Mathematical Modeling

Transcription

1 6/19/2015 CCGPS Advanced Algebra CCGPS Advanced Algebra Mathematical Modeling Mathematical Modeling Have you ever wondered how a plane flies or what makes your MP3 player actually work? Have you compared cell phone plans or the best place to invest your money? What if you were a year older or an inch shorter, or you lived in a different state? There are so many little things that affect our lives and most of them fit some kind of mathematical model. Equations determine how a plane is built for maximum aerodynamics, the programming for a music player, how much you pay for your phone and the return on an investment. Cost of living is different in other areas of the country. There are equations relating height to health, occupational success, ability to compete in different sports and genetics. Understanding mathematical models helps us understand the world and our place in it. Essential Questions How are equations used to represent situations? What are constraints and how do they affect solutions? What do key features of the graph mean to the problem? How can we use properties to rewrite, solve and interpret functions and their inverses? Can geometry be applied to problem solving? Module Minute In mathematical modeling, variables are used to represent quantities like time, money, distance, height, speed, frequency, density, volume, and many more. Equations created with the variables have constraints and solutions related to the problem. For example, the solutions to a system of equations could tell you the time that two different investments return the same amount of money, while constraints would be how long the money must stay in the account. Key features of a graph give maximums and minimums, starting points, rates of change, break even points and many others that we have looked at in the previous modules. Rewriting, solving, finding inverses and applying the geometric concepts of these models helps us interpret the situation to find the solutions we need. Key Words Absolute Value The absolute value of a number is the distance the number is from zero on the number line. Amplitude The distance from the midline to a maximum or minimum point on a sine or cosine graph. 1/23

2 Asymptotes Vertical lines where the function is undefined and horizontal lines that the graph gets closer to without touching. Base (of a Power) The number or expression used as a factor for repeated multiplication Composition of functions Given two functions, their composite (combined functions) uses the output from one function as the input for the other function. Cross section A section made by cutting an object with a plane. Degree The exponent of a number or expression Degree of a Polynomial The largest exponent of x which appears in the polynomial Domain The set of x coordinates of the set of points on a graph; the set of x coordinates of a given set of ordered pairs. The value that is the input in a function or relation. End behavior The rise or fall of the ends of the graph Estimate A guess about the size, cost, or quantity of something. Even A function is even if it has y axis symmetry. Exponential A number written with an exponent. For example, 6 3 is called an exponential expression. Extrema Maximums and minimums of a graph Factor When two or more integers are multiplied, each integer is a factor of the product. "To factor" means to write the number or term as a product of its factors. Function A rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set. Graph of a Function The set of all the points on a coordinate plane whose coordinates make the rule of function true. Horizontal line test If every horizontal line intersect the graph at no more than one point, then the function is one to one. Integer The set of numbers..., 3, 2, 1,0,1,2,3,... Intercepts Points where a graph crosses an axis Interest The percent of the money on deposit (the principal) paid to a lender for the use of the principle Interval A regular distance or space between values. The set of points between two numbers. Logarithmic expression The inverse of an exponential expression Midline The horizontal line that represents the center of a trig graph. Odd A function is odd if it has origin symmetry. One to one A function in which each y also has only one x associated with it. Pattern A set of numbers or objects that are generated by following a specific rule. Period The interval of one non repeating section of a trig graph. Piecewise function a function that is defined differently for different domain values. Power The exponent of a number or expression, which indicates the number of times the number or expression is used as a factor. Polynomial An algebraic expression involving variables with nonnegative integer exponents with one or more unlike terms. Quadratic Function A function of degree 2 whose graph is a parabola. Range The y coordinates of the set of points on a graph. Also, the y coordinates of a given set of ordered pairs. The range is the output in a function or a relation. Rate A comparison of two quantities that have different units of measure. Step functions Greatest (or least) integer function, graph is a piecewise function consisting of steps. Substitute To replace one element of a mathematical equation or expression with another. Symmetry A mirror image across a line such as the x axis, y axis or across the origin. Three Dimensional Figure Figures that have length, width, and height. Two Dimensional Figure Figures that have length and width (no height). Unit A fixed amount that is used as a standard of measurement. Variable A letter or symbol used to represent a number. Vertical line test If a vertical line passes through a graph at most once, the graph is a function. x intercept The value on the x axis where a graph crosses the x axis. y intercept The value on the y axis where a graph crosses the y axis. Zeros the roots of a function, also called solutions or x intercepts. A handout of these key words and definitions is also available in the sidebar 2/23

3 What To Expect New Functions Handout Functions and Graphs Quiz Uses of Math Discussion Working with Functions Quiz Inverse Functions Handout Geometry and Modeling Quiz Ted's Quest for a Tablet Project Mathematical Modeling Test To view the standards from this unit, please download the handout from the sidebar. Functions In past modules, we have looked at various functions. First, review what a function is; a rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set. Recall the vertical line test; that a vertical line can pass through the graph at most once. Also recall that the domain is all of the possible values for x and the range is all of the possible values for y. Functions we have studied are polynomial, rational, radical, exponential, logarithmic and trigonometric. We will be using these throughout the module and here we will look at some new functions. A piecewise function is a function that is defined differently for different domain values. An example is You can see that different x values are defined differently. In order to evaluate a function, you simply find the correct definition for the x value and then use that function. To find, first locate 3 in the domain, in the first section. Then evaluate the corresponding expression, "x 2, using 3. Another type of function (which could be written as a piecewise function) is a step function. A step function is a greatest integer function; which is defined by the greatest integer less than or equal to the value. The symbol for this function is [ ] or [[ ]]. (There is a least integer function that is also a step function, but we will not work with that kind here.) Here is an example f(x) =[[x 2]]. Evaluating these functions often involve fractions or decimals for x values. 3/23

4 Inside the double bracket, simplify normally. When you get to the double bracket, take the greatest integer less than or equal to the value inside. The greatest integer less than or equal to 3/2 is 1. Watch the video showcase to see examples of these functions and how to graph them. For more examples, see Piecewise and Step in the sidebar. The third new type of function (which could also be written as a piecewise function) is an absolute value function. Recall the absolute value of a number is the distance the number is from zero on the number line, which is always positive. Therefore the parent absolute value function will have a range consisting of zero and positive numbers. Here is an example:. To evaluate, put 5 in for x. 5+2 = 3 = 3 so f( 5) = 3 The graph of an absolute value function has a V shape. The standard form for an absolute value function is, where (h, k) is the vertex. To graph an absolute value function, find the vertex and then pick x values to the left and right of the vertex and substitute those values in for x to get a corresponding y value. Then plot the points and draw the graph. For more, see Absolute Value in the sidebar. Let's look at how these functions can be used. New Functions Assignment Select the "New Functions" Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment. Graphs of Functions In Topic 1, you learned how to graph the new functions. Now we will review the key features and transformations of the graphs of all functions. These apply to the new functions, functions learned in previous modules and functions you may learn in the future. If you do not have a graphing calculator, CLICK HERE for an online one. Transformations are how the graph is moved vertically and horizontally, stretches and shrinks, and reflections. The following chart summarizes these transformations To Graph Draw by Change in function 4/23

5 Vertical Shifts Y = f(x) + k, k > 0 Y = f(x) k, k > 0 Horizontal shifts Y = f(x + h), h > 0 Y = f(x h), h > 0 Vertical Stretches Y = af(x), a > 1 Y = af(x), 0 < a < 1 Horizontal Stretches Y = f(ax), a > 1 Y = f(ax), 0 < a < 1 Reflections Y = f(x) Y = f( x) Raise graph by k units Lower graph by k units Shift graph left h units Shift graph right h units Vertical stretch, makes the graph steeper Vertical shrink, makes the graph less steep Horizontal shrink, makes the graph steeper Horizontal stretch, makes the graph less steep Reflects across the y axis Reflects across the x axis Add k to f(x) Subtract k from f(x) Replace x with (x + h) Replace x with (x h) Multiply f(x) by a constant Multiply f(x) by a constant Multiply x by a constant Multiply x by a constant Multiply f(x) by a negative Multiply x by a negative Here is an example with a step function. Parent function: Transformed function: Transformation " " reflects the graph across the x axis "1/3" shrinks the graph vertically " 2" in the double brackets moves the graph 2 units to the right "+1" moves the graph 1 unit up If you replace the parent function with any of the functions shown below, the transformations would be the same. 5/23

6 This would also be a good time to review the shapes of the graphs of our various functions. Let's start with different polynomial functions. See the following chart below Polynomial Function Example Degree Leading Coefficient Graph Constant 0 2 Linear 1 1 Quadratic 2 2 Cubic 3 1 Quartic 4 1 Now let's look at other functions, including our new ones. Type of function Example Graph 6/23

7 Rational Radical Exponential Logarithmic Piecewise Step Absolute Value And here are the trigonometric functions. Trig function Example Graph 7/23

8 Sine Cosine Tangent Key features of graphs include domain, range, intercepts, extrema, intervals of increasing and decreasing, intervals of positive and negative, symmetry (even and odd), end behavior, asymptotes, midline, amplitude and period. You may need to go back and review some of these; they are defined in the Key Terms. Some things to highlight in reviewing key features For more on graph types and features, see Functions With Practice in the sidebar. Scroll to the bottom for practice. Even, Odd, Neither? 8/23

9 Another key feature that hasn't been discussed yet is whether a function is even, odd, or neither. A function is even if it has y axis symmetry and odd if it has origin symmetry. Recalling the definitions of y axis symmetry and origin symmetry: The graph is even if f(x) = f( x). In other words, it contain both (x,y) and ( x,y). The graph is odd if f(x) = f( x). In other words, it contain both (x,y) and ( x, y). A shortcut for this that only works with polynomial functions is: If all the powers of the variable are even, the function is even. If all the powers of the variable are odd, the function is odd. But you must understand that a constant has a variable power of zero;. So powers are 3, 1, and 0. is neither odd nor even. The The best way to determine even or odd for a function is to look at the symmetry. Watch the video below to learn more about symmetry. For some practice with even and odd functions, go to Even & Odd Practice in the sidebar. To review symmetry related to even and odd, go to Symmetry in the sidebar. Absolute Function Let's look at an absolute function Transformations are right 3, up one, and a vertical stretch of 2. The graph is shown here. What are the key features? Domain SOLUTION Range SOLUTION Intercepts SOLUTION Extrema SOLUTION Intervals SOLUTION Symmetry SOLUTION End Behavior SOLUTION Odd, Even or neither SOLUTION We can also calculate the average rate of change, or slope, of the graph. Find 2 points and calculate the rise over the run. The slope for the other half of the V will be +2. If the graph was not linear, you could calculate an approximate rate of change for a section of the graph the same way. To review transformations, see Transform in the sidebar. To practice problems, go to Transformation Practice and scroll to the bottom. 9/23

10 Let's look at some real life applications. Self Check Mary knows she needs to plan for the future. She wants to invest in a savings account and is researching the best choice. All the banks use formulas with compounded interest. When an investment has compounded interest, the interest due is added to the principal. Therefore, the next payment or new principal contains the old payment and the new interest. The formula is given below A=amount, t= time in years, P=principal, n= # times compounded per year, and r= annual interest When using the formula above you have to use the following values for "n". See the table Compounded N = Annually 1 Semiannually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365 If you let the number of times the principal is compounded, or "n", increase without a limit, then it is continuously compounded. The formula is given below A=amount, t= time in years, P=principal, r= annual interest, and e is the natural base found on your calculator. Mary will use the information from each bank to see how much money she would have after 20 years assuming she invests $2000 to start with (principal). Use the information above to answer the questions below: Quiz 1: Functions and Graphs It is now time to complete the "Functions and Graphs" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Uses of Math Discussion It is now time to complete the " Uses of Math " discussion. A rubric for your discussion in located in the sidebar. 10/23

11 Pick one of the two topics below for this discussion. This unit talks about different types and representations of functions. You will also learn things that you can do with functions, transformations and operations. But why do we need to learn about functions? How do functions and math affect real life? Math is used in biology, agriculture, health, sports, economics and many other areas. Your job is to find specific examples of math used in the real world. Go to The Futures Channel website by CLICKING HERE. Find a real world movie that interests you. Watch the movie and post a summary describing how math is used. (Be sure to pick to pick a movie not used by another classmate.) OR Now that we have gone through most of this course, that question is, why learn all of this? And why take more math classes? In this course, you have been introduced to topics in Statistics and Trigonometry. We are going to explore careers related to these topics, so you can see why you should take future math classes. Research a career that uses Statistics or Trigonometry. Give information about this career. Describe the job. What education is needed for it? What is the approximate salary/pay for this job? Find the answers to these questions and/or other facts about the career you've chosen, that would interest your classmates. (Pick a career that no one else has used.) Was the career they picked interesting to you? Would you like to know more about it? Include questions you might have. What did you learn from the movie they posted about? Were you surprised by how math was used? Was it interesting? Operations and Composition of Functions Operations can be done with functions just like with expressions. Here is the notation we will use. The examples will use the functions f(x)= 2x + 3 and g(x) = x² x. Operation Notation Example Addition Subtraction Multiplication Division, cannot be simplified Also, in division remember that denominators cannot be zero, so g(x) 0. These "combinations" of functions use the same properties you learned in previous modules. You can find the domain of each by determining what values can be used for x, the same as in previous modules. Always simplify your answers, where possible. For more examples, see Add & Subtract, Multiply, and Divide in the sidebar. Self Check For 1 4 below, use the following information to match the operation with the resulting function and domain. Match the operation with the resulting function and domain given the information above: For 5 8 below, use the following: 11/23

12 Match the operation with the resulting function and domain given the information above: Composite Functions The next thing to look at is composite functions. Composition of functions is when we are given two functions, their composite (combined function) uses the output from one function as the input for the other function. We will see the common notation of f(x) and g(x), representing the two different functions. Notation for composite functions is The second notation allows us to see what the input is and what the output is. In f(g(x)), the output for the function g(x) is used as the input for f(x). In g(f(x)), the output for the function f(x) is used as the input for g(x). The domain of composite functions is determined by the domains of the original functions, not the resulting function. The purpose of composing functions is often to evaluate the result for a specific number. For example, if f(x) = 2x + 3 and g(x) = x² x, we might want to find. To do this, we don't have to find f(g(x)).we can simply follow order of operation, doing what is in the ( ) first. So we find g(3) by putting 3 in for x in the g(x) function. g(3) = 3² 3 = 9 3 = 6 Now we take g(3) which is 6 and put it in for x in the f(x) function. f(g(3)) = f(6) = 2(6) + 3 = 15 For more on composition of functions and for practice, see Composite and Evaluate Composite in the sidebar. Let's see how this is used. Aisha made a chart of the experimental data for her science project and showed it to her science teacher. The teacher was complimentary of Aisha's work but suggested that, for a science project, it would be better to list the temperature data in degrees Celsius rather than degrees Fahrenheit. a. Aisha found the formula for converting from degrees Fahrenheit to degrees Celsius. Use this formula to convert freezing (32 F) and boiling (212 F) to degrees Celsius. SOLUTION b.later Aisha found a scientific journal article related to her project and planned to use information from the article on her poster for the school science fair. The article included temperature data in degrees Kelvin. Aisha talked to her science teacher again, and they concluded that she should convert her temperature data again this time to degrees Kelvin. 12/23

13 The formula for converting degrees Celsius to degrees Kelvin is K = C Use this formula and the results of part a to express freezing and boiling in degrees Kelvin. SOLUTION c. Use the formulas from part a and part b to convert the following to K 238 F, 5000 F. SOLUTION In converting from degrees Fahrenheit to degrees Kelvin, you used two functions, the function for converting from degrees Fahrenheit to degrees Celsius and the function for converting from degrees Celsius to degrees Kelvin, and a procedure that is the key idea in the composition of functions. We now explore how the temperature conversions from Item 1, part c, provide an example of a composite function. d. The definition of composition of functions indicates that we start with a value, x, and first use this value as input to the function g. In our temperature conversion, we started with a temperature in degrees Fahrenheit and used the formula to convert to degrees Celsius, so the function g should convert from Fahrenheit to Celsius. What is the meaning of x and what is the meaning of g(x) when we use this notation? SOLUTION e. In converting temperature from degrees Fahrenheit to degrees Kelvin, the second step is converting a Celsius temperature to a Kelvin temperature. The function f should give us this conversion; thus, f(x) = x What is the meaning of x and what is the meaning of f (x) when we use this notation? SOLUTION f. Calculate What is the meaning of this number? SOLUTION g. Calculate, and simplify the result. What is the meaning of x and what is the meaning of? SOLUTION h. Calculate using the formula from part d. Does your answer agree with your calculation from part c? SOLUTION i. Calculate, and simplify the result. What is the meaning of x? What meaning, if any, relative to temperature conversion can be associated with the value of? SOLUTION 13/23

14 Working with Functions Quiz It is now time to complete the "Working with Functions" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Inverse Functions In this topic, we will be looking at the inverse of a function. In order to do that, we must review what makes a function and the concept of one to one. Remember the definition of a function for every one number in the domain, there is one unique number in the range. In other words, each x can have only one y associated with it. You have used the vertical line test to determine if a graph is a function. The test states that if every vertical line intersects the graph at no more than one point, then it is a function. What is a one to one function? This is a function in which each y also has only one x associated with it. In other words, one x relates only to one y and one y relates only to one x. The test to determine one to one is the horizontal line test. The horizontal line test requires that every horizontal line intersect the graph at no more than one point. Only functions that are one to one have an inverse. For the inverse of a function, the domain and range values switch creating a new "function." If f(x) has the points (1, 4), (2, 5) and (3, 6), then the inverse function, denoted will have the points (4, 1), (5, 2), and (6, 3). With a function in equation form, find the inverse by switching x and y and then solving for y. Example 14/23

15 To verify that functions are inverses, show that Example Using the function and inverse above Note "Verify" is essentially a proof, so you must include each step as you simplify. Graphs of Inverse Functions The graphs of inverse functions are symmetric across the line y = x. Using the function and inverse above, the graph is 15/23

16 The video showcase will walk you through inverses. Also recall that exponential functions and logarithmic functions are inverses. Here is an example of finding the inverse of a logarithmic function. 16/23

17 Many functions don't have inverses, though sometimes the domain can be restricted to a section of the graph that is one to one. In Trigonometry, you will learn about the inverses of trig functions. Let's see how this is used. Cryptography Inverse functions are used by government agencies and other businesses to encode and decode information. These functions are usually very complicated. A simplified example involves the function. If each letter of the alphabet is assigned a numerical value according to its position (A = 1, B = 2,..., Z = 26), the word ALGEBRA would be encoded by putting the numbers for each letter into the function, getting The "message" can be decoded by finding the inverse function and plugging the encoded numbers in to find the numbers corresponding to the letters. a. What is the inverse of this function? SOLUTION b. What numbers do you get when you put the encoded number into the inverse? SOLUTION c. What are the letters that match these numbers? SOLUTION For more on inverses, see Find Inverses and Inverses in the sidebar. Also in the sidebar, Inv and Comp will review inverses and composition of functions. Boundless is a site that covers several of the topics we have been studying in this module. Scroll down to find a topic you want to review. Inverse Functions Assignment 17/23

18 It is now time to complete the "Inverse Functions" assignment. Please download the assignment from the sidebar. Record your answers in a separate document. Submit your completed assignment. Geometry and Modeling In this topic, we will look at real life situations using many of the functions that you have been studying in this course. There are models in real life situations for ALL of the functions but it wouldn't be practical to show you all of them. So a variety will be shown here. One important concept in modeling is to be able to rearrange equations for different variables. Here is a fairly simple example Jeff wants to fence in several small rectangular pens for his horses. Because of the lengths of the fence panels that he has and the distance along the property line, he wants the pens to have these widths 30 feet, 50 feet, and 90 feet. For his purposes, the areas of the pens must be 2400 sq ft, 5000 sq ft, and sq ft respectively. What will be the lengths of the pens? The formula for area of a rectangle is. To find the lengths, Jeff would have to solve 3 equations. Instead, he could rearrange the formula so the is by itself; then he just has to plug in the other values. The process of solving an equation for a specific variable is the same as solving any equation. Here, Jeff would divide both sides of the equation by w to get. What will be the lengths of the pens? SOLUTION Self Check The following equations are formulas for specific situations. Solve them for the variable indicated. Note Some of the answers involving square roots have ±, others do not. In real life there are often quantities that cannot be negative. Geometric Shapes Geometric shapes are included in many real world models. Many 3 dimensional models include the crosssection of the shape. A cross section is a section made by cutting an object with a plane. What shape would you get if you took a horizontal cross section of the following? 18/23

19 Let's look at some examples. Example 1 When sending a rectangular package through the U.S. Postal Service, the combined length and girth (perimeter of the cross section) cannot exceed 108 inches. Example 2 a. What shape is the girth? SOLUTION b. How do you find the perimeter of that shape? SOLUTION c. Can you send a box that is inches? SOLUTION d. Does it matter which way you take the girth? SOLUTION The official diameter of a tennis ball, as defined by the International Tennis Federation, is at least inches and at most inches. Tennis balls are sold in cylindrical containers that contain three balls each. To model the container and the balls in it, we will assume that the balls are 2.7 inches in diameter and that the container is a cylinder the interior of which measures 2.7 inches in diameter and = 8.1 inches high. Example 3 (a) Lying on its side, the container passes through an X ray scanner in an airport. If the material of the container is opaque to X rays, what outline will appear? With what dimensions? SOLUTION (b) If the material of the container is partially opaque to X rays and the material of the balls is completely opaque to X rays, what will the outline look like (still assuming the can is lying on its side)? SOLUTION (c) The central axis of the container is a line that passes through the centers of the top and bottom. If one cuts the container and balls by a plane passing through the central axis, what does the intersection of the plane with the container and balls look like? (The intersection is also called a cross section. Imagine putting the cut surface on an ink pad and then stamping a piece of paper. The stamped image is a picture of the intersection.) SOLUTION (d) If the can is cut by a plane parallel to the central axis, but at a distance of 1 inch from the axis, what will the intersection of this plane with the container and balls look like? SOLUTION (e) If the can is cut by a plane parallel to one end of the can a horizontal plane when the can is upright what are the possible appearances of the intersections? SOLUTION Mary (from Topic 1 who owns the wedding dress boutique) needs open topped boxes to store her excess inventory at year's end. Mary purchases large rectangles of thick cardboard with a length of 78 inches and width of 42 inches to make the boxes. Mary is interested in maximizing the volume of the boxes and wants to know what size squares to cut out at each corner of the cardboard (which will allow the corners to be folded up to form the box) in order to do this. 19/23

20 (a) Volume is a three dimensional measure. What is the third dimension that the value x represents? SOLUTION (b) Using the table below, choose five values of x and find the corresponding volumes. SOLUTION x Length Width Volume You tested several different values of x above, and calculated five different volumes. There is a way to guarantee that you use dimensions that will maximize volume, and now we're going to work through that process. (c) Write an equation for volume in terms of the three dimensions of the box. SOLUTION (d) Graph the equation from part (c). Remember to adjust your window, thinking about the domain and range. SOLUTION (e) From your graph, what are the values of the three dimensions that maximize the volume of the box? 20/23

21 What is the maximum volume of the box? SOLUTION Example 4 Professor Strange is designing a part for his time machine. The piece is a right triangle shape as shown. Example 5 a. What will be the horizontal length of this piece? SOLUTION b. What is the area of the piece? SOLUTION c. In the time machine, this piece will spin around the vertical side of the rectangle. What shape will be created as it spins? SOLUTION d. What will the volume of the spinning shape be? SOLUTION e. This piece in his machine will actually heat up. If it needs to reach 45 BTU's per cubic inch to make the time transfer, how much heat will have to be added to the piece? SOLUTION As shown in the diagram, you are standing 20 feet away from a tree, and you measure the angle of elevation to be. How tall is the tree? SOLUTION Example 6 A roller coaster is modeled by the function a. Graph the function (remember to adjust your window). SOLUTION b. What are the heights, to the nearest foot, of each of the humps? SOLUTION c. On what intervals, above the x axis, will the roller coaster be increasing speed? SOLUTION d. If the amusement park with this roller coaster covers 6 acres which is sq dekameters. If there are 5,432 people in the park at noon, how many people per square dekameter is that, to the nearest tenth? ( a dekameter is 10 meters) SOLUTION 21/23

22 The sidebar has several videos that show modeling with Algebra in the real world. Geometry and Modeling Quiz It is now time to complete the "Geometry and Modeling" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Module Wrap Up Module Checklist In this module you were responsible for completing the following assignments. Review New Functions Handout Functions and Graphs Quiz Uses of Math Discussion Working with Functions Quiz Inverse Functions Handout Geometry and Modeling Quiz Ted's Quest for a Tablet Project Mathematical Modeling Test A portion of this content is from cnx.org Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and extra resources. Re watch videos and visit the extra resources in the sidebars as needed. Then, continue to the next page for your final assessment instructions. Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a standardized test. Please follow the directions closely. Remember that you may have to use prior knowledge from previous units in order to answer the question correctly. If you have any questions or concerns, please contact your instructor. 22/23

23 Final Assessments Mathematical Modeling Test It is now time to complete the "Mathematical Modeling" Test. Once you have completed all self checks, assignments, and the review items and feel confident in your understanding of this material, you may begin. You will have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly. Ted's Quest for a Tablet Project Select Ted's Quest for a Tablet Project Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment. A rubric is available in the sidebar. 23/23