? About 10 cups e) If you increased the height of the carton by 6.4 cm, how many more cups could you fit in the carton? Remember that you should not have a "part" of a cup. 6.4 = 1.8 which would be rounded to 1 cups..5 f) In general, if the height of the carton were increased by d centimeters, how many more cups could you fit in the carton? d more cups 9. How many cups could you fit in a carton of height 36 cm? 50 cm? 47 cups, 75 cups 10. The function S(n)=.5n+1.5 expresses the height of a stack of cups in terms of the number of cups in the stack. Now write a function g(h) that expresses the number of cups in a stack in terms of the height, h, of the stack. g(h )= h - 5 11. The slope of a line represents a rate of change. What is the slope of S?.5 In the language of rate of change, this means that the height increases.5 cm for each increase of 1 cup. What is the slope of g? Express this as a rate of change. ; The number of cups increases by for each 1 cm increase in height. S and g are inverses; that is, their ordered pairs are the reverse of each other. What is the relationship of their slopes? They are reciprocals Use the concept of rate of change to explain why this is so. One is the change in cups over change in cm and the other is just the reverse. Page 50 of 73 Columbus Public Schools 7/0/05
Transformations C Name #1-: Using the standard viewing window (ZOOM 6), graph the given parent function. Then graph each of the given functions. (You should have only two graphs at a time.) Sketch the graph and describe how the parent function could be moved to create the new function. 1. Parent function: f (x) = x f (x) = x + f (x) = x 3. Parent function: f(x) = x f (x) = x 4 f (x) = x + 5 3. In general, if you have a parent function, y = f( x ), how will the graph of y = f (x) + c be related to the parent function if c is positive? if c is negative? Page 51 of 73 Columbus Public Schools 7/0/05
C When you start with a parent function and change it, it is called a transformation. These transformations are called vertical shifts. In #1, the transformations were vertical shifts of up and down 3. In #, the transformations were vertical shifts of down 4 and up 5. 4. Without using a calculator, sketch the graph of f (x) = x 3. Then sketch the graph of f (x) = x 3 3. Check your answer by graphing on your calculator. What kind of transformation is this? 5. Without using a calculator, sketch the graph of f (x) = 1. Then sketch the graph of x f (x) = f (x) = 1 +. Check your answer by graphing on your calculator. What kind of x transformation is this? #6-7: Using the standard viewing window (ZOOM 6), graph the given parent function. Then graph each of the given functions. (You should have only two graphs at a time.) Sketch the graph and describe how the parent function could be moved to create the new function. 6. Parent function: f (x) = x 3 f (x) = (x ) 3 f (x) = (x + 3) 3 Page 5 of 73 Columbus Public Schools 7/0/05
7. Parent function: f (x) = x C f (x) = (x 1) f (x) = (x + 4) 8. In general, if you have a parent function, y = f (x), how will the graph of y = f (x b) be related to the parent function if b is positive? In general, if you have a parent function, y = f (x), how will the graph of y = f (x b) be related to the parent function if b is negative? Notice that if b is a positive number (like 3) the argument of the function will have a "-" e.g. x-3. If b is a negative number (like -) the argument of the function will have a "+" e.g. x+. These transformations are called horizontal shifts. In #6, the transformations were horizontal shifts of right and left 3. In #7, the transformations were vertical shifts of right 1 and left 4. #9-10: Using the standard viewing window (ZOOM 6), graph the given parent function. Then graph each of the given functions. (You should have only two graphs at a time.) Sketch the graph and describe how the parent function could be moved to create the new function. 9. Parent function: f (x) = x f (x) = (x 4) f( x) = ( x+ 1) Page 53 of 73 Columbus Public Schools 7/0/05
10. Parent function: f (x) = x C f (x) = x + 3 f (x) = x 5 Consider the graph of f (x) = (x ) + 4. The parent function is f (x) = x and there are two transformations, a horizontal shift of to the right, and a vertical shift of up 4. The vertex of the parabola is at (, 4). Identify the parent function and the transformations for the following functions. Sketch the graph without using a calculator. 11. f (x) = 1 x + + 1 1. f x x ( ) = 3 13. f (x) = x + 4 + Page 54 of 73 Columbus Public Schools 7/0/05