Download MAT521A Academic Mathematics 2A

MAT621B Grade 12 Academic Math
FINAL EXAMINATION OUTLINE: DECEMBER 2013
GENERAL INFORMATION:
[1]
The examination is valued at 35% of the course mark and the class work is valued at 65% of
the course mark.
[2]
The examination will be written at the following times:
Wednesday, December 18 and Thursday, December 19, 2013
SNOW DATE Friday December 20, 2013
Note that any student who does not show up for the final examination must advise the school
before the end of the school day on their test day. Otherwise, that student will not be given the
opportunity to write and shall not receive credit for this course.
[3]
Extra help can be given any day afterschool or during both lunches. Start studying now.
The tutorial will be held at the following times:
Monday December 16, 2013 from 3:00 to 5:00 in Room 603
[4]
A.
621B
Corrected examinations will be available for viewing after the exam.
All students are encouraged to check over their exams for any possible marking errors or
discrepancies.
OBJECTIVE QUESTIONS [approx 65 points]
[20]
1.
State whether each of the following statements is TRUE or FALSE. {20 Questions}
 There will be at least one similar question from each of the chapter tests.
[27]
2.
For each of the following, place the NUMBER of the correct answer in the space
provided. {26 Questions}
 There will be at least one similar question from each of the chapter tests.
[18]
3.
For each of the following, place the correct answer in the space provided. {18 Questions}
There will be at least one similar question from each of the chapter tests.
B.
WRITTEN EXERCISES
DIRECTIONS: For all problems in this section, show all work in the space provided and clearly indicate
the answers.
Part I: TRANSFORMATIONS
[4]
1.
Consider the function

y  x . Determine the equation of its image in each case. {3 Questions}
Similar Question: Test #1 – Question #7 (page 5)
[8]
2.
A function will be given such as y  x . Describe all transformations in words. {3 Questions}
(Use HT, VT, HE, HC, VE, VC, reflection in answer)
 Similar Question: Test #1 – Question #6 (page 5) and Test #2- Question #4 (page3)
[6]
3.
[3]
4.
[6]
5.
Algebraically, find the inverse of each of the following functions (The inverse is the equation
of the reflected graph over the line y  x ). {2 Questions}
 Similar Question: Test #1 – page 2: Question #8 and page 6: Question # 8 and #9
Sketch the following graph reflected over the x-axis or y axis or line y=x {1 question}.
 Similar Question: Test #1 – Question # 3 and #4 (page 4)
Match the equations to the graphs. (Graphs from all units covered)
MAT621B Final Examination Outline
Page 2
Part 2: RADICAL FUNCTIONS
[3]
1.
[4]
2.
[2]
3.
Solve a radical eqauation with one variable algebraically. {1 question}.
 Similar Question: Test #2 – Question #8 (page 5)
Solve a radical eqauation algebraically, with two x variables. {1 question}.
This question will result in quadrtatic eqution that must be factored or use the quadratic
formula to solve for x. There will be two solutions, be sure to check for extraneous solutions.
 Similar Question: Test #2 – Question #9 (page 5)
Determine the equation of the transformed graph {1 question}.
 Similar Question: Test #2 – Question #11 (page 6)
Part 3: POLYNOMIAL FUNCTIONS
[3]
1.
[3]
2.
[3]
3.
2 x 3  3x 2  5 x  6
6
 2 x 2  3x  4 
x3
x3
Divide polynomials using synthetic division {1 question}. Ex.
 Similar Question: Test #3 – Question #1 on page 3
Try example given above
Divide polynomials using long division {1 question}.
 Similar Question: Test #3 – Question #2 on page 3
Solve a polynomial equation by factoring. {1 question}.
 Similar Question: Test #3 – Question #6 on page 5 .
3
2
 Example : x  6 x  11x  6  0 Use factor theorem and synthetic division and factor again. Show work
(x  1)(x  2)(x  3)  0
[6]
4.
X=1
x=2
x=3
Determine the equation of the polynomial function in factored form given graph. {1 question}.
 Similar Question: Test #3 – Question #7 and 8 on page 6
PART 4: TRIGONOMETRIC FUNCTIONS (TEST #4 AND TEST #5)
[3]
1.
Determine the length of the arc of a circle given the radius and the angle cut by the arc {1 question}.
 Similar Question: Test #4 – Question #4 on page 4 (look at questions onassignment)
[6]
2.
[2]
3.
[4]
4.
Solve a trigonometric equation. Find all the exact solutions {2 question}.
 Similar Question: Test #4 – Question #1 a,b, c. d under Solving Trigonometric Equations on page 5 and 6
Given a graph, find the period of a graph {1 question}.
 Similar Question: Test #5 – Question #3 a) on page 1 bottom of page
Determine an equation of the form y  a cos b  x  c   d having a given amplitude,
[4]
5.
[4]
6.
[5]
7.
period, phase shift, and vertical displacement. {1 Question}
 Similar Question: Test #5 – Question #2 written exercises on page 2
Determine the amplitude, period, phase shift, and vertical displacement given an equation.
 Similar Question: Test #5 – Question #1 written exercises on page 2
Describe all transformations given an equation {1 Question}
 Similar Question: Test #5 – Question #3 written exercises on page 2
Determine the equation of a trigonometric graph usng both a sine and cosine function
Use y  a cos b( x  c)  d and y  a sin b( x  c)  d {1 Question}
 Similar Question: Test #5 – Question #7 written exercises on page 3
MAT621B Final Examination Outline
Page 3
PART 5: TRIGONOMETRIC IDENTITIES (TEST #6)
[5]
1.
Prove the identity tan x  sin x csc x  sec x algebraically
{1 question}.
 Similar Questions: Test #6 – Question #5 on page 5 and 6 (look at questions on assignment) See question below
as practice.
2
2
tan 2 x  sin x csc x  sec 2 x
sin 2 x 
1 
  sin x

2
sin x 
cos x 
sin 2 x  sin x 


cos 2 x  sin x 
Note: There are other ways to prove this identity
sec 2 x
sin 2 x  cos 2 x 

 1
cos 2 x  cos 2 x 
sin 2 x cos 2 x

cos 2 x cos 2 x
1
cos 2 x
sec 2 x
QED
[3]
2.

Find the EXACT value of sin 75 {1 question}.
 Similar Question: Test #6 – Question #2c on page 3
(use special angles on unit circle and sum or difference Identities)
Part 6: EXPONETIAL AND LOGARITHMIC FUNCTIONS
[6]
1.
[5]
2.
[3]
3.
Evaluate each of the following logarithmic expressions using logarithmic laws. {2 questions}
 Similar Question: Test #7 – Question #5 abc on page 4
What is the value of x? short answer, no work needed. {5 questions}
 Similar Question: Test #7 – Question #4 on page 2
Solve for x to the fourth decimal, if necessary. Show work {1 question}.
 Similar Question: Test #7 – Question #7 a on page 5
32 x  5  25
[9]
4.
Solve each of the following equations, giving exact answers and clearly indicate your answer
{3 questions}.
 Similar Question: Test #7 – Question #7 b on page 5 and Question #1 page 3
2 x 5
x 1
a. 7(2)
b. 9  81
 112
 Solve logarithmic equations see Test #7b (test that was done after Christmas)
a. log 4 x  log 4 ( x  2)  log 4 8
b. log 6 ( x  5)  log 6 6  log 6 x
Part 7: RATIONAL FUNCTIONS:
After Christmas not on exam
Part 8: FUNCTION OPERATIONS: After Chrismas not on exam
3x
MAT621B Final Examination Outline
C.
Page 4
WRITTEN PROBLEMS [25 points]
DIRECTIONS: Solve each of the following written problems, showing all work and clearly indicating the
answers. Answer ONLY 5 of the following wordproblems
1.
2.
3.
Exponential Growth Problem {1 Question}
 Similar Question: Test #7b – Solve for time
Compound Interest Problem {1 Question}
 Similar Question: Test #7 – Question #2 on page 5
Compound Interest Problem {1 Question}
 Solve for time
4.
Depreciation Problem {1 Question}
 Similar Question: Test#7 and Assignment
5.
Radian Measure Problem {1 Question}
 Similar Question: Test #4- Question #6 page 4
Example:
6.
Example:
A wagon tire is turning at a rate of 70 revolutions per minute (rpm). If the tire is 15 cm in radius, find the
distance that a sticker stuck on the edge of the tire travels in 180 seconds.
Ferris Wheel Question {1 Question }
 Similar Question: Test #5 – page 4 Questions #1 and #2 (We also did a booklet in class on word problems called
section 5.4: Solving Equations, the question from booklet that is on the exam is on page 5 o booklet
Questions # 4 and 5)
A ferris wheel has a radius of 24 m. It rotates once every 50 s. Passengers get on the ride at
2 m above ground level. Suppose you get on at the bottom and the wheel starts to rotate.
a) Sketch a graph of how your height above the ground varies during the first two cycles.
(label the x and y axis with proper units)
b) Write an equation that expresses your height as a function of the elapsed time . The heght of the ride is
dependent on time (t)
Height  a cos b(t  c)  d
c) Estimate your height above the ground after 18 s (make sure calculator in radian mode)
D.
GRAPHING [21 points]
DIRECTIONS: For all problems in this section, show all work in the space provided, clearly indicate the
answers, and sketch the graphs in the coordinate planes provided.
[5]
1.
[6]
2.
Transformed Graph {1 Question}
 Similar Question: Test #1 – Queston #10, 11 and 12
Radical function Graph {1 Question}
 Similar Question: Test #2 – Queston #5 on page 3
Graph the following function
y  2 3x  4  1, then determine the domain and range.
Show a table
of values with at least 4 points. Be sure that the starting point is indicated on the graph (h,k) is the
starting point ]
The starting point for this function is (4,1) Put x values into function that are x  4 and follow order of
operations to get the y value
[10]
3.
Trigonometric function Graph {1 Question }
 Similar Question Group: Test #5 – Graphing Section page 5 and 6 – Questions #1 an #2 (no phase shift given on
exam)
MAT621B Final Examination Outline
Page 5
COURSE MARK CALCULATOR
(this will not be accurate since not end of year)
What is your current mark?
= A
What mark would you like for the course?
= B
 Multiply A by 0.65:
= C
 Subtract B – C:
= D
 Divide D by 0.35:
%
This is the mark that you need to make on your final exam to get
the mark that you would like for the course. Please note that if
your answer is negative, it is not possible to get the desired mark
The highest mark that you can get in this course is:
 (A x 0.35) + 65
%
The unit circle, all trigonometric identities and all formulas will be given. Except for the compound interest formula
and students must know how to convert from radians to degrees and degrees to radians using the conversion factor.
Finally, I would like to thank all of you for being such a great group and I have enjoyed working with all of you and I wish you
well in all you do in the future, you are all great people, keep in touch and take care,
GOOD LUCK !!! HAVE A GOOD BREAK
FORMULA SHEET

P  P0c et
P  P0 (2)
et
dt
P  P0 (0.5)
et
ht
 b  b2  4ac
x
2a
radians  180o
a  r
period 
b
2
b
2
period
y  a cos bx  c  d
y  a sin bx  c  d
y  a cos 2
x  c   d
y  a sin 2
x  c   d
p
p
MAT621B Final Examination Outline
Page 6
Trigonometric Identities
Reciprocal Identities
Quotient Identities
Pythagorean Identities
csc x 
1
sin x
tan x 
sin x
cos x
cos2 x  sin2 x  1
sec x 
1
cos x
cot x 
cos x
sin x
1 tan2 x  sec2 x
cot x 
1
tan x
cot2 x  1  csc2 x
Sum Identities
Difference Identities
sin
 A  B   sin
cos
 A  B   cos
tan
 A  B 
A cos B  cos A sin B
A cos B  sin A sin B
tan A  tan B
1  tan A tan B
sin
 A  B   sin
cos
 A  B   cos
tan
 A  B 
A cos B  cos A sin B
A cos B  sin A sin B
tan A  tan B
1  tan A tan B
Double-Angle Identities
sin 2A  2 sin A cos A
cos 2A  cos2 A  sin2 A
tan 2 A 
2 tan A
1  tan2 A
cos 2A  2 cos2 A  1
cos 2A  1 2 sin2 A
Proving Trigonometric Identities
 To prove a trigonometric identity algebraically, simplify both sides of the identity separately into identical expressions.
 Do not move terms from one side of the identity to the other.
 It is usually easier to make a more complicated expression easier than to make a simple expression more complicated. Consider
changing the UGLY SIDE FIRST
Some strategies that can be used when proving identities are: (TIPS)

Use known quantities to make substitutions.

If quadratics are present, the Pythagorean identity, or one of its alternate forms, can often be used.

Rewrite each expression using sine and cosine only.

Multiply the numerator and the denominator by the conjugate of an expression.

Factor to simplify expressions.
MAT621B Final Examination Outline
Page 7