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\begin_body

\begin_layout Title
Math Precalculus (12H/4H) Review
\end_layout

\begin_layout Author
CHSN Review Project
\end_layout

\begin_layout Publishers
\begin_inset Graphics
	filename ccbysa.png
	scale 50

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents

\end_inset


\end_layout

\begin_layout Standard
\SpecialChar \textcompwordmark{}

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\begin_layout Standard
This review guide was written by Dara Adib.
 Prateek Pratel checked the 
\begin_inset Quotes eld
\end_inset

Polar and Complex Numbers
\begin_inset Quotes erd
\end_inset

 chapter 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "cha:polar"

\end_inset

 for errors.
\end_layout

\begin_layout Standard
This is a development version of the text that should be considered a work-in-pr
ogress.
\end_layout

\begin_layout Standard
This review guide and other review material are developed by the CHSN Review
 Project.
\end_layout

\begin_layout Standard
Copyright © 2008-2009 Dara Adib.
 This is a freely licensed work, as explained in the Definition of Free
 Cultural Works (
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

freedomdefined.org
\end_layout

\end_inset

).
 It is licensed under the Creative Commons Attribution-Share Alike 3.0 United
 States License.
 To view a copy of this license, visit 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

http://creativecommons.org/licenses/by-sa/3.0/us/
\end_layout

\end_inset

 or send a letter to Creative Commons, 171 Second Street, Suite 300, San
 Francisco, California, 94105, USA.
\end_layout

\begin_layout Standard
This review guide is provided 
\begin_inset Quotes eld
\end_inset

as is
\begin_inset Quotes erd
\end_inset

 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 performanc
e of this review guide or other documents which are referenced by or linked
 to in this review guide.
\end_layout

\begin_layout Chapter
Functions
\end_layout

\begin_layout Standard
This chapter was designed for a test on functions administered by Jeanine
 Lennon to her Math 12H (4H/Precalculus) class on January 4, 2008.
\end_layout

\begin_layout Section
Definitions
\begin_inset CommandInset label
LatexCommand label
name "cha:Definitions"

\end_inset


\end_layout

\begin_layout Description
function relation in which each first coordinate (usually 
\begin_inset Formula $x$
\end_inset

 value) cooresponds to only one last coordinate (usually 
\begin_inset Formula $y$
\end_inset

 value); passes vertical line test
\end_layout

\begin_layout Description
vertical
\begin_inset space ~
\end_inset

line
\begin_inset space ~
\end_inset

test test on a graph to determine if a relation is a function
\end_layout

\begin_layout Description
domain set of all first coordinates (usually 
\begin_inset Formula $x$
\end_inset

 values)
\end_layout

\begin_layout Description
range set of all last coordinates (usually 
\begin_inset Formula $y$
\end_inset

 values)
\end_layout

\begin_layout Description
restricted
\begin_inset space ~
\end_inset

domain values for which the function is undefined
\end_layout

\begin_deeper
\begin_layout Standard
i.e.
 
\begin_inset Formula ${\displaystyle x\mid x\in R,x\neq0,x\neq\pm5}$
\end_inset


\end_layout

\end_deeper
\begin_layout Description
inverse
\begin_inset space ~
\end_inset

of
\begin_inset space ~
\end_inset

a
\begin_inset space ~
\end_inset

relation reversed domain and range of a relation
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula ${\displaystyle R^{-1}=\{(y,x)\}}$
\end_inset


\end_layout

\end_deeper
\begin_layout Description
one-to-one
\begin_inset space ~
\end_inset

function function whose inverse is also a function; passes both vertical
 and horizontal line tests
\end_layout

\begin_layout Description
periodic
\begin_inset space ~
\end_inset

function function in a cycle such that 
\begin_inset Formula $f(x+p)=f(x)$
\end_inset

 where 
\begin_inset Formula $p$
\end_inset

 represents period
\end_layout

\begin_layout Description
amplitude 
\begin_inset Formula $\frac{1}{2}(max-min)$
\end_inset

 of a periodic function
\end_layout

\begin_layout Description
frequency number of cycles per 
\begin_inset Formula $360^{\circ}$
\end_inset

 (degrees) or 
\begin_inset Formula $2\pi$
\end_inset

 (radians)
\end_layout

\begin_layout Description
period duration of one cycle
\end_layout

\begin_layout Description
composite
\begin_inset space ~
\end_inset

function
\begin_inset space ~
\end_inset

(composition
\begin_inset space ~
\end_inset

of
\begin_inset space ~
\end_inset

functions) application of one function on the result of other function
\end_layout

\begin_layout Description
asymptote line that a graph approaches, but never intersects
\end_layout

\begin_layout Section
Operations on Functions
\end_layout

\begin_layout Subsection
Explanation
\end_layout

\begin_layout Standard
\begin_inset Formula $(f+g)(x)$
\end_inset

 is equivalent (and equal) to 
\begin_inset Formula $f(x)+g(x)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $\left(\frac{f}{g}\right)\left(x\right)$
\end_inset

 is equivalent (and equal) to 
\begin_inset Formula $\frac{f(x)}{g(x)}$
\end_inset

, 
\begin_inset Formula $g(x)\neq0$
\end_inset

 (to keep the denominator from being zero).
\end_layout

\begin_layout Subsection
For All Common Values
\end_layout

\begin_layout Standard
On the other hand, 
\begin_inset Formula $f+g$
\end_inset

 means finding 
\begin_inset Formula $f(x)+g(x)$
\end_inset

 for all common 
\begin_inset Formula $x$
\end_inset

 values.
 The answer would be displayed as a set of ordered pairs.
 For example: 
\begin_inset Formula $\{(a,b),(c,d),(e,g)...\}$
\end_inset

.
\end_layout

\begin_layout Standard
It should be understood that operations on functions can only occur when
 
\begin_inset Formula $x$
\end_inset

 is in the domain of both functions.
\end_layout

\begin_layout Subsubsection
Example
\end_layout

\begin_layout Standard
\noindent
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\noindent
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\begin_inset Formula $x$
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\begin_inset Formula $f(x)$
\end_inset


\end_layout

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</cell>
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\begin_inset Formula $g(x)$
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<row>
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\begin_inset Formula $-2$
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\end_layout

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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $7$
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\end_layout

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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $3$
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\end_layout

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</cell>
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<row>
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3$
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $1$
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $8$
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\begin_inset Text

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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $4$
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\begin_layout Plain Layout
\begin_inset Caption

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LatexCommand label
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\end_inset

common 
\begin_inset Formula $x$
\end_inset

 values
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:common-x-values"

\end_inset

 consists of common 
\begin_inset Formula $x$
\end_inset

 values between 
\begin_inset Formula $f(x)$
\end_inset

 and 
\begin_inset Formula $g(x)$
\end_inset

.
 Other values may be provided, but only the common 
\begin_inset Formula $x$
\end_inset

 values are important.
\end_layout

\begin_layout Standard
\noindent
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wide false
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

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\begin_inset Formula $\frac{f}{g}$
\end_inset


\end_layout

\end_inset
</cell>
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<row>
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-2$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $7$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{7}{3}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-1$
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $8$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
undefined
\end_layout

\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $4$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1$
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $4$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:y-values-for"

\end_inset


\begin_inset Formula $y$
\end_inset

 values for the operation 
\begin_inset Formula $\frac{f}{g}$
\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:y-values-for"

\end_inset

 adds a fourth column, which contains the 
\begin_inset Formula $y$
\end_inset

 values for the operation 
\begin_inset Formula $\frac{f}{g}$
\end_inset

.
 Since 
\begin_inset Formula $\frac{8}{0}$
\end_inset

 is undefined, it will not be present in the answer.
\end_layout

\begin_layout Description
Answer 
\begin_inset Formula $\{(-2,\frac{7}{3}),(-1,3),(1,4)\}$
\end_inset


\end_layout

\begin_layout Section
Composite Functions
\end_layout

\begin_layout Standard
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 
\begin_inset Formula $y$
\end_inset

 value) is determined by one function, it is substituted into the other
 function (usually 
\begin_inset Formula $x$
\end_inset

 value).
 The functions may or not be commutative, so the order of the functions
 in the composition should be taken into account.
\end_layout

\begin_layout Standard
\begin_inset Formula $f(g(x))$
\end_inset

 and 
\begin_inset Formula $(f\circ g)(x)$
\end_inset

 are equivalent.
 In both cases, 
\begin_inset Formula $g(x)$
\end_inset

 is determined first and the result is plugged into the 
\begin_inset Formula $x$
\end_inset

 value of 
\begin_inset Formula $f(x)$
\end_inset

.
\end_layout

\begin_layout Quote
If you find 
\begin_inset Formula $f(g(x))$
\end_inset

 to be easier, thank the mid-20th century mathematicians that came up with
 this notation after determining that 
\begin_inset Formula $(f\circ g)(x)$
\end_inset

 was too confusing.
\end_layout

\begin_layout Section
Reflections and Symmetry
\end_layout

\begin_layout Subsection
Reflection
\end_layout

\begin_layout Standard
When functions are reflected over a certain line or point (i.e.
 
\begin_inset Formula $x$
\end_inset

-axis or origin), coordinates (
\begin_inset Formula $(x,y)$
\end_inset

) of points in the function and the function itself (
\begin_inset Formula $y=f(x)$
\end_inset

) are affected as seen in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:reflection"

\end_inset

.
\end_layout

\begin_layout Standard
\noindent
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status open

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\noindent
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Reflection
\end_layout

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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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Coordinates
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Function
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<row>
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $r_{x-axis}(x,y)$
\end_inset


\end_layout

\end_inset
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\begin_inset Text

\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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<row>
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $r_{y-axis}(x,y)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(-x,y)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y=f(-x)$
\end_inset


\end_layout

\end_inset
</cell>
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<row>
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\begin_layout Plain Layout
\begin_inset Formula $r_{y=x}(x,y)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(y,x)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x=f(y)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $R_{O}(x,y)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(-x,-y)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y=-f(-x)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:reflection"

\end_inset

reflection
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Symmetry
\end_layout

\begin_layout Standard
Functions symmetric over a certain line or point (i.e.
 
\begin_inset Formula $x$
\end_inset

-axis or origin) contain both 
\begin_inset Formula $(x,y)$
\end_inset

 and the coordinates shown in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:reflection"

\end_inset

.
\end_layout

\begin_layout Subsubsection
Even and Odd Functions
\end_layout

\begin_layout Description
even
\begin_inset space ~
\end_inset

functions functions symmetric over the 
\begin_inset Formula $y$
\end_inset

-axis
\end_layout

\begin_layout Description
odd
\begin_inset space ~
\end_inset

functions functions symmetric over the origin
\end_layout

\begin_layout Subsection
Determining Symmetry
\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Description
x-axis negate the 
\begin_inset Formula $y$
\end_inset

 values
\end_layout

\begin_layout Description
y-axis negate the 
\begin_inset Formula $x$
\end_inset

 values
\end_layout

\begin_layout Description
y=x reverse 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 values (substitute the 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 values with each other)
\end_layout

\begin_layout Description
origin negate both the 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 values
\end_layout

\begin_layout Subsubsection
Example
\end_layout

\begin_layout Standard
Equation of graph: 
\begin_inset Formula $x^{2}+xy=4$
\end_inset


\end_layout

\begin_layout Paragraph
x-axis
\end_layout

\begin_layout Enumerate
Negate 
\begin_inset Formula $y$
\end_inset

 values: 
\begin_inset Formula $x^{2}-xy=4$
\end_inset


\end_layout

\begin_layout Enumerate
Simplify, if necessary (already simplified): 
\begin_inset Formula $x^{2}-xy=4$
\end_inset


\end_layout

\begin_layout Enumerate
Compare with graph equation: not equivalent
\end_layout

\begin_layout Enumerate
Not symmetric over the 
\begin_inset Formula $x$
\end_inset

-axis
\end_layout

\begin_layout Paragraph
y-axis
\end_layout

\begin_layout Enumerate
Negate 
\begin_inset Formula $x$
\end_inset

 values: 
\begin_inset Formula $(-x)^{2}-xy=4$
\end_inset


\end_layout

\begin_layout Enumerate
Simplify, if necessary: 
\begin_inset Formula $x^{2}-xy=4$
\end_inset


\end_layout

\begin_layout Enumerate
Compare with graph equation: not equivalent
\end_layout

\begin_layout Enumerate
Not symmetric over the 
\begin_inset Formula $y$
\end_inset

-axis
\end_layout

\begin_layout Paragraph
y=x
\end_layout

\begin_layout Enumerate
Reverse 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 values: 
\begin_inset Formula $y^{2}+yx=4$
\end_inset


\end_layout

\begin_layout Enumerate
Simplify, if necessary: 
\begin_inset Formula $y^{2}+xy=4$
\end_inset


\end_layout

\begin_layout Enumerate
Compare with graph equation: not equivalent
\end_layout

\begin_layout Enumerate
Not symmetric over the line 
\begin_inset Formula $y=x$
\end_inset


\end_layout

\begin_layout Paragraph
origin
\end_layout

\begin_layout Enumerate
Negate 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 values: 
\begin_inset Formula $(-x)^{2}+(-x)(-y)=4$
\end_inset


\end_layout

\begin_layout Enumerate
Simplify, if necessary: 
\begin_inset Formula $x^{2}+xy=4$
\end_inset


\end_layout

\begin_layout Enumerate
Compare with graph equation: equivalent
\end_layout

\begin_layout Enumerate
Symmetric over the origin
\end_layout

\begin_layout Section
Periodic Functions
\end_layout

\begin_layout Subsection
Definitions
\end_layout

\begin_layout Standard
See definitions 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "cha:Definitions"

\end_inset

.
\end_layout

\begin_layout Subsection
Effects of Different Equations
\end_layout

\begin_layout Standard
The bulleted examples below are compared with 
\begin_inset Formula $y=f(x)$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $y=2\times f(x)$
\end_inset

 doubles the amplitude
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=f(2x)$
\end_inset

 doubles the frequency; halves the period (periodic shrink)
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=f(\frac{1}{2}x)$
\end_inset

 halves the frequency; doubles the period (periodic stretch)
\end_layout

\begin_layout Subsubsection
Other Effects
\end_layout

\begin_layout Paragraph*
The following text in this section may be incorrect.
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Corrections and other feedback sent to the author or to the CHSN Review
 Project are greatly appreciated.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The 
\begin_inset Quotes eld
\end_inset

ceiling function
\begin_inset Quotes erd
\end_inset

 is not a periodic function.
 The bulleted examples below are compared with 
\begin_inset Formula $y=\lceil x\rceil$
\end_inset

.
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=2\lceil x\rceil$
\end_inset

 doubles 
\begin_inset Formula $y$
\end_inset

 values
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=\lceil2x\rceil$
\end_inset

 doubles 
\begin_inset Formula $x$
\end_inset

 values
\end_layout

\begin_layout Section
Inverses of Relations
\end_layout

\begin_layout Standard
A function is a relation, but a relation does not necessarily have to pass
 the vertical line test.
\end_layout

\begin_layout Subsection
Definitions
\end_layout

\begin_layout Standard
See definitions 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "cha:Definitions"

\end_inset

.
\end_layout

\begin_layout Subsection
Determining the Inverse of a Relation
\end_layout

\begin_layout Standard
Reverse the 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 values of an equation and if necessary, solve for 
\begin_inset Formula $y$
\end_inset

.
 In this way, the inverse of 
\begin_inset Formula $y=2x+1$
\end_inset

 will be 
\begin_inset Formula $y=\frac{x-1}{2}$
\end_inset

.
\end_layout

\begin_layout Subsection
Other Situations
\end_layout

\begin_layout Standard
You may not need to necessarily determine the equation of a relation's inverse.
 For example, one can use a 
\begin_inset Formula $y$
\end_inset

 value on a chart of values of a relation instead of the 
\begin_inset Formula $x$
\end_inset

 value of its inverse.
\end_layout

\begin_layout Section
Translations of Functions
\end_layout

\begin_layout Standard
To translate a function, values are either added or subtracted to part of
 a function.
 The examples below are compared with 
\begin_inset Formula $y=f(x)$
\end_inset

.
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=f(x)+a$
\end_inset

 moves the graph 
\begin_inset Formula $a$
\end_inset

 units up; add 
\begin_inset Formula $a$
\end_inset

 to 
\begin_inset Formula $y$
\end_inset

 values
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=f(x+b)$
\end_inset

 moves the graph 
\begin_inset Formula $b$
\end_inset

 units to the left; subtract 
\begin_inset Formula $b$
\end_inset

 from 
\begin_inset Formula $x$
\end_inset

 values
\end_layout

\begin_layout Standard
The examples below are compared with 
\begin_inset Formula $y=|x|$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $y=|x|+a$
\end_inset

 moves the graph 
\begin_inset Formula $a$
\end_inset

 units up; add 
\begin_inset Formula $a$
\end_inset

 to 
\begin_inset Formula $y$
\end_inset

 values
\end_layout

\begin_layout Itemize
\begin_inset Formula $y=|x+b|$
\end_inset

 moves the graph 
\begin_inset Formula $b$
\end_inset

 units to the left; subtract 
\begin_inset Formula $b$
\end_inset

 from 
\begin_inset Formula $x$
\end_inset

 values
\end_layout

\begin_layout Standard
Keep in mind that absolute value will cause the graph of an example like
 
\begin_inset Formula $y=||x|-c|$
\end_inset

 to be shaped like a W (assuming 
\begin_inset Formula $c$
\end_inset

 is positive).
\end_layout

\begin_layout Section
Asymptotes
\end_layout

\begin_layout Subsection
Definition
\end_layout

\begin_layout Standard
See definitions 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "cha:Definitions"

\end_inset

.
 For more information on asymptotes, see the 
\emph on
Math Calculus Review
\emph default
.
\end_layout

\begin_layout Subsection
Determining Asymptotes
\end_layout

\begin_layout Standard
To determine asymptotes algebraically, you must solve for either 
\begin_inset Formula $x$
\end_inset

 or 
\begin_inset Formula $y$
\end_inset

---isolate the variable.
\end_layout

\begin_layout Subsubsection
Vertical Asymptotes
\end_layout

\begin_layout Standard
Solve the equation for 
\begin_inset Formula $y$
\end_inset

 and determine the 
\begin_inset Formula $x$
\end_inset

 value where the denominator of the fraction would equal zero.
\end_layout

\begin_layout Subsubsection
Horizontal Asymptotes
\end_layout

\begin_layout Standard
Solve the equation for 
\begin_inset Formula $x$
\end_inset

 and determine the 
\begin_inset Formula $y$
\end_inset

 value where the denominator of the fraction would equal zero.
\end_layout

\begin_layout Subsection
Important Information
\end_layout

\begin_layout Standard
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, remember to factor numerators and denominators and attempt to
 simplify fractions as much as possible before determining asymptotes.
 Also understand that 
\begin_inset Formula $\frac{0}{0}$
\end_inset

 is not a valid equation or asymptote.
\end_layout

\begin_layout Chapter
Polar and Complex Numbers
\begin_inset CommandInset label
LatexCommand label
name "cha:polar"

\end_inset


\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Section
Polar Coordinates
\end_layout

\begin_layout Description
Cartesian
\begin_inset space ~
\end_inset

(rectangular)
\begin_inset space ~
\end_inset

coordinates a point, 
\begin_inset Formula $P$
\end_inset

, on a plane is described in terms of 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

, where 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 are the respective horizontal and vertical distances from the origin
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Form: 
\begin_inset Formula $(x,y)$
\end_inset


\end_layout

\begin_layout Description
polar
\begin_inset space ~
\end_inset

coordinates a point, 
\begin_inset Formula $P$
\end_inset

, on a plane is described by specifying the distance, 
\begin_inset Formula $r$
\end_inset

, from the origin and the angle, 
\begin_inset Formula $\theta$
\end_inset

, measured counter-clockwise from the positive 
\begin_inset Formula $x$
\end_inset

-axis to the line joining 
\begin_inset Formula $P$
\end_inset

 to the origin
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Form: 
\begin_inset Formula $(r,\theta)$
\end_inset


\end_layout

\begin_layout Description
coterminal
\begin_inset space ~
\end_inset

angles angles that coincide (when placed in standard position); added or
 subtracted multiples of 
\begin_inset Formula $360^{\circ}$
\end_inset

 (including 
\begin_inset Formula $360^{\circ}$
\end_inset

)
\end_layout

\begin_layout Description
reference
\begin_inset space ~
\end_inset

angles way to simplify the calculation of the values of trigonometric functions
 in different quadrants
\end_layout

\begin_layout Standard
Refer to Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:reference-angles"

\end_inset

, where 
\begin_inset Formula $\theta$
\end_inset

 is the angle and 
\begin_inset Formula $\beta$
\end_inset

 is the reference angle.
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="2">
<features>
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Quadrant
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Reference Angle (
\begin_inset Formula $\beta$
\end_inset

)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=\theta$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
II
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=180^{\circ}-\theta$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
III
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=\theta-180^{\circ}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
IV
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=360^{\circ}-\theta$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:reference-angles"

\end_inset

reference angles
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Converting Coordinates
\end_layout

\begin_layout Subsection
Polar-to-Cartesian
\end_layout

\begin_layout Standard
Polar coordinates are expressed in the form 
\begin_inset Formula $(r,\theta)$
\end_inset

, while Cartesian coordinates are expressed in the form 
\begin_inset Formula $(x,y)$
\end_inset

.
 The two following equations may be used to find the appropriate values
 for 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 based on 
\begin_inset Formula $r$
\end_inset

 and 
\begin_inset Formula $\theta$
\end_inset

, which may be substituted into 
\begin_inset Formula $(x,y)$
\end_inset

.
 There is no requirement for 
\begin_inset Formula $r$
\end_inset

 to be positive.
\end_layout

\begin_layout Standard
\begin_inset Formula $x=r\cos\theta$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $y=r\sin\theta$
\end_inset


\end_layout

\begin_layout Standard
Therefore, the form for conversion from polar coordinates to Cartesian coordinat
es is expressed as 
\begin_inset Formula $(r\cos\theta,r\sin\theta).$
\end_inset


\end_layout

\begin_layout Standard
It should be noted that the polar and Cartesian coordinates must be in the
 same quadrant (hint: reference angles).
\end_layout

\begin_layout Subsubsection
Explanation
\end_layout

\begin_layout Standard
These formulas are derived by inscribing a right triangle with hypotenuse
 
\begin_inset Formula $r$
\end_inset

 in a circle on a coordinate plane, such that 
\begin_inset Formula $\theta$
\end_inset

 is an angle from the positive 
\begin_inset Formula $x$
\end_inset

-axis to the line joining a point to the origin.
 If the point assumes the coordinates 
\begin_inset Formula $(x,y)$
\end_inset

, the following two equations are valid.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \cos\theta=\frac{x}{r}}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sin\theta=\frac{y}{r}}$
\end_inset


\end_layout

\begin_layout Standard
By multiplying both sides of the equations by 
\begin_inset Formula $r$
\end_inset

, the method for conversion is derived.
\end_layout

\begin_layout Standard
\begin_inset Formula $x=r\cos\theta$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $y=r\sin\theta$
\end_inset


\end_layout

\begin_layout Subsection
Cartesian-to-Polar
\end_layout

\begin_layout Standard
Cartesian coordinates are expressed in the form 
\begin_inset Formula $(x,y)$
\end_inset

 , while polar coordinates are expressed in the form 
\begin_inset Formula $(r,\theta)$
\end_inset

.
 The two following equations may be used to find the appropriate values
 for 
\begin_inset Formula $r$
\end_inset

 and 
\begin_inset Formula $\theta$
\end_inset

 based on 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

, which may be substituted into 
\begin_inset Formula $(r,\theta)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle r=\sqrt{x^{2}+y^{2}}}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \theta=\tan^{-1}\frac{y}{x}}$
\end_inset

, 
\begin_inset Formula $x\neq0$
\end_inset


\end_layout

\begin_layout Standard
Therefore, the form for conversion from Cartesian coordinates to polar coordinat
es is expressed as 
\begin_inset Formula $(\sqrt{x^{2}+y^{2}},\tan^{-1}\frac{y}{x})$
\end_inset

, 
\begin_inset Formula $x\neq0$
\end_inset

.
\end_layout

\begin_layout Standard
It should be noted that the Cartesian and polar coordinates must be in the
 same quadrant (hint: reference angles).
 If 
\begin_inset Formula $x=0$
\end_inset

, the second equation cannot be used to find 
\begin_inset Formula $\theta$
\end_inset

 since it will be undefined.
 Instead, one must determine if 
\begin_inset Formula $\theta$
\end_inset

 is 
\begin_inset Formula $90^{\circ}$
\end_inset

 or 
\begin_inset Formula $270^{\circ}$
\end_inset

, depending on whether 
\begin_inset Formula $y$
\end_inset

 is positive or negative, respectively.
\end_layout

\begin_layout Subsubsection
Explanation
\end_layout

\begin_layout Standard
These formulas are derived by inscribing a right triangle with hypotenuse
 
\begin_inset Formula $r$
\end_inset

 in a circle on a coordinate plane, such that 
\begin_inset Formula $\theta$
\end_inset

 is an angle from the positive 
\begin_inset Formula $x$
\end_inset

-axis to the line joining a point to the origin.
 If the point assumes the coordinates 
\begin_inset Formula $(x,y)$
\end_inset

, the following two equations are valid.
\end_layout

\begin_layout Standard
\begin_inset Formula $r^{2}=x^{2}+y^{2}$
\end_inset

 (Pythagorean Theorem)
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \tan^{-1}\frac{y}{x}}$
\end_inset

, 
\begin_inset Formula $x\neq0$
\end_inset


\end_layout

\begin_layout Standard
The square root of both sides is taken from the first equation based on
 the Pythagorean Theorem.
\end_layout

\begin_layout Standard
\begin_inset Formula $r=\pm\sqrt{x^{2}+y^{2}}$
\end_inset


\end_layout

\begin_layout Standard
The negative value of 
\begin_inset Formula $r$
\end_inset

 can be rejected for practical purposes.
\end_layout

\begin_layout Standard
\begin_inset Formula $r=\sqrt{x^{2}+y^{2}}$
\end_inset


\end_layout

\begin_layout Standard
The second equation can be rewritten to isolate 
\begin_inset Formula $\theta$
\end_inset

 on one side of the equation.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \theta=\tan^{-1}\frac{y}{x}}$
\end_inset

, 
\begin_inset Formula $x\neq0$
\end_inset


\end_layout

\begin_layout Section
Polar Inequalities
\end_layout

\begin_layout Standard
Polar inequalities are fairly simple.
 They are expressed in the following form.
\end_layout

\begin_layout Standard
\begin_inset Formula $a\leq r\leq b$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $c\leq\theta\leq d$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $a$
\end_inset

 represents the smallest value of 
\begin_inset Formula $r$
\end_inset

 in the range, while 
\begin_inset Formula $b$
\end_inset

 represents the largest value of 
\begin_inset Formula $r$
\end_inset

 in the range
\end_layout

\begin_layout Standard
\begin_inset Formula $c$
\end_inset

 represents the smallest angle of 
\begin_inset Formula $\theta$
\end_inset

 in the range, while 
\begin_inset Formula $d$
\end_inset

 represents the largest angle of 
\begin_inset Formula $\theta$
\end_inset

 in the range
\end_layout

\begin_layout Subsection
Etiquette
\end_layout

\begin_layout Standard
For math etiquette, one should follow the following guidelines when writing
 polar inequalities.
\end_layout

\begin_layout Standard
\begin_inset Formula $a\geq0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $b>0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $0^{\circ}\leq c<360^{\circ}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $0^{\circ}<d\leq360^{\circ}$
\end_inset


\end_layout

\begin_layout Standard
In other words, the radii should be positive, while the angles should be
 between 
\begin_inset Formula $0^{\circ}$
\end_inset

 and 
\begin_inset Formula $360^{\circ}$
\end_inset

.
\end_layout

\begin_layout Section
Graphing Polar Equations
\end_layout

\begin_layout Subsection
Graphing
\end_layout

\begin_layout Subsubsection
Manually
\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Subsubsection
Graphing Calculator
\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Enumerate
Change 
\begin_inset Quotes eld
\end_inset

Mode
\begin_inset Quotes erd
\end_inset

 to radians or degrees as necessary
\end_layout

\begin_layout Enumerate
Change 
\begin_inset Quotes eld
\end_inset

Mode
\begin_inset Quotes erd
\end_inset

 from 
\begin_inset Quotes eld
\end_inset

FUNC
\begin_inset Quotes erd
\end_inset

 (function) to 
\begin_inset Quotes eld
\end_inset

POL
\begin_inset Quotes erd
\end_inset

 (polar)
\end_layout

\begin_layout Enumerate
Adjust 
\begin_inset Quotes eld
\end_inset

Window
\begin_inset Quotes erd
\end_inset

 as necessary
\end_layout

\begin_layout Standard
\begin_inset Quotes eld
\end_inset

Steps
\begin_inset Quotes erd
\end_inset

 (of 
\begin_inset Formula $\theta$
\end_inset

) correspond to the number of calculations done to graph the equation.
 A lower step means more calculations, while a higher step means less calculatio
ns.
 More calculations mean more accuracy in graphing (less distortion), but
 longer periods to graph.
 Less calculations mean less accuracy in graphing (more distortion), but
 shorter periods to graph.
\end_layout

\begin_layout Subsection
Polar-to-Cartesian
\end_layout

\begin_layout Standard
Generally, a polar equation is written in the form 
\begin_inset Formula $r=a\sin\theta$
\end_inset

 or 
\begin_inset Formula $r=a\cos\theta$
\end_inset

, while an equation in the Cartesian coordinate plane is written in the
 form 
\begin_inset Formula $x^{2}+y^{2}=ay$
\end_inset

 or 
\begin_inset Formula $x^{2}+y^{2}=ax$
\end_inset

.
 The two following equations may be used to find the appropriate coefficients
 of 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 based on the coefficients of 
\begin_inset Formula $\cos\theta$
\end_inset

 and 
\begin_inset Formula $\sin\theta$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \cos\theta=\frac{x}{r}}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sin\theta=\frac{y}{r}}$
\end_inset


\end_layout

\begin_layout Standard
The appropriate coefficient of 
\begin_inset Formula $x^{2}+y^{2}$
\end_inset

 may be determined based on the coefficient of 
\begin_inset Formula $r^{2}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $r^{2}=x^{2}+y^{2}$
\end_inset


\end_layout

\begin_layout Standard
Substitution is key in these cases.
\end_layout

\begin_layout Subsubsection
Example
\end_layout

\begin_layout Standard
An example of a conversion from a polar equation to a Cartesian equation
 follows.
 
\begin_inset Formula $r=2\cos\theta$
\end_inset

 is the polar function, while 
\begin_inset Formula $x^{2}+y^{2}=2x$
\end_inset

 is the determined equation (in the Cartesian coordinate plane).
\end_layout

\begin_layout Standard
\begin_inset Formula $r=2\cos\theta$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle r=2\left(\frac{x}{r}\right)}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $r^{2}=2x$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $x^{2}+y^{2}=2x$
\end_inset


\end_layout

\begin_layout Standard
No further work is required on this determined equation.
\end_layout

\begin_layout Section
Complex Numbers
\end_layout

\begin_layout Standard
The rectangular form of a complex number is 
\begin_inset Formula $a+bi$
\end_inset

.
 The polar form of a complex number is 
\begin_inset Formula $r\,\mathrm{cis}\,\theta$
\end_inset

.
 
\begin_inset Formula $r\,\mathrm{cis}\,\theta$
\end_inset

 is an abbreviation for 
\begin_inset Formula $r(\cos\theta+i\sin\theta)$
\end_inset

.
 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.
\end_layout

\begin_layout Subsection
Conversions
\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Subsubsection
Cartesian-to-Polar
\end_layout

\begin_layout Standard
The Cartesian complex number 
\begin_inset Formula $a+bi$
\end_inset

 can be expressed as the following polar number:
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sqrt{a^{2}+b^{2}}[\cos(\tan^{-1}\frac{b}{a})+i\sin(\tan^{-1}\frac{b}{a})]}$
\end_inset

, or abbreviated as 
\begin_inset Formula ${\displaystyle \sqrt{a^{2}+b^{2}}\,\mathrm{cis}\,(\tan^{-1}\frac{b}{a})}$
\end_inset

.
\end_layout

\begin_layout Standard
It should be noted that the Cartesian and polar coordinates must be in the
 same quadrant (hint: reference angles).
\end_layout

\begin_layout Paragraph
Explanation
\end_layout

\begin_layout Standard
The absolute value of a complex number, expressed as 
\begin_inset Formula $|z|$
\end_inset

, represents the length of the vector of a graphed complex number.
\end_layout

\begin_layout Standard
\begin_inset Formula $|z|=\sqrt{a^{2}+b^{2}}$
\end_inset

 (Pythagorean Theorem)
\end_layout

\begin_layout Standard
Since the length of the vector is equivalent to 
\begin_inset Formula $r$
\end_inset

 (the radius), 
\begin_inset Formula $r=\sqrt{a^{2}+b^{2}}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Polar-to-Cartesian
\end_layout

\begin_layout Standard
The polar complex number 
\begin_inset Formula $r\,\mathrm{cis}\,\theta$
\end_inset

 can be expressed as the Cartesian number 
\begin_inset Formula $r\cos\theta+(r\sin\theta)i$
\end_inset

.
 It should be noted that the polar and Cartesian coordinates must be in
 the same quadrant (hint: reference angles).
\end_layout

\begin_layout Subsection
Operations
\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Subsubsection
Cartesian
\end_layout

\begin_layout Standard
FOIL may be used to multiply two complex numbers in Cartesian form.
\end_layout

\begin_layout Standard
In general, if 
\begin_inset Formula $z_{1}=a_{1}+b_{1}i$
\end_inset

 and 
\begin_inset Formula $z_{2}=a_{2}+b_{2}i$
\end_inset

, then 
\begin_inset Formula $z_{1}\times z_{2}=a_{1}a_{2}+a_{1}b_{2}i+a_{2}b_{1}i-b_{1}b_{2}$
\end_inset

.
 The last term is negative and does not contain any imaginary numbers since
 
\begin_inset Formula $i^{2}=-1$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Polar
\end_layout

\begin_layout Standard
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.
\end_layout

\begin_layout Paragraph
Multiplication and Division
\end_layout

\begin_layout Standard
In general, if 
\begin_inset Formula $z_{1}=r_{1}\,\mathrm{cis}\,\theta_{1}$
\end_inset

 and 
\begin_inset Formula $z_{2}=r_{2}\,\mathrm{cis}\,\theta_{2}$
\end_inset

, then 
\begin_inset Formula $z_{1}\times z_{2}=r_{1}\times r_{2}\,\mathrm{cis}\,(\theta_{1}+\theta_{2})$
\end_inset

.
 Likewise, under the same conditions, 
\begin_inset Formula $\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\,\mathrm{cis}\,(\theta_{1}-\theta_{2})$
\end_inset

.
\end_layout

\begin_layout Paragraph
De Moivre's Theorem
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $n$
\end_inset

 be any integer, then 
\begin_inset Formula $(r\,\mathrm{cis}\,\theta)^{n}=r^{n}\,\mathrm{cis}\, n\theta$
\end_inset

.
\end_layout

\begin_layout Standard
To determine the roots of a complex number, 
\begin_inset Formula $n$
\end_inset

 can be substituted with 
\begin_inset Formula $\frac{1}{x}$
\end_inset

, where 
\begin_inset Formula $x$
\end_inset

 represents the 
\begin_inset Formula $x^{th}$
\end_inset

 root .
 A complex number raised to the power of 
\begin_inset Formula $\frac{1}{x}$
\end_inset

 will have 
\begin_inset Formula $x$
\end_inset

 number of roots.
 This is due to the fact that equivalent complex numbers in polar form (with
 coterminal angles) will have reference angles 
\begin_inset Formula $360^{\circ}$
\end_inset

 apart.
 When 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $360^{\circ}$
\end_inset

is divided by 
\begin_inset Formula $x$
\end_inset

, 
\begin_inset Formula $x$
\end_inset

 unique angles will be created (between 
\begin_inset Formula $0^{\circ}$
\end_inset

 and 
\begin_inset Formula $360^{\circ}$
\end_inset

), with 
\begin_inset Formula $\nicefrac{360^{\circ}}{x}$
\end_inset

 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.
\end_layout

\begin_layout Standard
In general, 
\begin_inset Formula ${\displaystyle (r\,\mathrm{cis}\,\theta)^{\frac{1}{n}}=r^{\frac{1}{n}}\,\mathrm{cis}\,\frac{\theta+360K}{n}}$
\end_inset

, where 
\begin_inset Formula $K=0,1,2,3...n-1$
\end_inset

.
\end_layout

\begin_layout Chapter
Sequences and Series
\end_layout

\begin_layout Standard
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 
\begin_inset Formula $n$
\end_inset

, and refers to the total number of terms in a sequence or series as 
\begin_inset Formula $k$
\end_inset

.
\end_layout

\begin_layout Section
Arithmetic Sequences
\end_layout

\begin_layout Standard
Arithmetic sequences (also known as lists and progressions) have a constant
 difference.
 Terms increase or decrease by adding or subtracting a fix quantity.
 The constant difference is known as 
\begin_inset Formula $d$
\end_inset

, which is the difference between any two consecutive terms.
\end_layout

\begin_layout Standard
Arithmetic sequences use the following equation to determine the 
\begin_inset Formula $n^{th}$
\end_inset

 term.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle t_{n}=t_{1}+(n-1)d}$
\end_inset

 (provided on test unlabeled)
\end_layout

\begin_layout Standard
\begin_inset Formula $t_{1}$
\end_inset

 represents the first term, while 
\begin_inset Formula $n$
\end_inset

 represents the position of a term (i.e.
 first, second, third).
\end_layout

\begin_layout Standard
Rewritten in the following equation, one may determine the number of terms
 in an arithmetic sequence.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle n=\frac{t_{n}-t_{1}}{d}+1}$
\end_inset


\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Note Multiples of a number are an arithmetic sequence.
\end_layout

\begin_layout Subsection
Example
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Sequence 
\begin_inset Formula $2,5,8,11,14,17$
\end_inset


\end_layout

\begin_layout Standard
The above sequence is an example of an arithmetic sequence, and 
\begin_inset Formula $d=3$
\end_inset

.
\end_layout

\begin_layout Subsection
Arithmetic Mean
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Sequence \SpecialChar \ldots{}

\begin_inset Formula $a,x,b$
\end_inset

\SpecialChar \ldots{}

\end_layout

\begin_layout Standard
If the above sequence is given, 
\begin_inset Formula $x$
\end_inset

 is considered the arithmetic mean of 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

.
 Therefore, 
\begin_inset Formula $x=\frac{a+b}{2}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Explanation
\end_layout

\begin_layout Standard
If one attempts to solve determine the value of 
\begin_inset Formula $x$
\end_inset

 based on 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

, it would be determined that 
\begin_inset Formula $d=x-a$
\end_inset

 and 
\begin_inset Formula $d=b-x$
\end_inset

.
 Through substitution, 
\begin_inset Formula $x-a=b-x$
\end_inset

.
 This can in turn be rewritten as 
\begin_inset Formula $x=\frac{a+b}{2}$
\end_inset

, the same as the arithmetic mean of 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

.
\end_layout

\begin_layout Subsection
Recursive Arithmetic Sequences
\end_layout

\begin_layout Description
recursive
\begin_inset space ~
\end_inset

equation an equation for sequences in which the value of a term is defined
 based on the preceding term(s).
\end_layout

\begin_layout Standard
Recursive formulas must have the following:
\end_layout

\begin_layout Enumerate
an initial condition, 
\begin_inset Formula $t_{1}$
\end_inset


\end_layout

\begin_layout Enumerate
a recursive equation
\end_layout

\begin_layout Standard
A recursive arithmetic formula is a different method of expressing an arithmetic
 sequence, instead of using standard form.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Standard
\begin_inset space ~
\end_inset

form 
\begin_inset Formula ${\displaystyle t_{n}=t_{1}+(n-1)d}$
\end_inset


\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Recursive
\begin_inset space ~
\end_inset

form 
\begin_inset Formula ${\displaystyle t_{n}=d+t_{n-1}}$
\end_inset


\end_layout

\begin_layout Section
Geometric Sequences
\end_layout

\begin_layout Standard
Geometric sequences have a constant ratio.
 All terms are connected by a common ratio.
 The constant ratio is known as 
\begin_inset Formula $r$
\end_inset

, which is the quotient determined by the division of any two consecutive
 terms.
\end_layout

\begin_layout Standard
Geometric sequences use the following equation to determine the 
\begin_inset Formula $n^{th}$
\end_inset

 term.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle t_{n}=t_{1}\times r^{n-1}}$
\end_inset

 (provided on test unlabeled)
\end_layout

\begin_layout Standard
\begin_inset Formula $t_{1}$
\end_inset

 represents the first term, while 
\begin_inset Formula $t_{n}$
\end_inset

 represents the 
\begin_inset Formula $n^{th}$
\end_inset

 term.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Note Geometric sequences with alternating signs of terms have negative 
\begin_inset Formula $r$
\end_inset

 (
\begin_inset Formula $-r$
\end_inset

) values.
 Consequently, when a term and its position in a sequence (
\begin_inset Formula $n$
\end_inset

) are provided and 
\begin_inset Formula $n$
\end_inset

 is even, multiple values for 
\begin_inset Formula $r$
\end_inset

 may be determined.
 In this case, 
\begin_inset Formula $r$
\end_inset

 and 
\begin_inset Formula $-r$
\end_inset

 (
\begin_inset Formula $\pm r$
\end_inset

) may both be correct values for 
\begin_inset Formula $r$
\end_inset

.
\end_layout

\begin_layout Subsection
Example
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Sequence 
\begin_inset Formula $4,8,16,32,64,128$
\end_inset


\end_layout

\begin_layout Standard
The above sequence is an example of a geometric sequence, and 
\begin_inset Formula $r=2$
\end_inset

.
\end_layout

\begin_layout Subsection
Geometric Mean
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
Sequence \SpecialChar \ldots{}

\begin_inset Formula $a,x,b$
\end_inset

\SpecialChar \ldots{}

\end_layout

\begin_layout Standard
If the above sequence is given, 
\begin_inset Formula $x$
\end_inset

 is considered the geometric mean of 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

.
 Therefore, 
\begin_inset Formula $x=\sqrt{a\times b}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Explanation
\end_layout

\begin_layout Standard
If one attempts to solve determine the value of 
\begin_inset Formula $x$
\end_inset

 based on 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

, it would be determined that 
\begin_inset Formula $r=\frac{x}{a}$
\end_inset

 and 
\begin_inset Formula $r=\frac{b}{x}$
\end_inset

.
\end_layout

\begin_layout Standard
Through substitution, 
\begin_inset Formula $\frac{x}{a}=\frac{b}{x}$
\end_inset

.
 This can in turn be rewritten as 
\begin_inset Formula $x=\sqrt{a\times b}$
\end_inset

, the same as the geometric mean of 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

.
\end_layout

\begin_layout Section
Series
\end_layout

\begin_layout Description
series the sum of a sequence, list, or progression
\end_layout

\begin_layout Standard
\begin_inset Formula $S_{k}$
\end_inset

 represents the summation of 
\begin_inset Formula $k$
\end_inset

 number of terms in a sequence.
\end_layout

\begin_layout Standard
The formulas follow, and are different for arithmetic and geometric series.
\end_layout

\begin_layout Subsection
Arithmetic Series
\end_layout

\begin_layout Standard
The following formula can be used to determine the value of an arithmetic
 series, 
\begin_inset Formula $S_{k}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle S_{k}=\frac{k(t_{1}+t_{k})}{2}}$
\end_inset

 (provided unlabeled on test as 
\begin_inset Formula ${\displaystyle S_{n}=\frac{n(t_{1}+t_{n})}{2}}$
\end_inset

)
\end_layout

\begin_layout Standard
\begin_inset Formula $k$
\end_inset

 represents the total number of terms in a sequence or series.
\end_layout

\begin_layout Subsection
Geometric Series
\end_layout

\begin_layout Standard
The following formula can be used to determine the value of a geometric
 series, 
\begin_inset Formula $S_{k}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle S_{k}=\frac{t_{1}(1-r^{k})}{1-r}}$
\end_inset

 (provided unlabeled on test as 
\begin_inset Formula ${\displaystyle S_{n}=\frac{t_{1}(1-r^{n})}{1-r}}$
\end_inset

)
\end_layout

\begin_layout Standard
\begin_inset Formula $k$
\end_inset

 represents the total number of terms in a sequence or series.
\end_layout

\begin_layout Subsubsection
Infinite Geometric Series
\end_layout

\begin_layout Standard
The sum of an infinite geometric series would be expressed by the following
 statement.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle S=t_{1}+t_{1}r+t_{1}r^{2}+t_{1}r^{3}}$
\end_inset

\SpecialChar \ldots{}

\end_layout

\begin_layout Standard
Infinite geometric series either converge or diverge.
\end_layout

\begin_layout Subsubsection
Converging Infinite Geometric Series
\end_layout

\begin_layout Standard
Converging infinite geometric series have a sum, and the value of 
\begin_inset Formula $r$
\end_inset

 in the series fits in 
\begin_inset Formula $-1<r<1$
\end_inset

.
 In other words, 
\begin_inset Formula $r$
\end_inset

 must be between 
\begin_inset Formula $-1$
\end_inset

 and 
\begin_inset Formula $1$
\end_inset

, non-inclusive.
\end_layout

\begin_layout Standard
The following formula would be used to determine the sum of the infinite
 geometric series.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle S=\frac{t_{1}}{1-r}}$
\end_inset

, 
\begin_inset Formula $-1<r<1$
\end_inset

 (provided on test, possibly without domain)
\end_layout

\begin_layout Paragraph
Interval of Convergence
\end_layout

\begin_layout Standard
In some cases 
\begin_inset Formula $r$
\end_inset

, may be provided or can be determined in terms of another variable.
 For example, 
\begin_inset Formula $r$
\end_inset

 may equal 
\begin_inset Formula $x+1$
\end_inset

.
 In this case, one simply substitutes 
\begin_inset Formula $r$
\end_inset

 with 
\begin_inset Formula $x+1$
\end_inset

 into the domain provided for converging infinite geometric series.
\end_layout

\begin_layout Standard
\begin_inset Formula $-1<r<1$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $-1<x+1<1$
\end_inset


\end_layout

\begin_layout Standard
To isolate the variable, 
\begin_inset Formula $x$
\end_inset

, one may subtract 
\begin_inset Formula $1$
\end_inset

 from all sides of the relation.
\end_layout

\begin_layout Standard
\begin_inset Formula $-2<x<0$
\end_inset


\end_layout

\begin_layout Standard
It has now been determined that if 
\begin_inset Formula $r=x=1$
\end_inset

, the interval that convergence will occur (resulting in a non-infinite
 sum) is between 
\begin_inset Formula $-2$
\end_inset

 and 
\begin_inset Formula $0$
\end_inset

, non-inclusive.
\end_layout

\begin_layout Subsubsection
Diverging Infinite Geometric Series
\end_layout

\begin_layout Standard
If the value of 
\begin_inset Formula $r$
\end_inset

 in the infinite geometric series does not fit in 
\begin_inset Formula $-1<r<1$
\end_inset

, then the sum of the series is infinity and is said to diverge.
\end_layout

\begin_layout Section
Sigma Notation
\end_layout

\begin_layout Standard
Summations may also be expressed through sigma notation.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sum_{i=n}^{m}x_{i}}$
\end_inset


\end_layout

\begin_layout Standard
In the above expression of sigma notation, 
\begin_inset Formula $i$
\end_inset

 is the index of summation, 
\begin_inset Formula $n$
\end_inset

 is the lower limit of summation, and 
\begin_inset Formula $m$
\end_inset

 is the upper limit of summation.
 It is possible than 
\begin_inset Formula $m$
\end_inset

 may be infinity (
\begin_inset Formula $\infty$
\end_inset

), consequently resulting in no limit on summation.
\end_layout

\begin_layout Standard
The sigma notation specified above would be equivalent to the following
 expression.
\end_layout

\begin_layout Standard
\begin_inset Formula $x_{m}+x_{m+1}+x_{m+2}+\cdots+x_{n-1}+x_{n}$
\end_inset


\end_layout

\begin_layout Standard
Sigma notation can be used for both arithmetic and geometric series
\end_layout

\begin_layout Subsection
Arithmetic Series
\end_layout

\begin_layout Standard
Sigma notation has the following form for arithmetic series.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sum_{n=t_{1}}^{t_{k}}dn}$
\end_inset


\end_layout

\begin_layout Standard
The constant difference is known as 
\begin_inset Formula $d$
\end_inset

, which is the difference between any two consecutive terms.
 
\begin_inset Formula $k$
\end_inset

 represents the total number of terms in a sequence or series.
 
\begin_inset Formula $n$
\end_inset

 represents the position of a term (i.e.
 first, second, third).
 It is possible than 
\begin_inset Formula $t_{k}$
\end_inset

 may be infinity (
\begin_inset Formula $\infty$
\end_inset

), consequently resulting in no limit on summation.
\end_layout

\begin_layout Standard

\end_layout

\begin_layout Subsection
Geometric Series
\end_layout

\begin_layout Standard
Sigma notation has the following form for geometric series.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sum_{n=t_{1}}^{t_{k}}(t_{1})(r)^{n-1}}$
\end_inset


\end_layout

\begin_layout Standard
The constant ratio is known as 
\begin_inset Formula $r$
\end_inset

, which is the quotient determined by the division of any two consecutive
 terms.
 
\begin_inset Formula $k$
\end_inset

 represents the total number of terms in a sequence or series.
 
\begin_inset Formula $n$
\end_inset

 represents the position of a term (i.e.
 first, second, third).
 It is possible than 
\begin_inset Formula $t_{k}$
\end_inset

 may be infinity (
\begin_inset Formula $\infty$
\end_inset

), consequently resulting in no limit on summation.
\end_layout

\begin_layout Subsection
Alternating Signs
\end_layout

\begin_layout Standard
Arithmetic series with alternating signs of terms are in fact both arithmetic
 and geometric series, with 
\begin_inset Formula $d$
\end_inset

 defined and 
\begin_inset Formula $r=-1$
\end_inset

.
 As a result, the format used for arithmetic sequences is represented by
 the following expression.
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sum_{n=t_{1}}^{t_{k}}(dn)(-1)^{k-1}}$
\end_inset


\end_layout

\end_body
\end_document