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MAV REVISION LECTURE. MATHEMATICAL METHODS UNITS 3 AND 4. Presenter: MICHAEL SWANBOROUGH Flinders Christian Community College. EXAMINATION 1. Short-answer questions (40 marks) Questions are to be answered without the use of technology and without the use of notes Time Limit:

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    2. EXAMINATION 1 • Short-answer questions (40 marks) • Questions are to be answered without the use of technology and without the use of notes • Time Limit: • 15 minutes reading time • 60 minutes writing time

    3. EXAMINATION 2 • Part I: Multiple-choice questions • 22 questions (22 marks) • Part II: Extended response questions: • 58 marks • Time limit: • 15 minutes reading time • 120 minutes writing time

    4. Examination Advice General Advice • Answer questions to the required degree of accuracy. • If a question asks for an exact answer then a decimal approximation is not acceptable. • When an exact answer is required, appropriate workingmust be shown.

    5. Examination Advice General Advice • When an instruction to use calculus is stated for a question, an appropriate derivative or antiderivative must be shown. • Label graphs carefully – coordinates for intercepts and stationary points; equations for asymptotes. • Pay attention to detail when sketching graphs.

    6. Examination Advice General Advice • Marks will not be awarded for questions worth more than one mark if appropriate working is not shown.

    7. Examination Advice Notes Pages • Well-prepared and organised into topic areas. • Prepare general notes for each topic. • Prepare specific notes for each section of Examination 2. • Include processsteps as well as specific examples of questions.

    8. Examination Advice Notes Pages • Include key steps for using your graphic calculator for specific purposes. • Be sure that you know the syntax to use with your calculator (CtlgHelp is a useful APP for the TI-84+)

    9. Examination Advice Strategy - Examination 1 • Use the reading time to carefully plan an approach for the paper. • Momentum can be built early in the exam by completing the questions for which you feel the most confident. • Read each question carefully and look for key words and constraints.

    10. Examination Advice Strategy - Examination 2 • Use the reading time to plan an approach for the paper. • Make sure that you answer each question in the Multiple Choice section. There is no penalty for an incorrect answer. • It may be sensible to obtain the “working marks” in the extended answer section before tackling the multiple choice questions.

    11. Examination Advice Strategy - Examination 2 • Some questions require you to work through every multiple-choice option – when this happens don’t panic!! • Eliminate responses that you think are incorrect and focus on the remaining ones. • Multiple Choice questions generally require only one or two steps – however, you should still expect to do some calculations.

    12. Examination Advice Strategy - Examination 2 • If you find you are spending too much time on a question, leave it and move on to the next. • When a question says to “show” that a certain result is true, you can use this information to progress through to the next stage of the question.

    13. Revision Quiz

    14. where a, b and c are three different positive real numbers. The equation has exactly a) 1 real solution b) 2 distinct real solutions c) 3 distinct real solutions d) 4 distinct real solutions e) 5 distinct real solutions Question 1 B

    15. a) b) c) d) e) Question 2 The range of the function with graph as shown is B

    16. Bonus Prize!!

    17. is the sum of the For the equation solutions on the interval Question 4 a) b) E c) d) e)

    18. Question 5 What does V.C.A.A. stand for? a) Vice-Chancellors Assessment Authority b) Victorian Curriculum and Assessment Authority c) Victorian Combined Academic Authority d) Victorian Certificate of Academic Aptitude e) None of the above B

    19. The linear factors of the polynomial are Question 1 ANSWER: B

    20. a) b) Question 4

    21. Functions and Their Graphs Vertical line test - to determine whether a relation is a function A represents the DOMAIN B represents the CODOMAIN (not the range!)

    22. Interval Notation Square brackets [ ] – included Round brackets ( ) – excluded

    23. Maximal (or implied) Domain The largest possible domain for which the function is defined • A function is undefined when: • a) The denominator is equal to zero • The square root of a negative number is present. • The expression in a logarithm results in a negative number.

    24. So the maximal domain is: Consider the function

    25. Using Transformations When identifying the type of transformation that has been applied to a function it is essential to state each of the following: NATURE– Reflection, Dilation, Translation MAGNITUDE(or size) DIRECTION

    26. 1. Translations a) Parallel to the x-axis – horizontal translation. b) Parallel to the y-axis – vertical translation. To avoid mistakes, let the bracket containing x equal zero and then solve for x. If the solution for x is positive – move the graph x units to the RIGHT. If the solution for x is negative – move the graph x units to the LEFT.

    27. Note: A dilation of a parallel to the y-axis is the same as a dilation of parallel to the x-axis. • 2. Dilations • a) Parallel to the y-axis – the dilation factor is the number outside the brackets. This can also be described as a dilation from the x-axis. • Parallel to the x-axis – the dilation factor is the reciprocal of the coefficient of x. This can also be described as a dilation from the y-axis.

    28. a) Reflection about the x-axis b) Reflection about the y-axis c) Reflection about both axes d) Reflection about the line 3. Reflections

    29. Determine the graph of Question 6

    30. ANSWER: A Reflection about the y-axis

    31. Graph of Reflection: Translation: Translation: Question 7 Reflected in the x-axis, Translated 2 units to the right, Translated 1 unit down ANSWER: B

    32. Graphs of Power Functions

    33. Square Root Functions • The graph is: • translated 2 units in the positive x direction • translated 1 unit in the positive y direction

    34. Question 9 The rule of the graph shown could be ANSWER: D

    35. Vertical: Horizontal: Graphs of Rational Functions Question 10 The equations of the horizontal and vertical asymptotes of the graph with equation ANSWER: E

    36. a) b) c) d) e) Question 12 The graph shown could be that of the function f whose rule is ANSWER: A

    37. Absolute Value Functions Question 14 ANSWER: D

    38. a) Sketch the graph of Question 15 Part of the graph of is shown below.

    39. From the graph, solve b) Find the set of values of x for which

    40. For the composite function to be defined When the composite function is defined Composite Functions

    41. Step 4: Remember that: Investigating Composite Functions Step 1: Complete a Function, Domain, Range (FDR) table. Step 2: Check that the range of g is contained in the domain of f . Step 3: Substitute the function g(x) into the function f (x).

    42. Question 16

    43. Inverse Functions Key features: The original function must be one-to-one Reflection about the line y = x Domain and range are interchanged Intersections between the graph of the function and its inverse occur on the line y = x

    44. To find the equation of an inverse function Step 1: Complete a Function, Domain, Range (FDR) table. Step 2: Interchange x and y in the given equation. Step 3: Transpose this equation to make y the subject. Step 4: Express the answer clearly stating the rule and the domain.

    45. Question 18 ANSWER: A

    46. Question 19 Graph of the inverse function ANSWER: C

    47. Question 21

    48. a) exists because the function f is one-to-one b) i)

    49. b) ii)