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Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Introduction to Functions
Function Notation
A function is like a machine. When an x-value in the domain of the function f enters, the machine
produces the output f (x) . The output f (x) is determined by the rule of the function.
Vertical Line TestEx 1) Which are functions:
a)
b)
c)
d)
x
-1
0
1
2
y
y
1
2
x

x
-2
-2
2
-2
-2
Ex 2) If g(x) = 2x2 + 3, evaluate when x = 4.
Expanding
Expand: a)
 2 x  1 x  2
b)
 3 x  1 3 x  1
Factoring
a) x  x  20
2
2
b) 2 x  2 x  24
2
c) 3 x  11 x  14
y
5
5
5
5
Domain and Range
Number System
Symbol
Natural Numbers

Description
The natural numbers start at 1 and increase by one to infinity. For example:
1, 2,3, 4,
Whole Numbers
W
The whole numbers start at 0 and increase by one to infinity. For example:
0,1, 2,3, 4,
Integers
Z
The integers are consist of all the positive natural numbers, their negatives and
the number zero.
The set of integers start at 0 and increase by one in one direction to infinity, and
decrease by one in the other direction to negative infinity. For example
, 4, 3, 2, 1, 0,1, 2,3, 4,
The symbol Z is used to represent the set of integer numbers. The German
word for numbers is Zahlen hence the usage of the symbol Z .
Real Numbers
R
The real numbers are the set of all positive and negative integers, fractions and
irrational numbers (numbers that have decimal parts that do not terminate or
repeat)
Domain –
Range –
Ex) State the domain and range of each of the following:
Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Quadratics
Investigating Quadratic Functions
Complete the table below and graph the data. Label the following words on your graph: axis of
symmetry, zeros, x-intercept, vertex, minimum/maximum value.
x
y
-2
12
-1
5
0
0
1
-3
2
-4
3
-3
First
Differences
Second Differences
Vertex Form
y = a (x – h)2 + k
Graph the following by using the vertex and step
Example: Graph the following equation: y = -3(x – 4)2 + 5
Vertex:
Optimal (Max/Min) Value:
Axis of Symmetry:
Step:
Factored Form
y = a (x – s) (x – t)
Sketch the graph of the following equation by filling out the table below: y = -2x2 – 4x + 16
Zeros:
Step:
Axis of Symmetry:
Optimal (Max/Min) Value:
Vertex:
Application
1. An outdoor gear company models the profit on its newest backpack using the function
, where x is the number of backpacks, in thousands, that the company
produces and P(x) is the profit, in tens of thousands of dollars.
a. What is the break even point for the backpacks (where does the company make ZERO profit)?
[2]
b. What is the maximum profit the company can earn from the backpacks?
[3]
c. What is the y-intercept of the profit function? Explain the meaning of this value in the context
of the question.
2. Find the equation for the data in all three forms. How far are you from the CBR at 3.25 seconds?
Time (s)
0
0.5
1
1.5
2
2.5
3
3.5
4
Distance from
Motion Detector
(m)
0
8.75
15
18.75
20
18.75
15
8.75
0
Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Quadratics 2 – Completing the Square
Introduction Problem
You throw a paper airplane from a height of 4 m above the ground. The height of the airplane as a
function of time is modeled by the function: h(t) = t2 + 10t + 4.
Find the maximum height of the airplane
Completing the Square Using Tiles
Can we convert the following into vertex form by factoring? y = x2 + 10x + 4
h = ______________________
k = ______________________
Completing the Square Using Algebra
Find the minimum of the following:
y = -2x2 – 16x + 3
I.
Remove the common factor (the a-value) on both variable terms
II.
Find the constant that must be added and subtracted to make a perfect square. Divide the
number on the x-term by 2, and square it. COMPLETE THE SQUARE!
III.
Rewrite the expression by adding, then subtracting, this value after the x-term inside the
brackets.
IV.
Group the three terms that form the perfect square. Move the subtracted value outside the
brackets by multiplying it by the common constant factor (a-value).
V.
Factor the perfect square and collect like terms.
The Quadratic Formula
A diver dives from a cliff at a height of 14 m above the sea. The height of the diver above the sea at
time t in seconds for the dive can be modeled by the function:
h(t) = 2x2 – 12x + 14
Find the time when the diver hits the water. Solve using two methods.
x 
 b  b 2  4ac
2a
Behaviour At Roots
If b 2  4ac is positive, there are two different real roots.
If b 2  4ac is zero, there are two equal real roots
If b 2  4ac is negative, there are no real roots.
Ex) Determine the nature of the roots of these equations:
x2  9 x  7  0
4 x 2  36 x  81  0
Determining An Equation From A Graph
Find the equation for the quadratic below in (i) vertex, (ii) standard and (iii) factored form
Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Trigonometry
Right Triangle Trigonometry
B
Pythagorean Theorem
a2 + b2 = c2
Trigonometric Ratios (SOH CAH TOA)
sin A=OPP
cos A = ADJ
tan A = OPP
HYP
HYP
ADJ
c
a
C
for example to find an angle from sin A = OPP/HYP
A = sin-1(OPP/HYP)
A
b
Non Right Triangle Trigonometry
C
a
a
b
A
Sine Law
c
sin A
B
Which Path to Choose?

b
sin B

c
sin C
Cosine Law
a 2  b 2  c 2  2bc cos A
Examples
Triangle
B
What Law?
Pythagorean Theorem | SOH CAH TOA |
Sine Law | Cosine Law
If I was finding the length of side a, I would use
25 cm
a
________________________________
20°
A
40 cm
C
P
If I was finding the length of side p, I would use
________________________________
124
74
Q
R
p
If I was finding the length of side x, I would use
________________________________
M
If I was finding the value of angle K, I would use
83
________________________________
L
74
110

K
Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Trigonometric Functions
Period Functions
Draw the graph of the walk on the grid provided below:
CBR
1m
Hula-Hoop
Modeling the Sine Function
Draw the graph of y = sin(x) on the grid provided below:
Transformations of y = sin(x)
y  a sin( x  c)  d
Ex) Graph the following: y = 2 sin(x – 60o) – 3
Ex) Find TWO possible equations for the graph below.
Equation 1:
Equation 2:
 What are the transformations that have happened to this sine graph?
Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Exponential Functions
Identifying Models: Linear, Quadratic and Exponential
Identify the type of model the data fits.
x
1
y
First
Differen Second
ces Differen
1
ces
x
1
y
First
Differe Second
0 nces Differen
ces
x
0
y
First
Differe Second
2 nces Differen
ces
2
3
2
1
1
4
3
5
3
4
2
8
4
7
4
9
3 16
5
9
5 16
4 32
Graphs of Exponential Functions
Exponential Growth
Exponential Decay
Exponential Growth and Decay Problems
Ex 1) A new form of bacteria doubles in population every 20 minutes. Initially there were 3 bacteria.
a. Write an equation to represent this problem.
b. Use your equation to find the number of bacteria after 10 hours.
Ex 2) You have invested $1000 in the bank. The bank pays you 3.4% of interest per year.
c. Write an equation to determine the amount of money in your account after x years
d. How much money will you have in the bank after 15 years?
Ex 3) The half-life of radioactive radium is 7 days. If you originally had 224 mg of this radioactive
substance, how much would be present after one week?
Laws of Exponents
Law
Example
x1 = x
61 = 6
x0 = 1
70 = 1
x-1 = 1/x
4-1 = 1/4
xmxn = xm+n
x2x3 = x2+3 = x5
xm/xn = xm-n
x4/x2 = x4-2 = x2
(xm)n = xmn
(x2)3 = x2×3 = x6
(xy)n = xnyn
(x2y)3 = x6y3
(x/y)n = xn/yn
(x/y)2 = x2 / y2
x-n = 1/xn
x-3 = 1/x3
1/x-n = xn
1/x-3 = x3
Exam Review Sheet
Grade 11 Functions and Applications
MCF3M1
Financial Mathematics
Simple Interest
I = Prt
IInterest Earned
PPrinciple
rInterest Rate in decimal (i.e. 4%=0.04)
tThe length of the investment in years (i.e.36 months, t=3)
Compound Interest
A = P(1 + i)n
or
P = A(1 + i)–n
A=FV=Amount or FV
P=PV=Present Value (What you put down today. This value is “-“ on the graphing calculator)
i=interest rate per compound period (i.e. 4%/a compounded monthly: i=0.04/12)
n=number of compounds total (i.e. you invest for 3 years compounded semi-annually:
n=3x2=6compounds)
Compound Interest using the TVM Solver
Press Menu8:Finance1:Finance Solver
Time in years multiplied by compounding periods
Interest rate in percent as stated in question
Present Value (value is Negative if money is given)
PMT stays 0
Future Value
P/Y  Compounds per Year
C/Y Compounds per Year
Annuities using the TVM Solver
Press Menu8:Finance1:Finance Solver
Total Number of Compounds
Interest rate in percent as stated in question
Present Value (value is Negative if money is given)
PMT-Payment per compound period
Future Value
P/Y  Payments per Year
C/Y Compounds per Year
Quadratics