Download Chapter 1: Introducing Functions

Chapter 1: Introducing Functions
1.1 Functions: Concept and Notation
Domain: the set of all possible input or “x” values … x is the domain
ex: D  x x  3, x  R in words this means: the domain is the set of all x values such that x is greater
than or equal to 3 and x is an element of (or belongs to the set of) Real numbers.
Range: the set of output or “y” values …. y is the range
ex: R  y y  0, y  I  this would mean: the range is the set of all y values such that y is less than or
equal to 0 and y is an element of Integers
f(x) is called function notation and represents an expression for determining the value of the function f for any
value x. f(3) means what is the value of the function when x = 3 . f(x) is another way of representing y.
Function: A function is a relation such that for every value of x there is one and only one value for y.
Vertical line test: determines if the graph is a function. If a vertical line crosses the graph at more than one
position, the relation is NOT a function.
see example page 7
1.2 Function Notation
Remember that something like f(2) simply means what is the value (or height) of a function when x = 2.
All you need to do is substitute 2 in for all values of x and solve.
1.3 Parent Functions 1.4 Domain and Range
You should be able to graph all parent functions for the basic functions shown on page 28. The domain and
range of a function depend on the type of function. For example you can not take the square root of a negative
number so you need to make sure that the domain of a radical ensures that you do not have a negative number.
Also, for rational functions you can not have a zero in the denominator so that makes a restriction on the
domain. What makes the denominator equal to zero becomes what x can NOT be!
See page 34 “Need to Know”
1.5 Inverse Functions
f (x ) is the notation for a function and f 1 ( x) is the notation for the inverse function
How to find an inverse function: change f(x) to y then switch the variables (x and y) and solve for y. Now you
can replace y by f 1 ( x) and you have found the inverse function.
Neat properties of inverse functions: 1. The graph of an inverse function is the reflection of the original
function about the line y = x. 2. If you are given the co-ordinates of a function to find the inverse just switch
the x and y coordinates and you have its inverse. 3. The domain of the original function becomes the range of
the inverse and the range of the original function becomes the domain of the inverse.
Horizontal line test: determines if the inverse will be a function. If a horizontal line crosses the graph at more
than one position, the inverse is not a function.
See key ideas and Need to Know on page 46
1.6 Transformations and Function Notation
1

y  af k x  d   c ex: y  3 f  x  2  4 What you need to remember when performing
2

transformations is that the changes to the x co-ordinate all happen INSIDE the brackets and you perform the
opposite operation than you would intuitively think you should use (x’s are WEIRD). The changes to the y coordinates are all OUTSIDE the brackets and you perform the operations as you see them. Remember that
BEDMAS rules still apply here. Find the co-ordinates of the original function that you need to use (first point,
last point and places where there is a change in direction, then for the above example: you would take the x
co-ordinate of the original function f(x) divide by ½ and subtract 2 and for the y co-ordinate you would take the
y co-ordinate of the original function f(x) multiply is by (-3) and subtract 4. EASY AS PIE!
x
The mapping rule becomes  x, y   (ay  c,  d ) See Need to know pages 58 and 69 AND the big handout
k
Chapter 2: Equivalent Algebraic Expressions
You should be able to simplify algebraic expressions involving polynomials that are added, subtracted,
multiplied or divided.
It is important, as well, to be able to factor . See the handout on factoring that I gave you for all the types of
factoring that you need to know. It is especially important to be able to factor when multiplying, dividing
adding and subtracting rational expressions in order to simplify and be able to find the common denominators
when adding and subtracting rational expressions.
See examples pages 111, 119,
2.6 Multiplying and Dividing Rational Expressions
The same rules apply as above only you must make sure to invert and multiply the second rational expression
and state any new restrictions (i.e. restrictions for the variables in the top AND the bottom of the second
expression) see examples 2-4 pages 118-120.
2.7 Adding and Subtracting Rational Expressions
Probably the most difficult as you need to find the lowest common denominator (LCD) before you can simplify.
Remember that once you find the LCD that you need only to work with the numerators. When simplifying
rational expressions we must remember to only divide out terms that are being multiplied. It is a common
mistake to “cancel” terms that are being added and subtracted. Sometimes you will need to factor the
numerator and/or the denominator to simplify. You may also be asked to state the restrictions on the variables
which means you are looking for the value of the variable that makes the denominator equal to zero.
See examples 2-4 pages 126, 127
Chapter 3: Quadratic Functions
3.1 Properties of Quadratic Functions
Quadratics can be in standard from, factored form and vertex form, each one gives us different information.
See Key ideas and Need to Know on page 145
3.2 Extending Algebra Skills: Completing the Square
2
Completing the square gives you the vertex form of a quadratic equation y  ax  h   k
See page 148, 149 examples 1 for a complete description of the method.
Finding the maximum or minimum value of a function by locating the vertex has many important applications.
You have a maximum if the parabola is concave down (a < 0) and a minimum if the parabola is concave up
( a > 0). The co-ordinates of the vertex are (h, k) from the equation above. Notice the sign of h changes for the
vertex. ex: y  2( x  3) 2  4 would be concave down (a = -2) so it has a maximum value of 4 when x equals
3 [ the vertex is at (3,4)].
3.3 The Inverse of a Quadratic Function
Given the graph of any parabola (whether concave up or concave down) we recognize that because the parabola
does NOT pass the horizontal line test which means that its inverse will NOT be a function it becomes
necessary for us to restrict the domain of the original function so that its inverse will be a function. In order to
do this we must know where the vertex is as this is the point that you will use for the restriction. We can then
find the inverse of each side of the parabola and the inverse will be a function.
see examples 2 page 157 for the how to!
3.4 Operations with Radicals
Adding, subtracting LIKE radicals (same number under the radicand), multiplying radicals as well as writing
radicals as mixed radicals. Example: 48  16  3  4 3 Radicals give you EXACT solutions.
3.5 Solving quadratic equations/ 3.6 Zeros of Quadratic Functions
Zeros of a function mean the x-intercepts or the roots of the equation.
2
standard form of a quadratic function: f ( x)  ax 2  bx  c
vertex form: y  ax  h   k
factored form of a quadratic function: f ( x)  a( x  p)( x  q)
In standard form you must use the discriminant b 2  4ac to determine the number of zeroes, zeros are clearly
visible as p and q when in factored form, and in vertex form you look at the position of the vertex and whether
or not the parabola is concave up or concave down. see need to know page 184 and examples in this section
pages 180 - 183
3.7 Families of Quadratics/ 3.8 Linear-Quadratic Systems
The main idea here is that you can draw many different quadratics through 2 zeros. The “a” value for the
quadratic determines the exact shape of the parabola. The “a” value can be found by substituting in a point on
the graph and using the factored form of a quadratic and its zeros. See page 191 for need to know information .
Linear quadratic systems is similar to linear systems that you studied in gr 10. The difference is that you are
working with a line and a quadratic instead of 2 lines. You set the line equal to the quadratic and solve.
See Need to Know on page 198
Chapter 4: Exponential Functions
4.2 Integer exponents
You can make a negative exponent positive by writing it as 1 over the number.
1
1 1
2
Ex: 4 
or 3  2  2 
if the number is a fraction you can “flip” it ex:  
4
9
3
3
See page 221 Need to know and examples page 220
1
1

3
2
4.3 Rational Exponents
1
1
x2  x ,
2
3
x 
 x
3
2
1
x 3  3 x , so in general x n  n x
take the cube root of x, then square it ;
m
 
m
3
5
x  (5 x ) 3 take the fifth root of x, then cube it
2
so in general x n  n x
ex 8 3  (3 8 ) 2 , which means the cube root of 8 (equals 2) squared, equals 4
see examples 3-6 on pages 225-227
4.4 Simplifying Algebraic expressions involving exponents.
You need to know your exponent rules. Try the examples on page 234.
4.5 Properties of Exponential Functions./4.6 Transformations
See page 242
See page 251 and the handout that we did with exponentials and transformations
4.6 Applications with Exponentials.
Growth and decay models. Working with equations
Chapter 5: Trigonometric Ratios
5.1 Trigonometric Ratios of Acute Angles
opp
opp
adj
hyp
hyp
adj
tan  
inverse functions: csc  
sin  
sec  
cos 
cot  
adl
hyp
adj
hyp
opp
opp
definitions: initial arm, terminal arm, standard position, positive angle, negative angle, principal angle, related
acute angle.
You should be able to sketch a sine, cosine function.
5.2 Using Special Triangles to Determine Exact values
When asked to determine an EXACT value you are being asked to use the “special” triangles.
These are the 45,60and 30 angles. You should be able to construct these without memorizing them by
using an isosceles triangle for the 45 angle and an equilateral triangle for both the 60and 30 angles.
See examples in Key ideas and Need to Know p 286
5.3 Exploring Trigonometric Ratios for angles greater than 90 degrees
First you need to review the CAST rule. example: sin   0.4255 0    360 What is  equal to? You
know that there will be two solutions because sine is positive in quadrant I and II. Using the inverse trig
function on your calculator you would find   25.2 which would give you one of the angles, the other would
be at 180  25.2  154.8
Try another one: cos x  0.655 Using your calculator you get 130.9 . That makes sense because cosine is
negative in quadrant II but where else is it negative? Quadrant III! So, to find the other angle we take
180  130.9  49.1 to first find the related angle and then add 180  49.1  229.1 Or you could take
360  130.9  229.1 . Which will also give you the third quadrant solution.
See the handout I gave you with more examples
5.5 Trigonometric Identities
See “key ideas” on page 309 for a complete description of the identities you should know and how to solve an
identity. The basic idea is to make one side equal to the other buy using identities. Always break down into
sine and cosine where possible, sometimes you need to factor, sometimes find a common denominator.
5.6 Sine Law/ 5.7 Cosine Law
Check out the formulae in “key ideas” for the sine and cosine law and when you should use them on page 317
and 325.
This section also introduced the “ambiguous case” which simply means that some times you aren’t given
enough information about the triangle and it is possible to draw it in two different ways. This is clearly
explained on page 312/313.
5.8 Solving Trigonometry Problems in Two and Three Dimensions
When solving word problems always check to see if you can use basic trig ratios (where there is a right
triangle). It is also advisable to add any information to your drawing that you can. Sometimes you may be able
to find another angle using the rule that all angles in a triangle add up to 180. When given the choice, use the
sine law as it is easier to use than the cosine law.
With 3-dimensional problems separate the information into horizontal and vertical triangles.
See example 1 page 328.
Chapter 6: Sinusoidal Functions
6.1 Periodic Functions
A cycle of a graph is the smallest self-repeating graph. The period is the length of one complete cycle.
The axis of the curve is the horizontal line that is halfway between the maximum and minimum value of a
periodic curve. The amplitude is the measurement of the height of the curve from the horizontal line. The
amplitude is always stated as a positive number.
6.4 Exploring Transformations
y  a sin k (  d )  c and y  a cos k (  d )  c are transformations of y  sin  and y  cos  .
Just like in transformations of functions if a>0 a vertical stretch occurs, if 0<a<1 a vertical compression occurs
and if a<0 you get a reflection in the  axis. +d causes a shift to the left d units, +c causes a vertical shift up c
units. The only tricky part is the “k” which determines the period of the function. For sine and cosine the
2
360
period becomes
or
You also have to be careful to factor out the k if it is inside the brackets.
k
k
See page 382/ 383
6.5 Modelling Periodic Phenomena
There are many examples of phenomena which follow a sine or cosine function. The moon phases, the rise and
fall of the ocean tides, hours of daylight etc. You can graph periodic sinusoidal data and then determine the
equation of the functions. See examples p 394.
Chapter 7 : Patterns of Growth: Sequences
7.1 Arithmetic Sequences:
Definitions: Sequence, finite sequence, infinite sequence recursive sequence, general term (p 416 )
See key ideas (p 423)
general term: t n  a  (n  1)d , where t n is the nth term
a is the first term or t1
see example 4 on pages 422
d is the common difference
7.2 Geometric Sequences : common ratio (p 426) see key ideas p 429
general term: t n  ar n 1 , where t n is the nth term
a is the first term or t1
see example 2 on pages 428
r is the common ratio
7.5 Arithmetic Series
Series: a series is the SUM of the terms of a sequence ex: S n  t1  t 2  t 3 ...  t n
n(t1  t n )
or if you know the first term, d and n(the # of terms) you can use:
2
n
S n  2a  (n  1)d  where a is the first term, d is the common difference
2
see examples # 2 and 3 on pages 450-451
The general term is: S n 
7.6 Geometric Series
The sum of the terms of a geometric sequence is a geometric series
t t
a r n 1
, r  1 where a is the first term, r is the
The general term is S n  n1 1 , r  1 or S n 
r 1
r 1
common ratio and n is the number of terms.
see examples #2-3 pages 456-457


This 6 page outline is to be used in conjunction with your notes, homework assignments
and tests to help you review for the final exam. It is not meant to be an all-inclusive
summary but another tool for you to use to prepare for the exam. Good luck.
N.B. The Finance sections will NOT be included on your exam. (YES!!)