1 - Robert Lindblom Math & Science Academy



PRE-CALCULUS: Semester 1 Exam Review (2012-2013)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ThursdayMinutesTime1508 - 8:504th508:55 - 9:452/A509:50 - 10:406/B5010:45 - 11:353/C5011:40 - 12:307/D5012:35 - 1:255501:30 - 2:208502:25 - 3:15Mo, Jan 28Tue, Jan 29We, Jan 30MinutesTime2/A7/ D81008 - 9:40156/B1009:55 - 11:353/C4Make Up 10011:50 - 1:30FINAL EXAM SCHEDULE:During this week you will be expected to display a cumulative and in-depth understanding of important mathematical concepts discussed in class during the first Semester. Your final exam is devoted to writing about mathematics. This test will measure your ability to:recognize major mathematical concepts, organize and express mathematical ideas clearly, develop and support the main idea, use appropriate math vocabulary and ability to define math terms in own words, use appropriate instruments and formulas to explain the concept and to solve problemsability to think critically using reasoning and evidenceability to apply the concepts and skills to solve application problemsDo not wait until the last minute! Organize and prioritize!Your test will consist of two portions: [I] Short Response and [II] Standardized Multiple Choice.[ I ] SHORT RESPONSE (Test Grade): This portion of the test will consist of 2 short response questions. Your short response is not graded on length, it will be graded on quality and clear demonstration of understanding of the concept presented. Your response on this portion of the test will be scored with 0-5 points per question. (see the grading rubric at the end of this document) [ II ] STANDARDIZED MULTIPLE CHOICE TEST (Final Exam Grade): This portion of the final exam will consist of 20 multiple choice questions involving concepts stated in the focus questions as well as Advanced Algebra pre-requisite skills. This test will be graded on point system (1 point per correct response). STUDENTS WILL BE ALLOWED TO USE THE GRAPHING CALCULATOR ON BOTH PORTIONS OF THE TEST.>>> TWO GRADES will be entered in grade book!<<<FINAL EXAM (Multiple choice portion grade) 10% of your overall grade in class. TEST (Short response portion grade) as a part of 40% of your overall grade in class. Good Luck and Happy Learning.SEMESTER 1 SHORT RESPONSE FOCUS QUESTIONS The following is the list of important Pre-Calculus focus questions we have covered during Semester 1. You will be expected to demonstrate mastery of these concepts by defining terms, writing theoretical explanations, creating and solving example problems, and supporting your statements with graphs, tables, and words.Three (3) questions will be randomly selected for you to answer any two of the three selected in the Short Response of the Exam.!!! Important note: Your Semester Exam test paper will only show the prompt of each focus question (text in bold font), it will NOT include the guiding questions listed on this review packet!It is highly recommended that you prepare a complete response for each focus question including example problem(s) you plan to use on your final exam. When writing a response, use the guiding questions to make sure you fully address the prompt, to make sure that your response is relevant, significant, and is supported by appropriate example problems to support your theoretical explanation. You should know at least in principle what problems you will be working with to demonstrate your understanding of each concept. Prepare 1 - 3 example problems, which you hope to explain while taking your test. It's a good idea to get the math right! Start preparing NOW!!! CHAPTER 1. FUNCTIONS AND THEIR GRAPHS. QUESTION 1. Analyze even and odd functions algebraically and graphically. To fully address this prompt, your response should include the following:Theoretical definition of even function and theoretical definition of an odd function. [Page 36]An example of one even, one odd, and one function that is neither related back to the theoretical definition. An equation and the graph for even, odd, and neither type of function. [Page 37 Example 10] NOTE: The response for this prompt would be considered complete and accurate if you used the same equation(s) for all parts of the prompt as applicable. QUESTION 2. Demonstrate composition of functions and explain the mathematical importance of composition of functions. To fully address this prompt, your response should include the following:Provide a theoretical explanation of what does composition of functions mean and when/why is it important. Discuss the differences between composition of (g o f )(1) vs. (g o f )(x) vs. (f o g )(x) [Page 54 Example 5, 6 ] Demonstrate composition of functions for various examples. NOTE: The response for this prompt would be considered complete and accurate if you used the same equation(s) for all parts of the prompt as applicable. QUESTION 3. Demonstrate finding inverse of functions algebraically and by graphing. To fully address this prompt, your response should include the following:Demonstrate how to find an inverse of a function algebraically and explain how can two inverse functions be verified algebraically using composition of functions. [Page 64 Example 3 ]Graph one function and its inverse in the same coordinate plane and describe how can inverse functions be verified graphically. [Page 65 Example 5] NOTE: The response for this prompt would be considered complete and accurate if you used the same equations for multiple parts of the prompt as appropriate. CHAPTER 2. QUADRATIC, POLYNOMIAL, and RATIONAL FUNCTIONS. QUESTION 4. Demonstrate finding zeroes and the vertex of a quadratic function by changing the standard form of the equation into a vertex form of the equation by completing the square. To fully address this prompt, your response should include the following:Use one specific example of a quadratic function in standard form and show how to change the equation into vertex form using the completing of the square. Identify the coordinates of the vertex in the vertex form of the equation. Find zeroes of this function by solving the vertex form for x. [Page 91 Example 2] QUESTION 5. Analyze graphs of polynomial functions using the leading coefficient test. Demonstrate sketching the graph of a polynomial function with real zeroes.To fully address this prompt, your response should include the following:Describe the four cases of polynomial functions using the leading coefficient test. [Page 101] Explain the use of the Intermediate Value Theorem to determine number of real zeroes. [Page 107, Page 135 Example 1]Demonstrate the use of rational Zeroes Test. [Page 119 Example 8] NOTE: You are required to use at least TWO different example problems to demonstrate finding real zeroes of polynomial functions. QUESTION 6. Analyze different types of zeroes for a polynomial function and demonstrate how to find non-real zeroes of a polynomial function. Describe various types of zeroes of polynomial functions. (REAL – rational or irrational NON-REAL – complex numbers. )Discuss the Fundamental Theorem of Algebra the degree of a function determines number of zeroes for that function. Use a specific example of a polynomial function with non-real zeroes. [Page 136 Ex(s) 2, 3, and 5]Demonstrate how to write equation of a polynomial function given its real and complex zeroes. [Pg 137 Example 4] QUESTION 7. Analyze graphs of rational functions that have vertical, horizontal, and slant asymptotes. To fully address this prompt, your response should include the following:Define Rational Function as f(x) = D(x) / N(x) [Page 142]Describe how to find asymptotes of rational functions. Describe the case of a hole in the graph. [Page 154 Example 4] Demonstrate graphing rational functions with vertical and horizontal asymptotes [Page 144 Example 2 ]Demonstrate graphing rational function with a slant asymptote [Page 155 Intro problem or Example 5] CHAPTER 3. EXPONENTIAL AND LOGARITHMIC FUNCTIONS. QUESTION 8. Compare and contrast graph of an exponential and logarithmic function. To fully address this prompt, your response should include the following:Explain the relationship between exponential and logarithmic function as inverse functions. Analyze the equation of an exponential and logarithmic equation. [Page188 Lesson Introduction] Use TWO different examples of graphs of exponential and logarithmic functions (which are inverses of one another) to analyze the graph of an exponential function versus the graph of a logarithm. [Page 190 Example 4] and one more example where equation involves two different transformations. Analyze the equation of your logarithmic function and explain why the domain of a logarithmic function is only defined if the log’s argument is >0. [ Page 193 Example 9]QUESTION 9. Define The Natural Logarithmic Function and The change of Base Formula using natural logarithmic function. To fully address this prompt, your response should include the following:Theoretically define natural logarithm by comparing y = ln x vs. x = e^y. [Page 192 Lesson Intro]Graph natural logarithmic function and identify its key points. [ Page 192 Figure 3.22List at least a few identities that are used to simplify logarithmic expressions [Page 193 Example 8]Demonstrate how to use the change of base formula using natural logarithm to evaluate an expression. [Page 199 Example 2] CHAPTER 4: TRIGONOMETRIC FUNCTIONS. QUESTION 10. Make a clear connection between the right triangle trigonometry and the unit circle to explain why the following are true: explain why To fully address this prompt, your response should include the following:Sketch a 30-60-90 special right triangle inside of the unit circle with a reference angle of 30° (reference angle has a vertex at the origin). Using ratios of sides for special right triangles, find the lengths of all sides of this triangle if the hypotenuse is = 1 unit since it represents the radius of a unit circle. Show that the adjacent side of the triangle represents the length along the x-axis value cosine Show that the opposite side of the triangle represents the length along the y-axis value of sineSketch a 45-45-90 special right triangle inside of the unit circle with a reference angle of 45° and repeat steps b) through d) For tangent show that vertical length does not represent a function tangent is undefined at 90 and 270. Also explain that since tan x = y/x or sin /cos at 90 degrees sine = 1 and cosine = 0; therefore, tan 90 = 1/0 is undefined.QUESTION 12. Using equations and graphs, demonstrate the following shifts of circular functions: vertical shift of a sine function period change of a cosine functionphase shift of a tangent function To fully address this prompt, your response should include the following:Write the standard form of trigonometric equation y = a sin b (x – c) + dFor each of the given transformations, identify the variable in the equation that represents the given transformation. Graph one example of each function. You can assume that the remaining coefficients remain unchanged from the common trigonometric function. Clearly explain the steps to finding the new KEY points of a function after the transformation was performed. SEMESTER 1 MULTIPLE CHOICE TEST. Preparing responses for all short response focus questions will assist you on multiple choice test as well; however, it is recommended that additionally, you work on several practice problems to be well prepared for this portion of the test. The following are additional recommended practice problems. Answers to all Practice Test questions are in the back of your textbook. CHAPTER 1 TEST Page 86/ # 3,4,5,7, 9, 10, 14 - 17CHAPTER 2 TEST Page 174/ 1-13CHAPTER 2 REVIEW EXRECISES Pg 172/ Ln 2.6 # 109-119 odd , Ln 2.7 # 131,133CHAPTER 3 TEST Page 244/ # 1- 22CHAPTER 4 TEST Page 338/ # 1- 15, 18SHORT RESPONSE TEST Scoring Rubric Each focus question will receive 0 to 5 points depending on the quality of the response as follows. 5 point score:Response to a focus question is outstanding, demonstrating clear and consistent understanding and mastery of mathematical concepts. Response exhibits skillful use of math vocabulary. Concept is explained using critical thinking, clear and appropriate examples, reasons, and other evidence of in depth understanding. Although it may have a few minor errors; there are no mathematical misconceptions and/or misleading information.4 point score:Response to a focus question is effective, demonstrating adequate understanding and mastery of mathematical concept, although it will have lapses in quality. Response demonstrates competent critical thinking, using adequate examples, reasons, and other evidence. There is an accumulation of multiple minor math errors throughout the response, but there are no major mathematical misconceptions. 3 point score: Response to a focus question is inadequate, but demonstrates developing understanding and mastery of mathematical concept, and is marked by one or more of the following weaknesses: i) demonstrates inconsistent critical thinking or uses inadequate examples, reasons, or other evidence ii) response is limited in its organization or focus iii) does not create example problems to demonstrate understanding of the concept. Response involves no more than one minor mathematical misconception. 2 point score: Response to a focus demonstrates weak understanding and mastery of mathematical concept, and is marked by the following weaknesses: i) response offers broad statements that are not adequately supported by explanation or example problems ii) response is seriously limited in its organization or focus, iii) response appears unplanned and unprepared and lacks in depth details or addresses less than half of the suggested guiding questions. Response involves no more than one major mathematical misconception. 1 point score: Response to a focus question is seriously limited, demonstrating little mastery of mathematical concepts, and is flawed by one or more of the following weaknesses: i) demonstrating weak critical thinking, providing inappropriate or insufficient examples, reasons, or other evidence ii) response is unfocused, or demonstrates serious problems with coherence or progression of ideas iii) response does not present any example problems to demonstrate understanding of the concept, iv) response does not address majority of the suggested guiding questions. Response contains an accumulation of minor and major mathematical misconceptions. 0 point score: Response to a focus question is missing or the response does not directly address the prompt. ................
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