Download FINAL EXAM REVIEW FOR MCR 3U

FINAL EXAM REVIEW FOR MCR 3U (Revised)
FUNCTIONS (11 UNIVERSITY)
Overall Reminders:
 To study for your exam your should redo all your past tests and quizzes
 Write out all the formulas in the course to help you remember them for the exam.
 Remember to bring your borrowed text book with you to the exam.
EXAM REVIEW: Rational Expressions



Factoring (common, difference of squares, trinomial)
Simplifying Rational Expressions
 Multiplication/ Division
 Addition/ Subtraction
Restrictions
1. Simplify.
a)  3 x 2  4 x  1   x 2  2 x  2 
b) 4 y 1  2y 1  y    y 2  2y  1  3 
c)  3z  1  2z 2  3z  4 
d) 2  a  2b   3  a  b  2a  b 
e)  x 2  5 x  3 
f)  x  2   x  1   x  4 
2
2
2
2
2
 x  4
2
2. Simplify and state any restrictions.
a)
18 p 2q 2
12 p 1q 3
b)
c)
2x  4
x2  4
d)
2x 2  3 x  1
3 x 2  2x  1
t 2  6t  9 3t  9
f) 2

t  6t  9 2t  6
4 x 4 y 6 x 3 y 2
e)

3x 2y 4
10 x 4
x2  y 2  x  y 
h)

2x 2  8 x
2xy
2t  4
2
t  6t  8
2
i)
25 x 2  9
8x  6
 2
2
4x  x  3 5x  8x  3
2x 2  5 x  2 2x 2  3 x  2
g)

2x 2  3 x  9 2x 2  x  3
j)
a2  6ab  8b2 a2  5ab  6b 2 a 2  8ab  12b 2
k) 2


a  9ab  20b2 a2  ab  6b 2 a 2  2ab  15b 2
l)
n)
4
5

3y  1 1  3y
3
2
 2
x  x  12 x  16
2
m)
4
2
 2
2m  m  1 m  2m  3
o)
ab
ab
2a  b


2
a
ab  b
b
2
1
a2  3a  10 a 2  2a  15

a2  a  12 a2  7a  12
EXAM REVIEW: Quadratics






Simplifying Radicals
Solving quadratic equations
 By factoring where possible
Using Quadratic Formula when factoring is not possible
Maximum and Minimum Values
Determine Quadratic Equation Given its Roots
Solve Linear-Quadratic Systems
Types of solutions (Nature of the Roots)
1. Simplify.
a)
b)  24
54
d) 2 12  4 20  3 27  5 45
g)
80
10
h)
2 5
3 2
e)
i)
3

c) 2  27

6  21
9  45
6
j)

f) 2 3  3 2
4  20
2
k)

5 4 3

2 3
2 3 3 2
2. Solve each of the following using the most efficient method. x C (No decimals)
a) 3y 2  7y  4  0
e)
1 2
y  4y  2
2
i) 5 x 2  4 x  20  0
b) x 2  4 x  11  0
c) 3 x 2  27  0
d) 2 x 2  3 x  3
f) 6 x 2  5 x  4
g) 3 x 2  4 x  7
h) 3 x 2  48  0
j) 2 x 2  5 x  3  0
k)
900 900

 25
x
x6
3. Determine the vertex by i) completing the square & ii) partial factoring
1
a) x2 + 3x + 1
b) y = 3x2 + 2x
c) y   x 2  2 x  4
3
d) y = 3x2 - 12x + 11
4. For each equation in Q.3, determine maximum or minimum value. Is it a maximum or minimum?
5. Find the minimum of product of two numbers whose difference is 12. What are the two numbers?
6. A sportswear store sells baseball caps with the local baseball team’s logo on them. Last year, the
store sold 600 caps at $15 each. The store manager is planning to increase the price. A
consumer survey shows that for every $1 increase, there will be a drop of 30 sales a year. What
should the selling price be to maximize the annual revenue?
7. The path of a thrown baseball can be modeled by the function h = -0.004d2 + 0.14d + 2 where ‘h’
is the height of the ball, in meters, and ‘d’ is the horizontal distance of the ball from the player, in
meters.
a) What is the maximum height reached by the ball?
b) What is the horizontal distance of the ball from the player when it reaches its maximum
height?
c) How far from the ground is the ball when the player releases it?
2
8. a) Find the equation of the quadratic function in standard form that has zeroes 2 and -3 and
containing the point (1, 8).
b) Sketch the graph. Include the y-intercept, x-intercept and vertex.
9. Find the equation of the quadratic function in standard form that has zeros 1 11 and that
contains the point (4, -6).
10. Determine the number of point(s) of intersection of y  3x  5 and y  3x 2  2 x  4 .
11. Determine algebraically the coordinates of the point(s) of intersection of y = 4x2-15x + 20 and y =
5x – 4.
12. For what value of ‘b’ will the line y = -2x + b be tangent to the parabola y = 3x2 + 4x – 1?
13. Find the value(s) of k so that each function has:
i.
two x-intercepts
ii.
one x-intercept
iii.
no x-intercepts
a) f  x   kx 2  9x  6
b) f  x   3x 2  kx  12
c) f  x   9x 2  3x  k
14. Determine the nature of the roots without solving the equation. (Type and number of solutions.)
a) x 2  5 x  7  0
b) 3 x  5 x 2  2  0
c) 4 x 2  20 x  25  0
3
EXAM REVIEW: Functions
 Domain and Range




Function Notation
Inverse Function y -1 or f –1(x)
Explanation of a transformation / Write an equation under given transformation
1

Mapping Notation:  x, y    x  d , ay  c 
b

1. If f  x   2x 2  5x  1, find:
a) f  0 
b) – f  3 
c) f  x  3 
d) x when f  x   0
2. State the domain and range of each of the following:
b) y  2  x  3   5
2
a) y   3  4 x  6
3. Given f  x   3x  8 , determine:
a) f  4 
b) x when f  x   14
c) y 
c) f 1  6
3
4
7x  5
d) x when f  x   f 1  x 
4. Determine the inverse of each of the following functions:
a) f  x   4x  3
b) g  x   4 x  8
c) h  x   2x 2  12x  14
d) k  x  
2
7
x 5
5. If f  x   2x 2  x and g  x   3  x , determine:
a) f  g  x  
b) g  f  4  
6. Describe the transformations of each function from the graph y  f  x  .
a) y  5f  x   3
b) y  f  x  3  4
d) y  f  7  x  5    1
e) y  f  8  x  7  
1 
c) y  4f  x   2
2 
1
f) y  f  2 x  6   9
3
7. Write the transformed equation if y  f  x  undergoes the following transformation:
It is reflected across the y-axis, stretched vertically by a factor of 3, compressed horizontally by a
factor of ¼, shifted 3 units down and translated 3 units to the left.
8. For each function in questions 4abc, sketch the graph and its inverse
9. Sketch each of the functions listed below. Include the parent function, a/k function and actual
function.
2
4
a) f ( x)  2 2( x  3)  1
b) f ( x)  2[3( x  1)] 2  3
c) f ( x) 
x2
1
d) f ( x)  | 4( x  2) | 2
2
4
EXAM REVIEW: Exponential Functions




Simplifying using exponent rules (in power form, radical form, and rational form)
Solving exponential equations
Exponential Growth and Decay
Exponential function
1. Evaluate. (No decimals)
2


30  40
a)  4 
b)
c) 27 3
d) 16 4
1
2
2. Simplify: (leave your answer in power form)
1
3
(ab)2 x y
a)
a xb y
 125  3
e) 

 8 
3
5 2  33
42  41
f)
( x a ) 4 ( xb ) 4
b) a  a
c) 3a b a 3b
a
x
x
30a 4b 2 (15a 2b5 )

3. Simplify, and then evaluate for a = -1, b = 2:
(5ab 4 ) 2
45a 3b6
(3a )(3a1 )(3a2 )
4. Prove:
 27
27a
x y
2 x y
2 x 7
5. Simplify each of the following. Write the answer using positive exponents.
2
1  2 p 1
a)
1  2 p 1
 3 
1
b)  2    2 
 25 
5 
1
 m5 
c) 

 2 
1
6
4
 
m
6. Evaluate (without calculators):
a) 27
2
3
 16
5
7. Solve. x  C
a) 62 x 1 
e) 2
x 2  20
2x
h) 3
1
36
 1
 
8
4
1
 
7
0
b)

32

0.4
c)  0.04 
c) 4 x 2 
b) 5 x 3  5 x 1  600
3x
 6(3x )  27  0
x2
2
1
64
6

d) 2
3
2
2
d) 32 x
7
2
2
 2
9 x
3
2
2
7
2

 243
 1 
g) 42 22 x 3  16 x 2 

 2
 
f)  42  22 x 3   16 x 2
* i) 5  8
1
5
 

 57 x
8. In 1976, a research hospital bought half of a gram of radium for cancer research. Assuming
the hospital still exists, how much of this radium will the hospital have in the year 6836,
if the half life of the radium is 1620 years?
9. There are 50 bacteria present initially in a culture. In 3min., the count is 204800.
What is the doubling period?
5
10. In a recent dig, a human skeleton was unearthed. It was later found that the amount of
in it had decayed to
 
1
8
14C
of its original amount. If 14C has a half life of 5760 years,
how old was the skeleton?
11. A house was bought 6 years ago for $175 000. If real estate values have been increasing at the
rate of 4% per year, what is the value of the house now?
12. Which set of properties does the function
have?
a. x-intercept is 0, y-intercept is 0
c. x-intercept is 1, no y-intercept
b. no x-intercept, y-intercept is 1
d. no x-intercept, no y-intercept
13. Which set of properties does the function
a. no x-intercept, y-intercept is 1
b. x-intercept is 1, no y-intercept
have?
c. no x-intercept, no y-intercept
d. x-intercept is 0, y-intercept is 1
14. Which exponential equation matches the graph shown?
y
6
(1, 6)
4
2
–4
–2
(0, 2)
2
4
x
–2
a.
b.
c.
d.
15. For the exponential function
a.
b.
c.
d.
, which of the following statements is not true?
The graph of the function is increasing.
The graph of the function is decreasing.
The domain is the set of real numbers.
The range is the set of real numbers greater than 0.
16. Which of the following transformations maps the function
a.
b.
c.
d.
onto the function
a horizontal shift 3 units to the right and a vertical shift 4 units up
a horizontal shift 3 units to the left and a vertical shift 4 units up
a horizontal shift 3 units to the right and a vertical shift 4 units down
a horizontal shift 3 units to the left and a vertical shift 4 units down
6
?
17. For the exponential function
, which properties does the graph of the function have?
a. domain
,
range
,
asymptote at y = –1
b. domain
,
range
,
asymptote at y = –1
c. domain
,
range
,
asymptote at y = –1
d. domain
,
range
,
asymptote at y = 1
18. Which graph represents the function
a.
?
c.
y
–4
y
20
40
10
20
–2
2
4
x
–4
–2
–10
–20
–20
–40
b.
2
4
x
2
4
x
d.
y
–4
y
20
40
10
20
–2
2
4
x
–4
–2
–10
–20
–20
–40
x
1
19. Sketch the function f ( x)  5   9 . Include a graph of the parent function, the a/k function
2
and the actual function. Determine the domain, range and equation of the asymptote.
7
EXAM REVIEW: Trigonometry









Primary Trig. Ratios, Sine Law and the Ambiguous Case, Cosine Law
Reciprocal Ratios
Angles in Standard Position (Co-terminal Angles, Primary Trig. Ratios)
Special Triangles and the CAST Rule
Solving Trigonometric Equations – Finding the Angle Measure
Prove Trigonometric Identities
Periodic Functions
Transformations: Amplitude, Period, Phase Shift, Vertical Translation, Reflection (Vertical and
Horizontal)
Graphing Trigonometric Functions ( y  a sin b  x  d   c or y  a cos b  x  d   c )
b) cot 315
1. Evaluate the following: a) csc170.1o
c) sec( 240)
2. If sec  135 , make a diagram and find the other five trigonometric ratios, if  is acute.
3. If cot  
2
21
and  is obtuse, find the exact values of the other five ratios
and the approximate value of  ,
4. Determine the exact value.
a) sin 180
b) cos 210
c) tan( 225)
e) cos 300 cos 270  sin 150 sin 270
5. If sin   
1
2
d) sin( 45) cos(135)  tan 30 tan 420
f) 5sec30tan60
, find all possible values of 0    360 .
6. If csc   2 , find all possible values of 0    360 .
7. Prove the following identities.
a) 1  sin2 x 1  tan2 x   1
b) sin x  tan x  cos x tan x  2cos x tan x  tan x
c) sin2   sin4   cos2   cos4 
d)
1  sin x
cos x
2


cos x
1  sin x cos x
f)
2
1
1


2
sin x 1  cos x 1  cos x
e)
tan2 
 sin2 
tan2   1
g) tan2 x  cos2 x  sin2 x 
1
cos2 x
h) sec cos  sec sin   1  tan 
8
8. For the given triangle, sin A is equal to which expression?
a.
c.
b.
d.
9. What reference angle should be used for an angle of 300° in standard position?
a. 30°
c. 60°
b. 45°
d. 300°
10. Which set of primary trigonometric ratios is correct for the angle 225°?
a.
c.
b.
d.
11. An 8-m ladder must be placed against a wall such that the base of the ladder makes a 60° angle
to the ground. How far up the wall will the ladder reach, to the nearest tenth of a metre?
a. 16.0 m
c. 9.2 m
b. 4.0 m
d. 6.9 m
12. Which set of angles has the same terminal arm as 70°?
a. –20°, –110°, –200°
c. –290°, –650°, –1010°
b. 0°, –70°, –140°
d. –110°, –290°, –470°
13. The point (–
ratios for the angle?
igonometric
a.
c.
b.
d.
9
14. Which pair of angles between 0° and 360° satisfies
a. –60° and –240°
c. 120° and 240°
b. 60° and 240°
d. 300° and 240°
?
15. Determine two angles between 0° and 360° that have a secant of
a. 45° and 315°
c. 45° and 135°
b. 225° and 135°
d. 45° and 225°
16. Which strategy would be best to use to solve for x?
a. sine law
b. cosine law
c. primary trigonometric ratio
d. none of the above
17. Determine the measure of x, to the nearest degree.
a. 52°
b. 38°
c. 32°
d. 58°
18. Determine the length of x, to the nearest tenth of a centimetre.
a. 44.8 cm
b. 66.8 cm
c. 64.6 cm
d. 42.6 cm
19. Determine the measure of x, to the nearest tenth of a degree.
a. 48.4°
b. 41.6°
c. 20.8°
d. 35.5°
10
.
20. The value of a periodic function at x = 6 is y = 3. What is the period of the function if the next xvalue along the positive x-axis where the function has the same y-value is x = 9?
a. 2
c. 4
b. 3
d. 6
21. A sinusoidal function has an amplitude of 4 and a period of 720°. What is a possible equation for
this function?
a.
c.
b.
d.
22. Which equation can be used to model the graph of the function shown?
a.
c.
b.
d.
23. Determine the amplitude, the period, the phase shift, and the vertical shift of the function
with respect to
.
a. amplitude 4
c. amplitude 4
period 90°
period 180°
phase shift left 30°
phase shift left 30°
vertical shift up 1 unit
vertical shift up 1 unit
b. amplitude –4
d. amplitude –4
period 90°
period 180°
phase shift right 30°
phase shift right 30°
vertical shift down 1 unit
vertical shift down 1 unit
24. Determine the amplitude, the period, the phase shift, and the vertical shift of the function
with respect to
a.
amplitude
period 1440°
phase shift left 180°
vertical shift down 1 unit
.
c. amplitude 3
period 90°
phase shift left 45°
vertical shift up 1 unit
11
b. amplitude 3
period 180°
phase shift right 180°
vertical shift up 1 unit
d.
amplitude
period 1440°
phase shift right 45°
vertical shift down 1 unit
25. Determine the domain and range of the function
a. domain
.
,
range
b. domain
,
range
c. domain
,
range
d. domain
,
range
26. What are the minimum and maximum values of the function
a.
b.
maximum , minimum
maximum
, minimum
c.
d.
?
maximum , minimum
maximum
, minimum
27. Consider the function
.
a) Determine the amplitude, the period, the phase shift, and the vertical translation of the function
with respect to y = sin x.
b) Graph the function over the interval
.
28. Consider the function
.
a) Determine the amplitude, the period, the phase shift, and the vertical translation of the function
with respect to y = cos x.
b) Graph the function over the interval
.
29. In ABC, c = 11 cm, b = 7 cm, and
.
a) Sketch possible diagrams for this situation.
b) Determine the measure of <C in each diagram.
c) Find the measure of <A in each diagram.
d) Calculate the length of BC in each diagram
12
EXAM REVIEW: Sequences and Series

Arithmetic Sequences / Series
tn  a   n  1 d

n
2a   n  1 d 
2
Geometric Sequence / Series
tn  ar


sn 
n 1
Sn 

Sn 
n
(a  t n )
2

a r n 1
r 1
Recursion Formula
Pascal’s Triangle
1. Find tn and t12 for each of the following sequences:
a) 9, 15, 21, …
b) –1, 2, –4, 8, …
2. Find the sum of each of the following series:
a) 251 + 243 + 235 +…–205
b) –4 – 12 – 36– … –8748
c) 21 + 23 + 25 + … + 43
d) 1280 – 640 +320 – … + 5
3. Write the first 4 term of each of the following sequences:
a) tI = –6; tn = tn – 1 +5; n > 1
b) t1 = –2; t2 = –1; tn = tn – 1 x tn – 2
4. In an arithmetic sequence, the 3rd term is 25 and the 9th term is 43. How many terms are less
than 100?
5. The sum of the first 6 terms is 297 and the sum of the first 8 terms is 500. Determine the 5 th term
if the sequence is arithmetic.
6. Insert 3 terms between 6 and 7776 so that you create a geometric sequence.
7. Insert 3 terms between 41 and 97 so that you create an arithmetic sequence.
8. For (1 – x)11 : a) find t3
b) how many terms are in the expansion?
c) explain where the numerical coefficients of the expansion are coming from
4
1 3
8. a) Expand    using binomial theorem; take the coefficients from Pascal triangle n = 4
4 4
b) Evaluate the sum.

1 
9. Expand (Leave your final answer in power form): a)  2 z 3  2 
3y 

13
5

b) 3 d  d 3

3
EXAM REVIEW: Finance


Simple Interest:
I = Prt
n
Compound Interest: A  P 1  i 
A=P+I
A = P + Prt
n
R 1  i   1

A 
i

Annuity:

n
R 1  1  i  

Present Value Annuity: P  
i
1. In order to have $7250 for tuition in a year and a half, you plan to deposit money into a savings
account every three months. If that account pays 5.2% compounded quarterly,
a) what will your regular deposits be?
b) how much interest would you have earned?
2. Leo’s parents want him to be able to withdraw an allowance of $2700 every 6 months. How much
should they invest into an account today that pays 8.5% compounded semi-annually so that Leo
can withdraw his allowance for the next 5 years?
3. Determine:
a) the amount $8500 will grow to if invested at 8% compounded quarterly for ten years.
b) the principal that must be invested now at 6% compounded annually to be worth $10 000 in 5
years.
c) The total accumulated amount of $500 invested every month at 7% compounded monthly for
8 years.
4. How long will it take, to the nearest year, for $800 to grow to $1000 in an account that pays 4.5%
annually compounded semi-annually?
5. If a savings account compounds daily, what would the annual interest rate need to be for $500 to
grow to $650 in 3 years?
6. Batman borrows $500 at an annual rate of 8.25% simple interest to enroll in a driver’s education
course. He plans to repay the loan in 18 months. What amount must he pay back?
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