Plaintext
Math Precalculus (12H/4H) Review
CHSN Review Project
Contents
Functions 3
Polar and Complex Numbers 9
Sequences and Series 15
This review guide was written by Dara Adib. Prateek Pratel checked the “Polar and Complex Num-
bers” chapter on page 9 for errors.
This is a development version of the text that should be considered a work-in-progress.
This review guide and other review material are developed by the CHSN Review Project.
Copyright © 2008-2009 Dara Adib. This is a freely licensed work, as explained in the Definition of Free
Cultural Works (freedomdefined.org). It is licensed under the Creative Commons Attribution-
Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons.
org/licenses/by-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite
300, San Francisco, California, 94105, USA.
This review guide is provided “as is” without warranty of any kind, either expressed or implied.
You should not assume that this review guide is error-free or that it will be suitable for the particular
purpose which you have in mind when using it. In no event shall the CHSN Review Project be
liable for any special, incidental, indirect or consequential damages of any kind, or any damages
whatsoever, including, without limitation, those resulting from loss of use, data or profits, whether or
not advised of the possibility of damage, and on any theory of liability, arising out of or in connection
with the use or performance of this review guide or other documents which are referenced by or
linked to in this review guide.
2
Functions
This chapter was designed for a test on functions administered by Jeanine Lennon to her Math 12H
(4H/Precalculus) class on January 4, 2008.
Definitions
function relation in which each first coordinate (usually x value) cooresponds to only one last coor-
dinate (usually y value); passes vertical line test
vertical line test test on a graph to determine if a relation is a function
domain set of all first coordinates (usually x values)
range set of all last coordinates (usually y values)
restricted domain values for which the function is undefined
i.e. x | x ∈ R, x 6= 0, x 6= ±5
inverse of a relation reversed domain and range of a relation
R−1 = {(y, x)}
one-to-one function function whose inverse is also a function; passes both vertical and horizontal
line tests
periodic function function in a cycle such that f(x + p) = f(x) where p represents period
amplitude 21 (max − min) of a periodic function
frequency number of cycles per 360◦ (degrees) or 2π (radians)
period duration of one cycle
composite function (composition of functions) application of one function on the result of other
function
asymptote line that a graph approaches, but never intersects
Operations on Functions
Explanation
(f + g)(x) is equivalent (and equal) to f(x) + g(x).
f f(x)
g (x) is equivalent (and equal) to g(x) , g(x) 6= 0 (to keep the denominator from being zero).
3
x f(x) g(x)
−2 7 3
−1 3 1
0 8 0
1 4 1
Table 0.1: common x values
x f(x) g(x) f
g
−2 7 3 7
3
−1 3 1 3
0 8 0 undefined
1 4 1 4
Table 0.2: y values for the operation gf
For All Common Values
On the other hand, f + g means finding f(x) + g(x) for all common x values. The answer would be
displayed as a set of ordered pairs. For example: {(a, b), (c, d), (e, g)...}.
It should be understood that operations on functions can only occur when x is in the domain of both
functions.
Example
Table 0.1 consists of common x values between f(x) and g(x). Other values may be provided, but
only the common x values are important.
Table 0.2 adds a fourth column, which contains the y values for the operation gf . Since 80 is undefined,
it will not be present in the answer.
Answer {(−2, 37 ), (−1, 3), (1, 4)}
Composite Functions
A composition of functions (composite function) consists of the application of one function on the
result of other function. In other words, a function is applied to another function. After a value
(usually y value) is determined by one function, it is substituted into the other function (usually x
4
Reflection Coordinates Function
rx−axis (x, y) (x, −y) y = −f(x)
ry−axis (x, y) (−x, y) y = f(−x)
ry=x (x, y) (y, x) x = f(y)
RO (x, y) (−x, −y) y = −f(−x)
Table 0.3: reflection
value). The functions may or not be commutative, so the order of the functions in the composition
should be taken into account.
f(g(x)) and (f ◦ g)(x) are equivalent. In both cases, g(x) is determined first and the result is plugged
into the x value of f(x).
If you find f(g(x)) to be easier, thank the mid-20th century mathematicians that came up
with this notation after determining that (f ◦ g)(x) was too confusing.
Reflections and Symmetry
Reflection
When functions are reflected over a certain line or point (i.e. x-axis or origin), coordinates ((x, y)) of
points in the function and the function itself (y = f(x)) are affected as seen in Table 0.3.
Symmetry
Functions symmetric over a certain line or point (i.e. x-axis or origin) contain both (x, y) and the
coordinates shown in Table 0.3.
Even and Odd Functions
even functions functions symmetric over the y-axis
odd functions functions symmetric over the origin
Determining Symmetry
To algebraically determine symmetry over a certain line or point, replace the values listed below.
Then, simplify the equation and determine if the two equations are equivalent. If the equations are
equivalent, the graph is symmetric over the specified line or point.
x-axis negate the y values
y-axis negate the x values
y=x reverse x and y values (substitute the x and y values with each other)
origin negate both the x and y values
5
Example
Equation of graph: x2 + xy = 4
x-axis
1. Negate y values: x2 − xy = 4
2. Simplify, if necessary (already simplified): x2 − xy = 4
3. Compare with graph equation: not equivalent
4. Not symmetric over the x-axis
y-axis
1. Negate x values: (−x)2 − xy = 4
2. Simplify, if necessary: x2 − xy = 4
3. Compare with graph equation: not equivalent
4. Not symmetric over the y-axis
y=x
1. Reverse x and y values: y2 + yx = 4
2. Simplify, if necessary: y2 + xy = 4
3. Compare with graph equation: not equivalent
4. Not symmetric over the line y = x
origin
1. Negate x and y values: (−x)2 + (−x)(−y) = 4
2. Simplify, if necessary: x2 + xy = 4
3. Compare with graph equation: equivalent
4. Symmetric over the origin
Periodic Functions
Definitions
See definitions on page 3.
Effects of Different Equations
The bulleted examples below are compared with y = f(x)
• y = 2 × f(x) doubles the amplitude
• y = f(2x) doubles the frequency; halves the period (periodic shrink)
• y = f( 21 x) halves the frequency; doubles the period (periodic stretch)
6
Other Effects
The following text in this section may be incorrect.1 The “ceiling function” is not a periodic
function. The bulleted examples below are compared with y = dxe.
• y = 2dxe doubles y values
• y = d2xe doubles x values
Inverses of Relations
A function is a relation, but a relation does not necessarily have to pass the vertical line test.
Definitions
See definitions on page 3.
Determining the Inverse of a Relation
Reverse the x and y values of an equation and if necessary, solve for y. In this way, the inverse of
y = 2x + 1 will be y = x− 1
2 .
Other Situations
You may not need to necessarily determine the equation of a relation’s inverse. For example, one can
use a y value on a chart of values of a relation instead of the x value of its inverse.
Translations of Functions
To translate a function, values are either added or subtracted to part of a function. The examples
below are compared with y = f(x).
• y = f(x) + a moves the graph a units up; add a to y values
• y = f(x + b) moves the graph b units to the left; subtract b from x values
The examples below are compared with y = |x|
• y = |x| + a moves the graph a units up; add a to y values
• y = |x + b| moves the graph b units to the left; subtract b from x values
Keep in mind that absolute value will cause the graph of an example like y = ||x| − c| to be shaped
like a W (assuming c is positive).
1 Corrections and other feedback sent to the author or to the CHSN Review Project are greatly appreciated.
7
Asymptotes
Definition
See definitions on page 3. For more information on asymptotes, see the Math Calculus Review.
Determining Asymptotes
To determine asymptotes algebraically, you must solve for either x or y—isolate the variable.
Vertical Asymptotes
Solve the equation for y and determine the x value where the denominator of the fraction would
equal zero.
Horizontal Asymptotes
Solve the equation for x and determine the y value where the denominator of the fraction would
equal zero.
Important Information
Some equations may not have any asymptotes. These include, but are not limited to, linear equations.
Beware, a linear equation may be obfuscated to appear to be another type of equation. Therefore, re-
member to factor numerators and denominators and attempt to simplify fractions as much as possible
before determining asymptotes. Also understand that 00 is not a valid equation or asymptote.
8
Polar and Complex Numbers
This chapter was designed for a test on polar and complex numbers administered by Jeanine Lennon
to her Math 12H (4H/Precalculus) class on March 5, 2008. Prateek Pratel checked this chapter for
errors.
Polar Coordinates
Cartesian (rectangular) coordinates a point, P, on a plane is described in terms of x and y, where
x and y are the respective horizontal and vertical distances from the origin
Form: (x, y)
polar coordinates a point, P, on a plane is described by specifying the distance, r, from the origin
and the angle, θ, measured counter-clockwise from the positive x-axis to the line joining P to
the origin
Form: (r, θ)
coterminal angles angles that coincide (when placed in standard position); added or subtracted
multiples of 360◦ (including 360◦ )
reference angles way to simplify the calculation of the values of trigonometric functions in different
quadrants
Refer to Table 0.4, where θ is the angle and β is the reference angle.
Converting Coordinates
Polar-to-Cartesian
Polar coordinates are expressed in the form (r, θ), while Cartesian coordinates are expressed in the
form (x, y). The two following equations may be used to find the appropriate values for x and y
based on r and θ, which may be substituted into (x, y). There is no requirement for r to be positive.
Quadrant Reference Angle (β)
I β=θ
II β = 180◦ − θ
III β = θ − 180◦
IV β = 360◦ − θ
Table 0.4: reference angles
9
x = r cos θ
y = r sin θ
Therefore, the form for conversion from polar coordinates to Cartesian coordinates is expressed as
(r cos θ, r sin θ).
It should be noted that the polar and Cartesian coordinates must be in the same quadrant (hint:
reference angles).
Explanation
These formulas are derived by inscribing a right triangle with hypotenuse r in a circle on a coordinate
plane, such that θ is an angle from the positive x-axis to the line joining a point to the origin. If the
point assumes the coordinates (x, y), the following two equations are valid.
x
cos θ =
r
y
sin θ =
r
By multiplying both sides of the equations by r, the method for conversion is derived.
x = r cos θ
y = r sin θ
Cartesian-to-Polar
Cartesian coordinates are expressed in the form (x, y) , while polar coordinates are expressed in the
form (r, θ). The two following equations may be used to find the appropriate values for r and θ based
on x and y, which may be substituted into (r, θ).
q
r = x2 + y2
y
θ = tan−1 , x 6= 0
x
Therefore, the form for conversion from Cartesian coordinates to polar coordinates is expressed as
( x2 + y2 , tan−1 yx ), x 6= 0.
p
It should be noted that the Cartesian and polar coordinates must be in the same quadrant (hint:
reference angles). If x = 0, the second equation cannot be used to find θ since it will be undefined.
Instead, one must determine if θ is 90◦ or 270◦ , depending on whether y is positive or negative,
respectively.
Explanation
These formulas are derived by inscribing a right triangle with hypotenuse r in a circle on a coordinate
plane, such that θ is an angle from the positive x-axis to the line joining a point to the origin. If the
point assumes the coordinates (x, y), the following two equations are valid.
r2 = x2 + y2 (Pythagorean Theorem)
y
tan−1 , x 6= 0
x
10
The square root of both sides is taken from the first equation based on the Pythagorean Theorem.
p
r = ± x2 + y2
The negative value of r can be rejected for practical purposes.
p
r = x2 + y2
The second equation can be rewritten to isolate θ on one side of the equation.
y
θ = tan−1 , x 6= 0
x
Polar Inequalities
Polar inequalities are fairly simple. They are expressed in the following form.
a≤r≤b
c≤θ≤d
a represents the smallest value of r in the range, while b represents the largest value of r in the range
c represents the smallest angle of θ in the range, while d represents the largest angle of θ in the range
Etiquette
For math etiquette, one should follow the following guidelines when writing polar inequalities.
a≥0
b>0
0◦ ≤ c < 360◦
0◦ < d ≤ 360◦
In other words, the radii should be positive, while the angles should be between 0◦ and 360◦ .
Graphing Polar Equations
Graphing
Manually
A polar equation can be graphed by hand. This work is tedious, and requires the creation of a table of
values. Instructions for graphing manually are not included, as the test this review report is designed
for does not require manually calculating values and graphing.
11
Graphing Calculator
A graphing calculator must be placed in polar mode. Instructions for modern Texas Instruments (TI)
graphing calculators follow, though they may be altered for use with other graphing calculators as
well.
1. Change “Mode” to radians or degrees as necessary
2. Change “Mode” from “FUNC” (function) to “POL” (polar)
3. Adjust “Window” as necessary
“Steps” (of θ) correspond to the number of calculations done to graph the equation. A lower step
means more calculations, while a higher step means less calculations. More calculations mean more
accuracy in graphing (less distortion), but longer periods to graph. Less calculations mean less accu-
racy in graphing (more distortion), but shorter periods to graph.
Polar-to-Cartesian
Generally, a polar equation is written in the form r = a sin θ or r = a cos θ, while an equation in the
Cartesian coordinate plane is written in the form x2 + y2 = ay or x2 + y2 = ax. The two following
equations may be used to find the appropriate coefficients of x and y based on the coefficients of cos θ
and sin θ.
x
cos θ =
r
y
sin θ =
r
The appropriate coefficient of x2 + y2 may be determined based on the coefficient of r2 .
r 2 = x 2 + y2
Substitution is key in these cases.
Example
An example of a conversion from a polar equation to a Cartesian equation follows. r = 2 cos θ is the
polar function, while x2 + y2 = 2x is the determined equation (in the Cartesian coordinate plane).
r = 2 cos θ
x
r=2
r
r2 = 2x
x2 + y2 = 2x
No further work is required on this determined equation.
Complex Numbers
The rectangular form of a complex number is a + bi. The polar form of a complex number is r cis θ.
r cis θ is an abbreviation for r(cos θ + i sin θ). The same rules and methods to convert from polar
form to Cartesian form and vice-versa apply to complex numbers in polar form as they apply to real
numbers in polar form.
12
Conversions
If a complex number is represented as an ordered pair, represent the complex number in the other
form as an ordered pair. If a complex number is not represented as an ordered pair, do not represent
the complex number in the other form as an ordered pair.
Cartesian-to-Polar
The Cartesian complex number a + bi can be expressed as the following polar number:
p b b p b
a2 + b2 [cos(tan−1 ) + i sin(tan−1 )], or abbreviated as a2 + b2 cis (tan−1 ).
a a a
It should be noted that the Cartesian and polar coordinates must be in the same quadrant (hint:
reference angles).
Explanation The absolute value of a complex number, expressed as |z|, represents the length of the
vector of a graphed complex number.
√
|z| = a2 + b2 (Pythagorean Theorem)
√
Since the length of the vector is equivalent to r (the radius), r = a2 + b2 .
Polar-to-Cartesian
The polar complex number r cis θ can be expressed as the Cartesian number r cos θ + (r sin θ)i. It
should be noted that the polar and Cartesian coordinates must be in the same quadrant (hint: refer-
ence angles).
Operations
Answer in exact form (i.e. with square roots) when specified or when using special angles (chart
available from the CHSN Review Project). Otherwise, decimal rounded form may be used.
Cartesian
FOIL may be used to multiply two complex numbers in Cartesian form.
In general, if z1 = a1 + b1 i and z2 = a2 + b2 i, then z1 × z2 = a1 a2 + a1 b2 i + a2 b1 i − b1 b2 . The last
term is negative and does not contain any imaginary numbers since i2 = −1.
Polar
The majority of operations can only be done in polar form. If a Cartesian complex number is provided
and a Cartesian (rectangular) answer is requested, the provided Cartesian complex number must be
first converted to polar form, the operation done, and finally the resulting polar complex number
converted back into Cartesian form.
Multiplication and Division In general, if z1 = r1 cis θ1 and z2 = r2 cis θ2 , then z1 × z2 = r1 ×
r2 cis (θ1 + θ2 ). Likewise, under the same conditions, zz12 = rr12 cis (θ1 − θ2 ).
13
De Moivre’s Theorem Let n be any integer, then (r cis θ)n = rn cis nθ.
To determine the roots of a complex number, n can be substituted with x1 , where x represents the xth
root . A complex number raised to the power of x1 will have x number of roots. This is due to the fact
that equivalent complex numbers in polar form (with coterminal angles) will have reference angles
360◦ apart. When 360◦ is divided by x, x unique angles will be created (between 0◦ and 360◦ ), with
360◦/x as the difference in the angles of the roots. As a result, a non-zero complex number will have
two square roots, three cube roots, four fourth roots, etc.
1 1 θ + 360K
In general, (r cis θ) n = r n cis , where K = 0, 1, 2, 3...n − 1.
n
14
Sequences and Series
This chapter was designed for a test on sequences and series administered by Jeanine Lennon to her
Math 12H (4H/Precalculus) class on March 14, 2008. This chapter refers to the position of a term as
n, and refers to the total number of terms in a sequence or series as k.
Arithmetic Sequences
Arithmetic sequences (also known as lists and progressions) have a constant difference. Terms in-
crease or decrease by adding or subtracting a fix quantity. The constant difference is known as d,
which is the difference between any two consecutive terms.
Arithmetic sequences use the following equation to determine the nth term.
tn = t1 + (n − 1)d (provided on test unlabeled)
t1 represents the first term, while n represents the position of a term (i.e. first, second, third).
Rewritten in the following equation, one may determine the number of terms in an arithmetic se-
quence.
tn − t1
n= +1
d
Note Multiples of a number are an arithmetic sequence.
Example
Sequence 2, 5, 8, 11, 14, 17
The above sequence is an example of an arithmetic sequence, and d = 3.
Arithmetic Mean
Sequence . . . a, x, b. . .
If the above sequence is given, x is considered the arithmetic mean of a and b. Therefore, x = a+ b
2 .
Explanation
If one attempts to solve determine the value of x based on a and b, it would be determined that
d = x − a and d = b − x. Through substitution, x − a = b − x. This can in turn be rewritten as
x = a+ b
2 , the same as the arithmetic mean of a and b.
15
Recursive Arithmetic Sequences
recursive equation an equation for sequences in which the value of a term is defined based on the
preceding term(s).
Recursive formulas must have the following:
1. an initial condition, t1
2. a recursive equation
A recursive arithmetic formula is a different method of expressing an arithmetic sequence, instead of
using standard form.
Standard form tn = t1 + (n − 1)d
Recursive form tn = d + tn−1
Geometric Sequences
Geometric sequences have a constant ratio. All terms are connected by a common ratio. The constant
ratio is known as r, which is the quotient determined by the division of any two consecutive terms.
Geometric sequences use the following equation to determine the nth term.
tn = t1 × rn−1 (provided on test unlabeled)
t1 represents the first term, while tn represents the nth term.
Note Geometric sequences with alternating signs of terms have negative r (−r) values. Con-
sequently, when a term and its position in a sequence (n) are provided and n is even,
multiple values for r may be determined. In this case, r and −r (±r) may both be correct
values for r.
Example
Sequence 4, 8, 16, 32, 64, 128
The above sequence is an example of a geometric sequence, and r = 2.
Geometric Mean
Sequence . . . a, x, b. . .
√
If the above sequence is given, x is considered the geometric mean of a and b. Therefore, x = a × b.
Explanation
If one attempts to solve determine the value of x based on a and b, it would be determined that r = ax
and r = bx .
√
Through substitution, ax = bx . This can in turn be rewritten as x = a × b, the same as the geometric
mean of a and b.
16
Series
series the sum of a sequence, list, or progression
Sk represents the summation of k number of terms in a sequence.
The formulas follow, and are different for arithmetic and geometric series.
Arithmetic Series
The following formula can be used to determine the value of an arithmetic series, Sk .
k(t1 + tk ) n(t1 + tn )
Sk = (provided unlabeled on test as Sn = )
2 2
k represents the total number of terms in a sequence or series.
Geometric Series
The following formula can be used to determine the value of a geometric series, Sk .
t (1 − rk ) t (1 − rn )
Sk = 1 (provided unlabeled on test as Sn = 1 )
1−r 1−r
k represents the total number of terms in a sequence or series.
Infinite Geometric Series
The sum of an infinite geometric series would be expressed by the following statement.
S = t 1 + t 1 r + t 1 r2 + t 1 r3 . . .
Infinite geometric series either converge or diverge.
Converging Infinite Geometric Series
Converging infinite geometric series have a sum, and the value of r in the series fits in −1 < r < 1. In
other words, r must be between −1 and 1, non-inclusive.
The following formula would be used to determine the sum of the infinite geometric series.
t
S = 1 , −1 < r < 1 (provided on test, possibly without domain)
1−r
Interval of Convergence In some cases r, may be provided or can be determined in terms of an-
other variable. For example, r may equal x + 1. In this case, one simply substitutes r with x + 1 into
the domain provided for converging infinite geometric series.
−1 < r < 1
−1 < x + 1 < 1
To isolate the variable, x, one may subtract 1 from all sides of the relation.
−2 < x < 0
It has now been determined that if r = x = 1, the interval that convergence will occur (resulting in a
non-infinite sum) is between −2 and 0, non-inclusive.
17
Diverging Infinite Geometric Series
If the value of r in the infinite geometric series does not fit in −1 < r < 1, then the sum of the series is
infinity and is said to diverge.
Sigma Notation
Summations may also be expressed through sigma notation.
X
m
xi
i =n
In the above expression of sigma notation, i is the index of summation, n is the lower limit of summa-
tion, and m is the upper limit of summation. It is possible than m may be infinity (∞), consequently
resulting in no limit on summation.
The sigma notation specified above would be equivalent to the following expression.
xm + xm+1 + xm+2 + · · · + xn−1 + xn
Sigma notation can be used for both arithmetic and geometric series
Arithmetic Series
Sigma notation has the following form for arithmetic series.
X
tk
dn
n =t 1
The constant difference is known as d, which is the difference between any two consecutive terms. k
represents the total number of terms in a sequence or series. n represents the position of a term (i.e.
first, second, third). It is possible than tk may be infinity (∞), consequently resulting in no limit on
summation.
Geometric Series
Sigma notation has the following form for geometric series.
X
tk
(t1 )(r)n−1
n =t 1
The constant ratio is known as r, which is the quotient determined by the division of any two consec-
utive terms. k represents the total number of terms in a sequence or series. n represents the position
of a term (i.e. first, second, third). It is possible than tk may be infinity (∞), consequently resulting in
no limit on summation.
18
Alternating Signs
Arithmetic series with alternating signs of terms are in fact both arithmetic and geometric series, with
d defined and r = −1. As a result, the format used for arithmetic sequences is represented by the
following expression.
X
tk
(dn)(−1)k−1
n =t 1
19