REVISION LECTURE

1. REVISION LECTURE MATHEMATICAL METHODS (CAS) UNITS 3 AND 4 Presenter: MICHAEL SWANBOROUGH Flinders Christian Community College

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. • It is assumed that students will provide exact answers to questions unless specified otherwise. • 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. • Correct mathematical notation is expected and should always be used. Calculator syntax should not be used.

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 CAS calculator for specific purposes. • Be sure that you know the syntax to use with your calculator.

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. The range of the function is a) c) e) b) d) Question 1 A

15. b) a) c) d) e) Question 2 The equations of the asymptotes of the graph of the inverse function are C

16. Bonus Prize!!

17. The smallest positive value of x for which is Question 4 a) b) B c) d) e)

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

19. The complete set of linear factors of the polynomial ANSWER: E VCAA 2004: 49% Question 1

20. Therefore is not exactly divisible by Use the remainder theorem Question 2

21. Infinite solutions No solution Unique solution Simultaneous Linear Equations Same gradients Different y-intercept Different gradients Same gradients Same y-intercept

22. Infinitely many solutions for the simultaneous linear equations Question 4

23. ANSWER: B VCAA 2008: 45%

24. 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!)

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

26. 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.

27. So the maximal domain is: Consider the function

28. 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

29. 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 in the positive x-direction (ie. to the right). If the solution for x is negative – move the graph x units in the negative x-direction (ie. to the left).

30. Note: A dilation of a from the x-axis is the same as a dilation of from the y-axis. • 2. Dilations • a) From the x-axis – the dilation factor is the number outside the brackets. • From the y-axis – the dilation factor is the reciprocal of the coefficient of x.

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

32. Determine the graph of Question 5

33. Reflection about the x-axis Dilation by a factor of from the y-axis

34. Translation of 1 unit in the positive y-direction ANSWER: E VCAA 2009: 66%

35. Graph of Reflection: Horizontal Translation: Vertical Translation: Question 6 Reflected in the x-axis, Translated 3 units to the right, Translated 4 units down ANSWER: B VCAA 2007: 95%

36. maps onto Dilation by a factor of from the y-axis Matrix Transformations Question 7 Translation of 2 units in the positive x-direction Translation of 3 units in the positive y-direction

37. Dilation: Translations: ANSWER: A VCAA 2006: 33%

38. Graphs of Power Functions

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

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

41. Vertical: Horizontal: Graphs of Rational Functions Question 10 The equations of the horizontal and vertical asymptotes of the graph with equation ANSWER: D VCAA 2008: 86%

42. Absolute Value Functions Question 12 ANSWER: D VCAA 2006: 75%

43. a) Sketch the graph of Question 13 Part of the graph of is shown below.

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

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

46. 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).

47. Question 14

49. 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

50. 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.