PRE CALCULUS 6.0 - THE YEAR IN REVIEW
Chapter 0
•
Know that the square root operation produces only positive values in order for the square root
operation to be a function (recall from our class discussions that the 4 = 2, not ± 2 and that
x 2 = x to ensure that the result of the absolute value operation is positive (refer to handout
for more details)).
Be able to rewrite an absolute value function as a piecewise defined function with an
appropriate domain for each rule.
Graphing by Pattern
• Based on a given function presented algebraically, be able to identify and state the types of
transformations (magnitude and direction) to the parent functions. Remember that there are
two types of transformations including: 1) deformations (stretch or compression via
multiplication -> shape changes) and reflections (mirror image reflected across x or y axis as
seen via a negative value of the leading coefficient) and/or 2) translations (shifts via addition
or subtraction -> shape stays the same). Also keep in mind that changes to the parent
function that occur "inside" the function (i.e. changes to the argument of the function)
represent changes in the horizontal direction (i.e. domain) and those that occur "outside"
function represent vertical changes (i.e. range).
• Know how to graph basic functions such as linear, quadratic, cubic, square root and absolute
value functions based on the pattern of the parent function coupled with transformation
related changes to that pattern. Graphing by creating a table of values is not necessary nor is
it encouraged at this stage; graphing by pattern is expected.
Solving Equations
• Be able to solve absolute value equations.
 For equations where the variable is only on the inside of the absolute value, consider
using the graphical distance approach to solve (recall central value and distance from
central value).
 If variables are both inside and outside of the absolute value, you must solve by the
more formal case analysis method where you establish domain criteria based on the
argument of the absolute value being positive or negative. Remember that your
algebraic answers to both cases must be inspected for validity by comparing to the
domain criteria for each case. If you ignore this comparison, you may end up
identifying answers which are in fact extraneous.
• Be able to solve quadratic equations using an appropriate method:
• Factoring and zero product property.
• Extracting the root (use only if there is no isolated linear term or if there is a linear
term but it is bound within a square). Remember that while this problem starts as a
quadratic equation, it ends as an absolute value equation because the square root of a
squared algebraic term always produces the absolute value of that algebraic term.
• Quadratic formula (you MUST know the formula forever more - no reminders). You
MUST be able to state solutions in simplest rational, radical form – no decimals and no
fractions within fractions (complex fractions).
• Completing the square.
• Know how to calculate the value of the discriminant for a quadratic equation and then be able
to characterize the nature of solutions to that quadratic equation based on the value of the
discriminant (i.e Real Rational, Real Irrational or Complex).
• Be able to solve absolute value equations (both distance and case analysis approaches).
• Be able to solve quadratic equations by a) factoring/zero product property, b) extracting the
root, c) quadratic formula or d) completing the square.
• Be able to solve equations containing radicals. Remember… BEWARE OF THE SQUARE! You
MUST check answers in order to identify extraneous answers.
• Be able to solve rational equations (equations that have variable terms in the
denominator). Remember to clear the fractions by multiplying all terms by the least common
denominator of those terms. Don't forget to consider restricted values which would cause
division by zero in the original problem.
•
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
•
Be able to solve equations that can be rewritten in quadratic form (i.e. the U-Sub method). In
order to use this approach, recall that the power of the middle term must be half that of the
first highest power. Don’t forget to substitute back to find the answer in terms of the original
variable.
• Be able to solve equations containing radicals. Remember… BEWARE OF THE SQUARE! You
MUST check answers in order to identify extraneous answers.
• Be able to solve rational equations (equations that have variable terms in the
denominator). Remember to clear the fractions by multiplying all terms by the least common
denominator of those terms. Don't forget to consider restricted values which would cause
division by zero in the original problem.
•
Be able to solve equations that can be rewritten in quadratic form (i.e. the U-Sub method). In
order to use this approach, recall that the power of the middle term must be half that of the
first highest power. Don’t forget to substitute back to find the answer in terms of the original
variable.
Solving Inequalities
• Remember that you must state solutions to inequalities in two ways 1) graphically on a
number line (don’t forget to shade solution region) and 2) stated in interval and/or set builder
notation as requested).
• Be able to solve absolute value inequalities.
• Be able to solve absolute value inequalities.
 For inequalities where the variable is only on the inside of the absolute value, consider
using the graphical distance approach to solve. Remember that this approach takes
advantage of our knowledge of transformations (specifically horizontal translations) to
identify a central value and the required distance away from that central value.
 If variables are both inside and outside of the absolute value, you must solve by the
more formal case analysis method where you establish domain criteria based on the
argument of the absolute value being positive or negative. Remember that your
algebraic answers to both cases must be inspected for validity by comparing to the
domain criteria for each case (i.e. the algebraic solution is “subject to” the domain
criteria). While the memorized process from prior years will get the job done (if done
correctly), completion of all components of the case analysis will demonstrate that you
know why the memorized process works. Case analysis is therefore a required
part of your solution for these types of absolute value inequalities.
• Be able to solve polynomial inequalities and graph solutions on a number line. When finding
critical points, always consider factoring first starting with greatest common factor, difference
of squares, sum or difference of cubes, trinomial methods and then grouping. See the
"Factoring Toolbox" in the Pre Calc Notesheets section if you need a little refresher on these
common factoring patterns. Remember that you MUST be able to show sign analysis on
a number line to identify the appropriate solution regions. You do not have to state
what values you tested but + or – signs above the number line are required as proof
that you in fact tested the regions.
• In the specific case on a quadratic inequality where there are no linear terms or if the linear
term is bound within a square, recall that a simpler approach to solving this type of problem is
by using the method of extracting the root. Remember that in this case, the quadratic
inequality will ultimately be transformed into an absolute value inequalities (after you take the
square root of both sides) which can then be easily solved using the distance approach for an
absolute value inequality.
• Be able to solve rational inequalities. Remember that you should NEVER NEVER EVER multiply
by variable expressions in an inequality to clear fractions since you do not know whether the
value of the variable expression is positive or negative and therefore do not know whether the
inequality sign stays the same or changes direction. As with other inequalities, you MUST be
able to find critical points and show sign analysis on a number line to identify the appropriate
solution regions.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
Chapter 1
Chapter 1.1 – Functions
•
Know the definition of, and be able to state similarities and differences between a relation and
a function.
• Know the definition of domain and range.
• Be able to determine domain and range from a relation or function presented as a set of
ordered pairs.
• Be able to determine if a set of ordered pairs may be classified as a function (no repeats in the
domain).
• Be able to determine if a relation presented as an equation in algebraic form represents a
function (solve for y in terms of x - remember that each value of the domain is "mapped" to
one and only one unique value of the range). Algebraic solution is required!
• Be familiar with function notation (i.e. f(x)) and be able complete function evaluations
of numerical or algebraic expressions.
• Be able to evaluate the difference quotient for a function (pp 80-81 - formula will be supplied
as necessary). Remember that the final stage of this simplification involves division by "h"
which may only be completed if "h" is not equal to zero. The notation/statement about this
restriction is a critical and required part of this evaluation. As necessary, you should also be
able to complete operations like “clearing the fractions” and/or “rationalizing the numerator” in
order to completely simplify difference quotient expressions (see more advanced problems in
Section 1.2, p 138, #’s 39-44).
• Know that the difference quotient represents the slope of a secant line for a given function
which in turn represents an average rate of change for that function on a given interval of the
domain. Recall our class example about the average rate of change of a position function
which results in an average change in position with respect to time (otherwise known as
velocity) over a given time interval (see Section 1.2, p. 130).
• Be able to evaluate a piecewise defined function. Remember that each piece, or rule, is valid
only over the stated interval of the domain for that rule which means that one and only one
rule may be applied for a given value of the domain.
• From a given graph, be able to determine the value of a function given a domain or determine
the value(s) of domain that would produce a given range value (see pp 119-120, #’s 19-34).
• Be able to find the domain of a function given in algebraic form. Some key considerations:
• the denominator cannot = zero
• the radicand of any even index radical must be greater than or equal to zero unless
that radical is in the denominator in which case, the radicand must be simply greater
than zero.
• there are no restrictions in the case of an odd index radical unless that radical is in the
denominator of an expression. In this case, the radicand in the denominator cannot =
zero.
• for rational functions (i.e. the ratio of two polynomial functions), keep in mind that the
overall domain for the expression must meet the requirements for BOTH the
numerator and denominator. As such, you must identify the intersection of the
domains for the expression in the numerator and the denominator.
• in the case of two or more added terms, be sure to consider the domain for each of
the terms individually and then collectively via the intersection of the individual
domains (p 84, problem 65).
Chapter 1.2 - Graphs of Functions
• Be able to identify domain and range of a relation or function based on a graph.
• Be able to determine if the graph of a relation represents a function (vertical line test).
• Be able to describe the "nature" of a function. Remember the key phrase: " As the domain
increases (domain “sweep” from left to right), the value of the function (i.e. range (y))
[increases, decreases, or is constant] over what specific interval of the domain.” In this
statement, notice that we do not need to use actual values of the function - just the behavior
of range values (i.e. increase, decrease or constant). At the same time, be sure to include the
actual numerical interval of the domain over which the function changes occur. Finally,
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
remember that the values of the domain where the nature of the function changes (i.e.
transitional values at relative local maximums or minimums) are excluded in the statement of
the domain interval since we are unable to determine how the function is changing at these
transitional points.
• Be familiar with the concept of relative local minimum and relative local maximum
values. You should also be able to locate where these minimums or maximums occur on a
graph.
• Be able to graph a piecewise defined function. Keep in mind that the pieces could be any of
the functions discussed so far this year including linear, quadratic, cubic, square root, absolute
value, etc.). Be sure to graph these pieces only where the domain for that rule is valid (i.e.
between the "firewalls.").
• Be able to develop the algebraic form of a piecewise function based on a given graph of the
piecewise function.
• Be able to determine whether a function is odd, even or neither based on graph. You will need
to be familiar with point (origin) and line (y-axis) symmetry in order to make this analysis.
• Be able to determine algebraically whether a function is odd, even, or neither based on a
function presented algebraically. Remember that this algebraic test requires general domain
replacement of (-x) for (x) and then whatever transformations are necessary to determine if
the new function (f(-x) is the same (even), opposite (odd) or a mixture of same and opposite
(neither).
Chapter 1-3 - Shifting, Reflecting and Stretching Graphs
• Graphing by pattern must be a mastered skill at this point based on our work in the
P-Chapter.
• Be able to complete graphical transformations of a function presented graphically to create a
new function. You must also be able to state the type and magnitude of the transformations
for the new function. Remember - you are just doing the arithmetic on the ordered pairs of
the given graph.
Chapter 1-4 –Combinations and Compositions of Functions
• Given functions presented algebraically, be able to complete Arithmetic Combinations for those
functions for variable expressions and/or numerical values.
• Be able to evaluate combinations of functions presented in ordered pair form, in tabular form,
or from functions presented graphically.
• Be able to determine the domain for Arithmetic Combinations. Remember that the domain of
combinations is simply the intersection of the domains for the given functions that are being
added, subtracted or multiplied. BUT… also keep in mind that for division of two functions,
you must consider not only the intersection of the domains for the given functions but you
must also include any additional restrictions that would occur when the function in the
denominator is equal to zero (i.e. for (f/g)(x), g(x)≠0).
• Given function presented algebraically, be able to complete Composition of Functions for those
functions for variable expressions or numerical values.
• Be able to evaluate composition of functions presented in the form of ordered pairs or in
tabular form. Remember the designation of "Partially Composed" for those functions for which
only some of the domain values of the inner function can be composed.
• Be able to identify the domain for Composition of Functions. Think about using the “function
box” diagram to help you keep track of the number flow through the composition. Be sure to
consider restrictions on the domain of the inner function along with any further restrictions as
imposed by the domain of the outer function on the range of the inner function. Remember
that "the output of the inner function is the input to the outer function." Say 20 times fast!
Chapter 1-5 – Inverse Functions
• Know the requirements of a 1:1 function from the standpoint of both domain and range values
(each x mapped to one unique y and each y mapped to one unique x) as well as graphically
(passes both VLT and HLT).
• Know that if a function is not 1:1, then an inverse function does not exist unless domain
restrictions are imposed. You should be able to develop appropriate domain restrictions as
necessary.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
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•
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Be able to generate the inverse of a given 1:1 function given in ordered pair, tabular form or
algebraic form. Keep in mind that the domain of the given function becomes the range of the
inverse and the range of the given function becomes the domain of the inverse.
If creating an inverse algebraically, remember to make notation "Create Inverse" so that it is
not assumed that x's magically become y's and vice versa. Also remember that the domain of
an inverse is a required part of your solution, even if the problem does not specifically ask
for it.
Be sure to use inverse notation (Remember the -1 in the “exponential” position is just notation
- not really an exponent).
Given the graph of a function, be able to graph its inverse.
Know the equation of the Identity Function (y = x) and its importance as a line of symmetry
for a function and its inverse.
Be able to prove algebraically via composition whether two functions are inverses of each
other. Keep in mind that composition in only one direction in not sufficient (recall our example
of
•
•
•
=
f ( x ) x2 ; =
g( x )
x;
( g  f=
)( x )
x)
(=
2
x
but
(f
 g )(=
x)
=
x2
x
which is
not equal to just x, the identity function). Be aware that this is the formal process for proving
two functions are inverses of each other. AVOID creating the inverse of one of the given
functions and comparing it to the other because it does not demonstrate that the composition
of a function with its inverse creates the identity function.
If you are asked to generate the inverse of a function, remember to consider domain
restrictions as necessary to maintain 1:1 functionality.
Be able to complete all the necessary higher level algebraic manipulations demonstrated in
class when completing compositions (e.g. clearing the fractions when you have a complex
fraction, etc.) See pp 180=181, problems 27, 28, 51, 52 as examples requiring this higher
level of algebra).
Given a 1:1 function, be able to evaluate the inverse of that function for a specific domain
value without creating the inverse (Recall example from class: Find
f ( x=
) 7x − 3;
since
f −1( 17 )
given
17 is in D f −1 , it is also in R f ∴ 17 = 7 x − 3 ; Solve for x). You should
be able to complete this evaluation for 1:1 functions presented algebraically, as sets of
ordered pairs, or in tabular form.
Chapter 10
Chapter 10.5 – Binomial Expansion Theorem
• Be able to use Pascal’s triangle to determine the coefficients for expansion of a binomial raised
to a power.
7!
• Be able to simplify the quotient of two factorials including numerical factorials (e.g.
) as
5!2!
n!
)
well as variable factorials (e.g.
( n − 1)!
•
•
•
Be familiar with the different notations for a combination and be able to evaluate a
combination.
Be able to expand powers of binomials through use of the Binomial Expansion Theorem. (Note
that use of Pascal’s Triangle will NOT be acceptable if requested to expand a binomial using
the Binomial Expansion Theorem).
Be able to find the value of a specific term in the expansion of a binomial without completely
expanding the binomial.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
Chapter 2
•
•
Chapter 2.1 – Quadratic Functions
Be able to identify the vertex (as an ordered pair), x and y intercepts (as ordered pairs) and
axis of symmetry (as equation x = __) for a quadratic function given either in general form
( f ( x ) = ax 2 + bx + c ) or standard "graphing" form ( f ( x ) = a ( x − h ) + k ).
2
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•
Be able to convert a quadratic function from general form to standard "graphing" form and
vice versa.
Be able to graph a quadratic function regardless of the given form for the function.
Be able to find the equation of a quadratic function given its vertex and a solution point.
Use the characteristics of the vertex to solve min/max problems related to quadratic models
(e.g. cost models, projectile motion models, etc.).
Be sure you understand what each part of the ordered pair for the vertex really represents for
the given situation.
You must be able to create a model for vertical projectile motion given initial height ( h0 ) and
1 2
gt where g
2
represents the acceleration due to gravity and is equal to 32ft/sec2 or 9.8m/sec2 depending
on whether you are working with British or Metric units of measure.
Chapter 2.2 – Polynomial Functions of Higher Degree
• Know the general characteristics of any polynomial function (smooth, continuous, no sharp
turns, no breaks).
• Be able to identify the end behavior of a function based on the degree of the function and the
sign on the leading coefficient (i.e. the leading coefficient test).
• Be able to identify any multiplicity of zeros
• Be able to identify zeros of a function by basic factoring methods or via the quadratic formula.
• Be able to generate a function with integral coefficients (i.e. no fractions) based on given
zeros of the function.
• Be able to graph a polynomial function with a fairly high degree of accuracy without use of a
graphing calculator through use of end behavior, zeros, y-intercept, multiplicity and additional
fine tuning (test point) calculations. Remember the strategy we discussed in class – first find
zeros, then complete fine tuning calculations, then use the values from the zeros and from the
fine tuning to set the scales on your graph and finally complete the graph with special
attention to multiplicity and end behavior.
Chapter 2.3 – Dividing Polynomials
• Be able to complete synthetic division based on a linear factor with a linear coefficient of 1 and
then write the quotient as a variable expression including any remainder. Remember that the
synthetic divisor is given by the zero associated with the given linear divisor. Also remember
to use placeholders in the case of missing terms.
• Be able to complete long division of polynomials.
Chapter 2.4 – Real Zeros of Polynomial Functions
• Be able to identify real zeros of a function by using basic factoring methods and the zero
product property when possible.
• Know and be able to use all of the previously discussed tools to find the zeros (Real rational
and/or Real irrational) of higher degree polynomial functions:
o GCF Factoring
o Quick Check for +1 or -1 as Zeros.
o Fundamental Theorem of Algebra (degree dictates total number of zeros).
o Rational Zero Test: given by (factors of p) / (factors of q). Remember, this gives a
list of possibilities but there is no guarantee that any will work.
o Descartes’ Rule of Signs: predicts the possible number of positive and negative REAL
roots based on sign changes in f(x) for positive Real and f(-x) for negative
Real. Remember that Real roots could be rational (and therefore would be in your p/q
list) or they could be irrational (meaning that you'll probably find them by using the
quadratic formula on a polynomial that has been depressed to a quadratic).
initial velocity ( v0 ) information. The model you must know is h(t ) =h0 + v0t −
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
Synthetic Division: allows us to determine whether a value is truly a zero of the
function. Remember that synthetic division is the same as evaluating the function
using the synthetic divisor with the result of that evaluation displayed in the synthetic
division remainder box. Formally, if you get zero as a remainder, you have
demonstrated the Factor Theorem. If you get a value other that zero as a remainder,
you have demonstrated the Remainder Theorem.
o Intermediate Value Theorem - remember that if the synthetic division remainder is not
zero, you have still gained information with regard to a point on the graph of the
depressed polynomial which can be used to help decide on a logical next guess for a
zero. For example, if a synthetic divisor of 1 yields a synthetic remainder of -2, then
we at least know that the point (1, -2) is on the graph of the depressed polynomial.
o Upper and Lower Bound Rules: helps you to decide on direction for your next guess
based the type of synthetic divisor and the signs on the synthetic division
quotient. Remember that the upper bound rule may only be considered if the
synthetic divisor is positive and accordingly, the lower bound rule may only be
considered if the synthetic divisor is negative. Also keep in mind that 0 may be
counted as positive or negative when inspecting the pattern of the quotient row.
o Factoring/Zero Product Property (ZPP) and Quadratic formula. You must be able to
find zeros of quadratic functions be either of these methods, without hesitation. No
mistakes in the quadratic formula will be tolerated!
• Keep in mind that if you are finding the zeros of a polynomial of degree four or higher, once
you find one zero, make sure that you base any further work on the depressed polynomial
with the overall goal of depressing the polynomial in a cascaded manner down to a
quadratic. Once you have a quadratic, factoring/ZPP or the quadratic formula may be used to
find the remaining zeros.
• When you generate the Fully Factored Form (FFF) of the polynomial, make sure that it will
expand (i.e. multiply out) to give the exact original function. If the FFF has fractions in it but
the original function does not, chances are that you factored out and lost a constant along the
way that would "clear the fractions".
Chapter 2.5 – Fundamental Theorem of Algebra (Polynomials with Complex Zeros)
• Be able to identify ALL zeros (Real and/or Complex) of a function by using all of the methods
previously used in Chapter 2.4.
• Be able to apply quadratic methods to find zeros of higher degree polynomials. We used
simple factoring by difference of squares and ZPP for higher degree binomial functions and the
"U-Substitution" method to handle higher degree trinomial functions.
• Be able to generate a polynomial function with integral coefficients (i.e. no fractions) given the
zeros of a function. (Reference p 257, #’s 9-16). Remember that if you are asked for a
polynomial with Real coefficients and you are given only one complex zero or one irrational
zero, you must automatically consider the conjugate of these zeros in developing your answer.
• Given one complex or irrational zero of a function, be able to generate the quadratic factor
from which the given zero and its conjugate would have been derived and then be able to
complete long division of the given polynomial by the newly generated quadratic factor to find
any remaining zeros. (Reference, p257, #’s 22, 24)
• Chapter 2.6 – Rational Functions
• You should be familiar with all of the graphing techniques developed earlier in Chapter 2 for
graphing of polynomials (e.g. use of zeros, multiplicity, end behavior, etc.).
o
•
•
•
Know the technical definition of a rational function based on the quotient of two polynomials
with the polynomial in the denominator not equal to zero.
Keep in mind that our earlier work focused on rational functions without a common factor
between the numerator and denominator. In the latter part of our work however, we focused
on rational functions that did have common factors between the numerator and denominator
which ultimately resulted in the existence of holes (or punctures) in the graph.
Remember that if you simplify away a common factor between the numerator and
denominator, you are required to state the simplified rational function with a restriction for the
zero associated with the factor that was removed. This additional restriction statement is
required because it informs whoever is using the simplified equation of a preexisting
restriction that is now hidden from view.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
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Be able to determine the vertical asymptotes (based on domain restrictions) and horizontal
asymptotes (end behavior as predicted by the limiting values of function as the domain
approaches +/- infinity) for a rational function. It is expected that you will be able to use
basic Calculus limit notation where appropriate to find horizontal asymptotes.
Be able to identify the conditions under which a slant asymptote would exist and be able to
determine the equation of the slant asymptote (via polynomial long division). You should be
able to justify why the remainder from long division is discarded (via limit analysis as x
approaches +/- infinity).
Remember that all asymptotes are given by equations of lines and therefore must be stated in
equation form. Also keep in mind that asymptotes are reference lines and therefore must
always be drawn with dotted or dashed lines (never solid lines).
Be able to determine when there is no vertical asymptote (i.e. when there are no restrictions
to the domain) and be familiar with the effect of no vertical asymptotes on the graph of the
associated rational function.
Be able to justify whether the rational function crosses the horizontal asymptote by setting the
rational function equal to the value of the horizontal asymptote. If you can solve for x, then it
crosses. If you obtain a false statement (e.g. -3 = 7), then it does not cross.
Be able to graph various types of rational functions. You should be able to define graphing
parameters including coordinates of x and y intercepts as well as equations of vertical,
horizontal and slant asymptotes. You should also consider determining odd/even/neither
function symmetry to support and possibly simplify your “fine tuning” calculations.
In the event that a graphing parameter does not exist, you must be also able to provide
justification for its non-existence.
Remember that besides defining the graphing parameters, the "fine tuning" calculations are
a critical and required component of the graphing process.
Chapter 3
Chapter 3.1 - Exponential Functions and Their Graphs.
• Know the basic definition of an exponential function ( f ( x ) = b x where b > 0 but not equal to 1
and x = any real number).
• Be able to differentiate between exponential decay versus exponential growth functions (refer
to WS 3.1 distributed at the beginning of the chapter). You should also be able to identify the
growth or decay rate (as a percentage) from the multiplier and vice versa.
• Be able to graph a "no frills added" version of an exponential function by pattern.
• Be able to graph an exponential function with various transformations. Remember - find the
"pseudo axes" based on translations and then graph from the "pseudo origin" by pattern. You
should also be able to state in words the types and magnitude of transformations involved.
• Be able to determine the domain and range of an exponential function (think about the
graph).
• Be able to identify any horizontal asymptotes of an exponential function via limit analysis. For
growth, you need to examine the limit as the domain approaches −∞ ; for decay, examine the
limit as the domain approaches +∞ . Keep in mind that exponential functions do not have a
vertical asymptote.
• Be familiar with the natural base, e. You do not have to reproduce the development of e via
the Calculus limit concept presented in class but you should at least be able to identify the
domain, range and y-intercept of the basic graph of f ( x ) = e x .
Chapter 3.2 - Logarithmic Functions and Their Graphs
• Know the basic definition of a logarithmic function ( f ( x ) = logb x ) where b > 0 but not equal to
•
•
1 and x > 0.
Know that logarithms and exponential functions are inverses of each other and as such,
composition of an exponential with its inverse logarithmic function yields the identity function,
x. Of course the same holds true for composition of a logarithmic function with an exponential
function.
Be able to convert from log form to exponential form and vice versa.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
•
( )
5
Be able to simplify logarithmic expressions without a calculator (e.g. =
log2 32 log
5 ).
=
2 2
Think prime factorization of the argument.
• Be able to graph "no frills added" logarithmic functions as well as those involving various
transformations. You should be able to identify and quantity the types of transformations.
• Be able to determine the domain and range of a logarithmic function (for domain, remember
that the argument of the log function must be greater than zero).
• Be able to identify any vertical asymptotes of a logarithmic function (think about using the
domain statement to identify the vertical asymptote as a "dividing" line between the domain
values and the restricted values).
• Be familiar with the natural log function in terms of what it represents (inverse of e x ;
log base e) and how it is graphed (same as other logs but now using a base of e).
Chapter 3.3 - Properties of Logarithms
• Know the product, quotient and power rules for logarithms and be able to use to rules to
o rewrite the sum, difference or constant multiples of logs in a log expression as a single
log, or
o expand a single log into the sum, difference or constant multiple of logs.
• Be able to use the change of base rule to find the value of "un-common" logarithms.
Chapter 3.4 - Solving Exponential and Logarithmic Equations
• Know and be able to apply all skills as developed in Chapters 3.1 - 3.3 (see below for the Ch
3.1-3.3 Mini Quiz study guide) to solve the higher level problems in Chapters 3.4-3.5.
• Be able to use the 1:1 property of exponents to solve simple exponential equations (e.g.
8 x = 322 x −1 ). Remember to complete prime factorization of the bases and apply the power of
power rule to generate an equation with the same base on both sides of the equation. If the
bases are then equal, the exponents must also be equal.
• Be able to solve various equations that contain exponential expressions (i.e. variable is in
exponential position). Remember that the first key step is to ISOLATE the exponential
term. The second key step is to CHANGE THE FORM from exponential to logarithmic so that
you can solve for the variable value.
• For exponential equations that follow the pattern of a squared term, linear term and constant
term, be able to use the method of u-substitution to solve the equation by quadratic
methods. Remember that once you solve for the value of u, you must then use your definition
of u to solve the equation in terms of the original variable (e.g. x).
• Be able to use the 1:1 property of logarithms to solve simple logarithmic equations (e.g.
logb x = logb y means that the arguments, x and y, must be equal).
•
Be able to solve various equations that contain logarithmic expressions. Remember to simplify
each side of the equation to a single term. If the simplified equation is one log term equal to
another log term (with the same bases), then set the arguments equal to each other and
solve. If the simplified equation is a log term equal to a number, then change the form and
solve.
• In all cases involving logarithmic expressions, BEWARE OF EXTRANEOUS ROOTS. Of critical
importance is that your solutions must yield positive arguments of all log terms in the
ORIGINAL equation. It does not matter if the solution value is positive or negative - it is the
value of the logarithmic argument that must be positive since the domain of log functions is
(0, + ∞) .
Chapter 3.5 - Solving Applications of Exponential Functions.
•
You must know and be able to apply the compound interest formula for periodic compounding
r

=
F P 1 + 
n


•
nt
(e.g. quarterly, monthly, daily, etc) as well as for continuous compounding,
the "Pert" rule F = Pe rt . You must know these formulas - they will NOT be supplied.
You must know and be able to apply the model for exponential growth and decay using the
natural base ( A = A0 e kt ). Again, this model will NOT be supplied. Possible applications
involve half life, doubling time, percentage increases or decreases, etc.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
•
•
You must be able to solve applications where no initial condition is provided (i.e. you do not
know the amount at time = 0). Remember to find k first.
Use your calculator memory to store intermediate calculated values. Do not use rounded
versions of these intermediate values in further calculations since they will propagate any
rounding error. If you are going to round, then round only your final answer and follow
directions carefully with regard to the extent of rounding (typically you can expect 3 decimal
places).
Chapter 10 (Again…)
Chapter 10 – Sequences and Series
• Be able to find the formula for the nth apparent term of a given sequence. Remember to look
for obvious patterns (perfect squares, reciprocals and any other special operation) before
completing analysis for arithmetic sequences (common difference) or geometric sequences
(common ratio).
• Recall that if the sequence is presented as a collection of fractions, consider developing a rule
just for the values in the numerator and then do the same for the values in the denominator
before pulling the two separate rules together into a single rule. In the same way, if there are
alternating terms, consider incorporating something like (-1) to a power into the overall rule in
order to achieve the necessary alternating signs.
• Be able to find the terms of a sequence defined recursively (like Fibonacci’s
sequence). Consider using a number line (as explained in class) to understand the relative
position of terms in a sequence that need to be used to calculate the value of new terms (e.g.
the value of a term is equal to 3 times the previous term minus two times the 2nd previous
term would be the interpretation of=
an 3an−1 − 2an−2 ). Remember that you must always be
supplied with a “seed” values to start the recursive sequence (e.g. for the Fibonacci sequence,
a1 = 1 and a2 = 1 would be given).
•
•
•
Be familiar with summation notation and be able to evaluate by 1) expansion of the
summation and/or 2) application of one of the summation formulas for arithmetic or geometric
sequences as applicable. Remember that if the summation is not arithmetic or geometric,
then you may have to simply expand the summation based on the lower and upper index of
summation and then add up the terms.
Be able to simplify factorial expressions including both numerical factorials like 7! as well as
factorials expressed in algebraic form like (2n + 1) ! .
=
dn + c ) and geometric
Know the general form of the rules (or formulas) for arithmetic ( a
n
a
(=
n
a1st ⋅ r n−1
)
sequences. Remember that arithmetic sequences behave in the same manner
as a linear function (i.e. the sequence "slope" is the common difference between terms as
represented by a change in the range ( an ) divided by the change in domain (n) along with the
projected value of what we called the " 0th " term (i.e. the " anth intercept")). For geometric
•
•
sequences, remember that the general form of the rule involves a common ratio and value of
the first term.
Given a sequence of numbers, be able to identify the type of sequence (arithmetic or
geometric) and then find the nth apparent term (i.e. formula) for the sequence.
Be able to find the sum of a Finite Arithmetic sequence using the appropriate summation
formula (recall the example of Gauss finding the sum of numbers from 1 to 100).
 n
=
Sn   ( a1st + alast )
2
•
Keep in mind that it is not possible to find the sum of an Infinite Arithmetic sequence.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
•
Be able to find the sum of a Finite Geometric sequence using the appropriate summation
formula.
1 − rn
Sn = a1st �
(1 − r )
(
•
Be able to find the sum of an Infinite Geometric sequence (a.k.a. sum of a Series) using the
appropriate summation formula.
=
S
•
)
a1st
(1 − r )
only if r < 1
Remember that this formula applies only when the absolute value of the common ratio is less
than 1.
Chapter 4
Chapter 4.2 – Right Triangle Trigonometry
• Be familiar with the Decimal Degree (DD) and Degree Minute Second (DMS) notation for angle
measurements.
• Know the relationship between DD and DMS notation and how to convert from one form to the
other using your calculator.
• Basic definitions for all six trigonometric functions as given by the ratio of sides in right
triangles.
• Be familiar with the reason why trigonometric ratios for a given angle are always the same
regardless of the size of the right triangle.
• Proper trigonometric notation (Don't forget to include angle measure or symbol of angle after
name of trig function).
• Given the dimensions of two sides of a right triangle, be able to find the third side and then
develop the values for all six trigonometric functions of an acute angle within the triangle. All
answers must be expressed in simplest rational, radical form (i.e. no decimals!).
• Given the value of one trigonometric function for an angle, develop a right triangle diagram to
model this trig ratio and then determine the values of the 5 remaining trigonometric functions.
• WITHOUT CALCULATOR, be able to develop the values of the six trigonometric functions for
the common angles (30⁰, 45⁰ and 60⁰). Values must be expressed in simplest rational, radical
form.
• WITHOUT CALCULATOR, given the value of a trigonometric ratio for the trig function of a
common angle, be able to identify the common angle.
• Be able to use the calculator to find the value of any of the six trigonometric functions for
uncommon angles. REMEMBER that you never use the reciprocal of an angle measure (i.e. if
asked for something like csc(22⁰), do not use sin(1/22⁰); instead calculate as 1/sin(22⁰)).
• Given the value of a trigonometric ratio for the trig function of an uncommon angle, be able to
find the angle using the appropriate inverse function on the calculator. All work must be
shown with proper inverse notation!!!
• Be familiar with the concepts of angle of depression and angle of elevation and be able to
incorporate into application problems. Remember that one of the sides of your reference
angle will always be the horizon.
• Be familiar with the concept of bearings and how to use bearings in application problems.
• In all problems, complete all algebraic transformations first to yield an EXACT answer before
using the calculator to complete final calculations and rounding. Remember that the use of
rounded numbers in the middle of a series of calculations will create greater levels of error in
your final answer (called propagation of error in Calculus).
Chapter 4.4 and 4.5 – Law of Sines and Cosines
• Be able to use proper notation and show all work for all problems including all transformations
to exact answer form (i.e. the final form you would use to plug into your calculator).
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
•
Follow directions with regard to rounding. Use the storage feature of your graphing calculator
as necessary and never round until the final calculation. Also remember to include units of
measure where appropriate.
• The basic statement of the Law of Sines (LOS) as applied to oblique (i.e. non-right)
triangles. The Law of Sines indicates that the ratio of the sine of an angle in a triangle to the
length of its opposite side is constant for any angle/opposite side pair within the triangle.
• Be able to "state the case presented" based on given information for a triangle (e.g. AAS,
ASA, SSA, etc).
• Be able to complete/solve triangles (i.e. find all of the missing pieces) using LOS and limited
given information about a triangle. Remember to avoid using LOS to find any angle which has
even the slightest potential of being obtuse since LOS does not predict obtuse angles.
• For SSA cases, know that all SSA cases are classified as ambiguous regardless of the outcome
of your analysis.
• Be able to complete analysis of triangles given SSA case information according the decision
process used in class. As reiterated numerous times in class, you must be able to complete
this process on your own without the aid of the decision chart distributed at the beginning of
this topic.
• Be able to solve triangles given SSA information. If two triangles are possible, you must be
able to solve for both the acute and obtuse cases. Remember that in these cases, the angle
that has the opportunity to be either acute or obtuse is the angle at the end of the far side "as
you travel through given information." Also remember that the relationship between the acute
and obtuse versions of this angle is that they are supplementary.
• Be able to solve various applications using LOS. You need to be able to solve bearing related
problems as well as problems which involve multiple adjacent triangles wherein solving for
something common to both triangles (e.g. length of a side) allows you to solve for other
specific information within one of the triangles (the "cascaded" solution process).
• Remember that your application of prior Geometry knowledge, especially with regard to angle
relationships, will be critical to solving many application problems.
• Keep in mind that if you end up with a right triangle within your solution, you should consider
using basic right triangle trigonometry relationships to solve the problem rather than spending
time to use LOS.
• Know and be able to apply the Law of Cosines to complete/solve triangles (i.e. find all of the
missing pieces) using limited given information about a triangle. Remember that LOC is "goof
proof" in the sense that it predicts both acute and obtuse angles (as compared to LOS that
predicts only acute angles). Also recall that after you have found the largest angle in a
triangle using LOC, most prefer to use LOS to solve for one of the two remaining angles and
then solve for the last one by difference. (Note that if you want to use LOC for everything,
feel free to do so - just be careful with your calculator entries.)
• To reiterate, if you do not have conclusive evidence that a given angle is acute, then solve for
that angle using LOC.
• Know and be able to apply the Triangle Inequality Theorem to determine whether a triangle
can be even be made before proceeding with analysis. If no triangle can be made, you should
document the reason for no solution by showing the inequality that demonstrates that the sum
of two sides is not greater than the third and therefore a triangle cannot be formed.
• Be able to apply LOC to solve various application problems as practiced during HW exercises.
• Be able to find the area of a triangle given SAS case information using the
1
formula Area = ab * sin(c) , etc. Recall the angle between the two given sides is called the
2
included angle.
• Be able to find the area of a triangle given SSS information through use of Heron's Formula.
Remember that calculation of the "s" value in Heron's Formula represents the semi-perimeter
so divide the sum of the lengths of the sides (i.e. perimeter) by 2, not by 3.
Combined Applications:
• Be able to apply any of the concepts in 4.4 and 4.5 to solve application problems. In this
case, you will have to decide for yourself on the "right tool for the right job."
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
Chapter 4.1, 4.3 Radian Measure, Trig Function of Any Angle
• Know what it means to draw an angle in standard position.
• Know what is meant by an initial side and terminal side of an angle in standard position.
• Know that positive angles are created by rotation of the terminal side in the counterclockwise
direction and that negative angles are created by rotation in the clockwise direction.
• Know the definition of a radian and be able to convert angle measures from one form to
another (i.e. convert from degrees to pi radians, from decimal radians to DMS, etc.)
• Be able to use your calculator to find the value of any trig function for an angle given in
degree, decimal radian or pi radian measure.
• Know the meaning of coterminal angles and be able to calculate both positive and negative
coterminal angles in degree measure as well as radian measure.
• Know how to calculate complements and supplements of a given angle. Remember that
complements and supplements are always positive. If a complement or supplement does not
exist, state "NONE."
• Know and be able to apply the arc length formula. REMEMBER: ALL ANGLE MEASURES THAT
GO INTO AND COME OUT OF THIS CALCULATION MUST BE IN RADIAN MEASURE.
• Know and be able to apply the definitions for linear speed and angular speed:
o AS = 2 π (or 360 degrees) * RS
o LS = 2 π *r * RS
o LS = r * AS (where AS is in radians/unit time)
• Remember the rotational speed is given in typical units like rpm or rps.
Chapter 5
Chapter 5.1 – Unit Circle
Be able to label all components of the Unit Circle including all common angles, both positive
and negative, in pi radian or degree measure, as well as the coordinates. You should also
know, at a minimum, decimal radian measure for quadrant angles.
• Know the key characteristics of the Unit Circle including why it is called the unit circle and the
significance of the coordinates associated with each common angle.
• Based on the values of the x and y coordinates on the edge of the unit circle, be able to
determine the sign of each trig function in any quadrant (A-S-T-C).
• Know how to use the period of the sine and cosine function to calculate values of trig functions
for angles that are greater than one complete revolution (Remember to use concept of
coterminal angles).
• Given the value of a trig function for a common angle, be able to identify all common angles
within one revolution that possess the given trig function value (i.e. find all angles between 0
and 2 π that have a value of sine = 1/2.)
• Given the coordinates of a point in the coordinate plane, be able to determine whether the
point is inside, outside or on the unit circle and then find the value of all trig functions for an
angle whose terminal side goes through that point. Keep in mind that if the point is not on the
unit circle, you can still calculate the values of trig functions by constructing an angle to the
closest x-axis and then use right triangle trig definitions (i.e. SOH CAH TOA).
• Given the value of a trig function for an uncommon angle, be able to identify all angles within
one revolution that possess the given trig function value (i.e. find all angles between 0
and 360 degrees that have a value of sine = 0.2468.)
Chapter 5.2 – Sine and Cosine Graphs
• Given a sine or cosine trigonometric function, be able to identify all transformations of the
given function including amplitude (vertical stretch/compression determined from "a"), period
changes (horizontal stretch/compression determined from "b"), phase shift (horizontal
translation determined from “h”; often referred to as “c" in trig textbooks), vertical shift
(vertical translation determined from "k"; often referred to as “d” in trig textbooks) and
vertical or horizontal reflections (based on sign of "a" and "b" respectively).
•
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
•
Be able to manually graph sine and cosine trigonometric functions with high degree of
accuracy. Generally, you will be asked to graph one cycle but be prepared to graph more if
requested.
• You must be able to create graphs using given scales or by developing your own scales. If
developing your own scales, look for common factors between the period, quartile and phase
shift values.
• Be able to state the domain and range, in interval notation, of given sine and cosine
trigonometric functions by inspection.
• Be able to develop sine and cosine trigonometric functions based on a given graph.
• Be able to develop both sine and cosine trigonometric models based on a given graph or a
given scenario like the Ferris Wheel.
• Be able to use models to make calculated predictions (e.g. height as a function of time for the
ferris wheel).
Chapter 5.3 – Tangent, Cotangent, Secant and Cosecant Graphs
• Be able to manually graph tangent, cotangent, secant and cosecant trigonometric functions
with high degree of accuracy. You should be prepared to graph one or more cycles as
requested.
• You must be able to create graphs using given scales or by developing your own scales. If
developing your own scales, look for common factors between the period, quartile and phase
shift values.
• Be able to develop a general equation for all asymptotes associated with a particular function.
• Be able to state the domain and range of given trigonometric functions by observation.
• Be able to determine key graphing parameters for given trigonometric function (i.e. period,
phase shift, vertical shift, vertical reflection (flip), etc.).
Chapter 6
Chapter 6.5 – Inverse Trigonometric Functions
Know nomenclature for inverse functions as arc functions.
Be able to manually graph the inverse sine, inverse cosine and inverse tangent functions.
You must be able to identify the domain and range of these three primary inverse
trigonometric functions.
• You must be able to evaluate inverse trig functions.
• You must be able to evaluate composition of "like type" trigonometric and inverse
trigonometric functions. Remember the key question that can be asked to make a
“sophisticated data based decision"(a.k.a. shortcut) about how to evaluate the problem: “Is
the domain of the innermost function within the range of the outermost function?” If yes,
then the answer is the domain of the innermost function. If no, then you must complete the
problem the “long way” starting with the evaluation of the inner function and then moving to
the outer function.
• You must be able to evaluate composition of "mixed type" trigonometric and inverse
trigonometric functions. Remember that this type of problem is completed via triangle
construction.
• Keep in mind that while π is often used to help express the measure of an angle, the reality is
•
•
•
that π is a symbolic representation of a number (3.14…). As such, a problem like sin−1 (π )
would have no solution because π (or 3.14…) is not within the domain of  −1,1 for the
inverse sine function. Similar reasoning should be used when interpreting problems with
arguments that contain the natural base e (or about 2.718…).
• Consider rereading information in Inverse Trigonometric Function packet that I distributed in
class for more details on thought process.
Chapter 6.1 –Trigonometric Identities
• All 5 major types of trigonometric identities and transformations of those identities.
• Be able to simplify trigonometric expressions through use of the trigonometric
identities. Remember that your target is the simplest form possible (i.e. a number or a trig
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
function, or a combination of both wherein all trig terms are in the numerator in the final
answer.
• Be able to use trig identities to verify trigonometric equations (i.e. verify
identities). Remember that all work is to be done on one side of the equations with a little bit
of "envisioning" work where necessary on the other side to present an intermediate target.
• Be able to use identities to find the values of trigonometric functions. Remember to be
sensitive to location of the terminal side of the angle.
Chapter 6.5 –Trigonometric Equations
•
Be able to solve trigonometric equations for specific intervals (like [0,2pi)) or general solutions
(infinite number of possibilities as represented by a minimum number of solution statements).
• You must be able to use algebraic methods and/or trigonometric identities to isolate the
trigonometric function(s). You must then use your knowledge of common angles (and/or
uncommon angles) to find all solutions requested.
• Algebraic methods include the use of factoring and zero product property, quadratic formula,
quadratic methods applied to non-quadratics (e.g. u-substitution to change a quartic to a
quadratic), etc. Don't forget to consider the typical hierarchy of factoring methods (see
Factoring Toolbox in Notesheets section as necessary to review hierarchy and factoring
patterns).
• Be able to solve trigonometric equations with multiple angles for specific solutions over a given
interval. Remember that the key is to first find general solutions for the multiple angle (like
2x, 3x, x/2, etx.), solve the general solutions for the single angle (x), and then use this
general solution equation to find specific solutions over the required given interval.
• Chapter 6.2 / 6.3 – Sum and Difference Formulas / Double Angle Formulas
• Be able to rewrite an angle as the sum or difference of two common angles.
• Know and be able to apply sum and difference formulas to find the exact value of trig
functions for uncommon angles (simplest radical form).
• Be able to use sum and difference formulas in order to simplify trigonometric expressions,
verify identities or solve trigonometric equations that involve trig functions of sums or
differences.
• Be able to find the value of a trig function based on given trig function values for 2 angles
(e.g. sin(theta)=2/3 and theta is in Quad I, cos(beta)= -1/5 and beta is in Quad IV -> find the
value of sin (theta + beta).
• Know and be able to apply double angle formulas to find the exact value of trig functions
(simplest radical form) based on given information (e.g. find the value of all trig functions for
the angle 2*gamma given info on the value of one trig function for the angle gamma.)
• Be able to use double angle formulas to solve trigonometric equations.
• Know and be able to apply half angle formulas to find the exact value of trig functions for an
uncommon angle. Remember that the determination of whether sine and cosine are positive
or negative depends solely on the location of the half angle.
• As per our previous work, you must be able to employ the problem solving strategy to find the
solutions for trig functions of multiple angles. Remember the strategy - solve for general
solution for the single angle and then apply your general solution for all values of n that
generate solution within the specified range of [0, 2pi).
• Also as per our previous work, if you are asked to solve an equation for an angle which is not
a common angle, you must be able to use your calculator (inverse function) to find the
appropriate answers on [0, 2pi) rounded to the specified number of decimal places.
Chapter 9
Chapter 9.1 – Conic Basics
• Be familiar with the origin of the conic sections as cross sections formed by the intersection of
a plane with a double end-to-end cone.
• Be able to identify the type of conic based on the format of a given equation.
• Know and be able to apply the standard form for the equation of a circle.
PRE CALCULUS 6.0 - THE YEAR IN REVIEW
Chapter 9.2 – The Parabola
• Be familiar with the formal definition of a parabola based on the set of all points in a plane
that are equidistant from the focus and the directrix.
• Know and be able to use the standard (graphing) form of a parabola to create a graph of the
parabola and/or to identify key graphing parameters (including the "p" value) for the parabola.
Remember that if the equation is quadratic in x, then the parabola opens up or down and if it
is quadratic in y, then it opens right or left.
• Know and be able to create the standard equation of a parabola given certain graphing
parameters (e.g. coordinates of focus and center, etc.)
• Be able to accurately sketch a parabola including the focus and the directrix.
• Know and be able to identify the coordinates of the focus as well as the equation of the
directrix.
Chapter 9.3 – The Ellipse
• Be familiar with the formal definition of an ellipse based on the set of all points in a plane
where the sum of distances from a point on the ellipse to two fixed points (foci) is constant.
• Know and be able to use the standard (graphing) form of an ellipse to create a graph of the
ellipse and/or to identify key graphing parameters for the ellipse. Remember that ellipse will
be oriented vertically if the larger denominator value is under the y term or oriented
horizontally if the larger denominator value is under the x term.
• Know and be able to create the standard equation of an ellipse given certain graphing
parameters (e.g. coordinates of foci and vertices, etc.)
• Be able to accurately sketch an ellipse including the foci, vertices, co-vertices, major axis and
minor axis.
• Know and be able to apply the proper relationship between the distance from the center to the
vertex, co-vertex and focus points. (Think triangle construction).
• Know and be able to identify the coordinates of the center, foci ,vertices, co-vertices, length of
major axis and length of minor axis.
Chapter 9.4 – The Hyperbola
• Be familiar with the formal definition of a hyperbola based on the set of all points in a plane
where the absolute value of the difference between distances from a point on the hyperbola to
two fixed points (foci) is constant.
• Know and be able to use the standard (graphing) form of a hyperbola to create a graph of the
hyperbola and/or to identify key graphing parameters for the hyperbola. Remember that
hyperbola will be oriented vertically if the first term is the y term or oriented horizontally if the
first term is the x term.
• Know and be able to create the standard equation of a hyperbola given certain graphing
parameters (e.g. coordinates of foci and vertices, etc.)
• Be able to accurately sketch an ellipse including the foci, vertices, co-vertices, transverse axis,
conjugate axis, asymptote "box" and corresponding asymptotes.
• Know and be able to apply the proper relationship between the distance from the center to the
vertex, co-vertex and focus points. (Think triangle construction).
• Know and be able to identify the coordinates of the center, foci ,vertices, co-vertices and
equation of asymptotes.