Subtest I: Algebra Practice Qs

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Are you prepared for Subtest I: Algebra?

Then, have a stab at the following!

Assess yourself: You should do OK on the Test if you could solve 3-4 of the 5 Qs below, taking about 20 minutes/Q!

5. a) Finds the equation of the parabola passing through (0,5), (2,5) and (-2,21).
b) Find the equation of the line with x-intercept 5/8 making a triangular area of 25/16 in the 4th Quadrant.
c) At what point(s) does the line intersect the parabola?

4. If 1, 1/2 and -3 are x-intercepts of a polynomial function f(x) = ax³ + bx² + cx + 3, find f(x) and graph it.

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3. If f(x) = x² - 7x + k, find ALL value(s) of k such that f(x) has
a) No x-intercepts
b) Exactly 1 x-intercept
c) 2 x-intercepts
Interpret your answer graphically.

2. On the same graph, plot the following functions:
a) f(x) = x² - 2x - 8
b) g(x) = r(f(x))
c) h(x) = 1/f(x)
Note: r ~ Square Root

1. Graph the rational function: f(x) = (x² - 6x + 8)/(x² - 3x) stating its
a) Domain
b) x- and y-intercepts
c) Vertical and Horizontal Asysmptotes


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SUBTEST II: Geometry Top 8 Topics to Master

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The following are the 8 MOST important topics and associated concepts/skills you ought to be a Guru at for the CSET Subtest II Geometry:

1. Definitions, Postulates, Properties and Theorems: Basic knowledge and understanding of these is tested throughout the Multiple Choice section of Subtest II, and of course, these are vital for constructing Geometric Proofs!

You shall find a fairly exhaustive list of Definitions, Postulates, Properties and Theorems at other links on this site - Subtest II: Writing Geometry Proofs the Painless Way! and SUBTEST II: Geometry - 42 Vital Theorems!)

2.Properties of Parallel Lines: Proofs regarding Alternate Interior Angles, Alternate Exterior Angles and Consecutive Interior Angles and their Converse (ie. conditions for 2 lines to be parallel).

You MUST know in a very fundamental fashion when to use the Theorems themselves, and when their Converses: it can mean the difference between getting a question right (and passing!) and not. While proofs of parallel lines themselves are not asked, these form the building blocks of more advanced proofs, for instance, for sophisticated problems on triangle congruence, parallelograms and similar triangles.

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3. Congruence of Triangles: You are all but assured ONE question on this topic on the Free Response section, and it might call for application in a direct fashion - for instance, having to prove that corresponding parts of 2 triangles are congruent - or in an indirect manner - embedded in a question on parallelograms, perhaps.

Not infrequently, you may be provided with two pairs of triangles, and you have to show
a) the congruence of 1 pair to prove that
b) their corresponding parts are congruent, and use this fact to prove
c) the congruence of the second pair of triangles, and employ this, finally, to prove that
d) corresponding parts (sides/angles) of this pair are congruent!

4. Triangle Theorems and Properties: These relate to sides and angles of triangles in general, and for Isosceles and Right Triangles, in particular. This also includes Perpendicular Bisector, Angle Bisector, Median, and Mid-Segment Theorems.

5. Quadrilaterals and Parallelograms: This topic is a terrific assessment of your skills regarding Parallel Lines and Congruence of Triangles as they are indispenseable here! You ought to master the proofs about properties of Parallelograms and Rhombuses. Those on Rectangles, Squares, Trapezoids and Kites can be treated depending on the time you have, though I strongly recommend it if it can 'squeezed in'.

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6. Transformations: This is a much neglected topic, but you ought have a robust grasp of elementary transformations like Reflection, Rotation and Translation, which are sometimes coupled with the Similarity concept of Dilation. Further, the Proofs on these transformations being isometries are VERY important!

7. Similar Polygons and Solids: Properties about their sides and angles; and Ratios of their Perimeters, Areas and Volumes (for similar solids). A passing acquaintance with Similar Triangles is sufficient.

8. Right Triangles: This topic concerns 2 Geometric Mean Theorems (using Similarity of Triangles), Pythagoras Theorem and Special Right Triangles (30-60-90 and 45-45-90), all of whose proofs I recommend that you know. Further, Right-Angle-Trigonometry is a fundamental skill area using which you can find the Areas and Perimeters of regular polygons, on which you're assured one, if not more questions!

As a general tip, Formulae of Areas of simple geometric figures and 3-dimensional solids are critically important and therefore useful to commit to memory, since they're not provided.

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How many Subtests should you take at one time?

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Each examination is designed to allow sufficient time for examinees to complete all subtests in a five-hour session. The subtests are not individually timed and you may spend as much time as you choose on each subtest.

Some examinees may finish testing well before the scheduled ending time of the test session. However, you should be prepared to use the entire test session. You will be allowed no more than the allotted time to complete the subtest(s).

Some of the factors that determine the number of Subtests you want to take at one sitting are:

a) Purpose of your appearance: Since most candidates have only the vaguest notions of what the CSET is about, many take one or more (usually 2) of the Subtests to simply 'get a feel' for the test. I strongly recommend it!

However, as I've stated in an earlier posting, even if you're appearing for 'experimental' purposes, go as best prepared as you can possibly be!

This way, you'd get a surer understanding of your strengths and weaknesses, and you wouldn't be wasting money and time simply sitting for a couple of hours, skimming the questions languidly, then throwing your hands up and retreating in some consternation!

b) How prepared you are: Each of us has a 'gut-feel' for our state of preparedness. Sure, there are many imponderables that relate to the test - for instance, have you studied enough about a certain topic? Could the emphasis on certain skills/concepts have changed from the previous time? - but ultimately, one either feels confident or not!

But how do you know if you should feel confident?!

Well, for one thing if you can answer at least 75% of the Qs on this site without much trouble, you should do just fine!

Likewise, if you've obtained my database of Qs and feel self-assured in solving most of them, you can feel upbeat!

So, the number of Subtests you should take depends on your comfort with the subject matter.

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In general, if you're reasonably well-prepared, you should plan to allocate about 3-4 minutes for each of the 30 MCQs. Similarly, for the Free Response section, you should budget for about 20 minutes (perhaps, even 30, if you're determined to solve a particularly nettlesome Q that you know you should get!)

Of course, there shall be a few Q (like, on Abstract Algebra / Number Theory: most guess on these Qs!) in the MCQ format that you'd be clueless about and shan't be able to possibly attempt. But to compensate for these, there shall be other Qs that simply consume more time because they're just more extensive, or because you made errors and the choices don't tally!

A similar case holds for the Free Response section. A good assumption is that you attempt just 2 of the 4 Qs.

All this implies that unless you have a robust grasp of Math and are confident - prior to the Test! - that you can do pretty much whatever they throw at you, you should plan to devote about 3+ hours on each Subtest. (I knew my stuff: it took me about one and a half hours / Subtest...)

This means that for most of you it's extremely prudent (financially and otherwise) to take only 1 Subtest per appearance. This way you shan't have to hurry up mid-way on the first, do a poor job of it, and then end up mucking the second one too for want of sufficient time!

Alternately, prepare extremely well for any one Subtest: spend an hour and a half or so browsing through the test where you're weak to get a good idea, and then concentrate the meat of your time on the Subtest you've mastered. This maximizes your chances of success!

Hankering for more CSET Math Qs? Click on Need More Practice Questions? To purchase a vast database of Qs that I've compiled from my experience as a test taker and CSET Single Subject Math instructor at Cal State, San Bernardino, email me at innovationguy@yahoo.com. The CSET Qs on this site are taken from that collection of questions!

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SUBTEST II: Geometry - 42 Vital Theorems

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Below are 42 absolutely indispensable Geometry Theorems you ought to master!


Segment Addition Postulate: Q is the midpoint of PR. Show that PQ and QR are each equal to ½ PR.

Congruent Supplements Theorem: If 2 angles are supplementary to the same angle (or to congruent angles) then they are congruent.

Vertical angles Theorem: Vertical angles are congruent

Congruent Complements Theorem: If 2 angles are complementary to the same angle (or to congruent angles) then they are congruent.

If two lines are perpendicular, then they intersect to form four right angles.

Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate interior angles are congruent.

Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate exterior angles are congruent.

Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of consecutive interior angles are supplementary.

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Alternate Interior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.

Alternate Exterior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the lines are parallel.

Consecutive Interior Angles Converse Theorem: If two lines are cut by a transversal such that the consecutive interior angles are supplementary, then the lines are parallel.

Triangle Sum Theorem: The sum of the measures of the interior angles is 180 degrees.

Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.

3rd Angles Theorem: If two angles of 1 triangle are congruent to two angles of another, then the 3rd angle is also congruent.

Angle-Angle-Side (AAS) Congruence Theorem: If two angles and an non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite of them are congruent.

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Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the end points of the segment.

Perpendicular Bisector Theorem Converse: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Angle Bisector Converse: If a point is in the interior of an angle, equidistant from its sides, then it lies on the bisector of the angle.

Mid Segment Theorems:
* The segments connecting the midpoints of two sides of a triangle is parallel to the third side and half the third side.
* Given: ABC with mid-segments DE, EF, FD. Prove: ADE is congruent to DBF

Triangle Inequality: If one side of a triangle is longer than another side, then the angle opposite the longest side is larger than the angle opposite of the shorter side.

Parallelogram Theorems:
* If a quadrilateral is a parallelogram, then its opposite sides are congruent.
* If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
* If a quadrilateral is a parallelogram, then its opposite angles are congruent.
* If a quadrilateral is a parallelogram, then its diagonals bisect each other.
* If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
* If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
* If an angle of a quadrilateral is supplementary to both its consecutive angles, then the quadrilateral is a parallelogram.
* If a parallelogram is a rhombus, then its diagonals are perpendicular.
* If the diagonals of a parallelogram are perpendicular, then its a rhombus.

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SSS Similarity Theorem: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.

SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the length of the sides including these angles are proportional, then the triangles are similar.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle, intersects the other two sides, then it divides 2 sides proportionally.

Pythagorean Theorem

Tangents to a Circle Theorem: If two segments from the same exterior point are tangents to a circle, then they are congruent.

Inscribed Angle Theorems:
* If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
* If two inscribed angles of a circle intersect the same arc, then the angles are congruent.
* The angle in a semicircle is a right angle (if one side of an inscribed triangle is the diameter of a circle, then the triangle is a right triangle).
* If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

Polygon Angle Theorems:
* The sum of the interior angles of an n-sided polygon is (n-2)180 degrees.
* The sum of the exterior angles of an n-sided polygon is 360 degrees.

Hankering for more CSET Math Qs? Click on Need More Practice Questions? To purchase a vast database of Qs that I've compiled from my experience as a test taker and CSET Single Subject Math instructor at Cal State, San Bernardino, email me at innovationguy@yahoo.com. The CSET Qs on this site are taken from that collection of questions!

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SUBTEST I: Algebra Prep Resources

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What books do I recommend for the Subtest I: Algebra/Number Theory portion of the Single Subject Math CSET?

The following topics are not required from a Precalculus text for CSET Subtest I:

* Trigonometry more sophisticated than ordinary ratios and knowledge of sin / cos / tan of 0-30-45-60-90
* Probability
* Permutations & Combinations
* Rotation of Axes re Conics
* Partial Fractions

First, any regular College-Level Algebra book of RECENT edition (1998+) should be adequate. Also, any Precalculus book (High School or College-Level) would be perfect. Typically, such books have titles with more than one of the following terms:

(Advanced / College-level) Algebra
Trigonometry
Analytical Geometry
PreCalculus

The following texts would be woefully inadequate:

a) Algebra I textbook
b) Algebra II textbook: such texts ONLY serve as an introduction to advanced Algebra concepts, but do not treat topics in the depth as required for the CSET.
c) College Algebra textbook for a basic foundational / remedial Algebra course

Here's how to judge if an Algebra book is sufficient for CSET purposes. Look for the following terms in the Index at the back. (While NOT all terms/concepts are required for Subtest I, they indicate how sophisticated the book is!):

i) Row Transformations or Row Operations or Row Reduced Matrix or Row Echelon Form or Transformations (some of them might be listed under the broad term: Matrix)
ii) Vectors, Orthogonal or Orthogonal Vectors
iii) Piece-wise defined function
iv) Augmented Matrix
v) DeMoivre's Theorem
vi) Oblique Asymptotes

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The following are books I have myself used with abundant profit:

a) Algebra & Trigonometry (with Analytical Geometry)

By: Swokowski/Cole
Pub: Brooks/Cole
Edition: 9th/10th
Comments: A terrific, lucidly written text!

b) Algebra and Trigonometry OR
College Algebra (3rd Edition) OR
Precalculus (3rd Edition)

By: Stewart/Redlin/Watson
Pub: Brooks/Cole
Comments: Another marvellous set of books; any ONE of the above would be fine as they cover the same grounds almost identically!

c) Precalculus with Limits: A Graphing Approach

By: Larson/Hostetler/Edwards
Pub: Houghton Mifflin
Edition: 3rd
Comments: A fairly popular High School Precalculus book; the 3rd edition layout is superior to the 2nd!

d) PreCalculus : With Unit Circle Trigonometry
By: David Cohen
Pub: Brooks/Cole
Edition: 3rd
Comments: A stupendous College-level text by a UCLA chap with some very clever exercises to test your grasp; a little daunting for the rookie, perhaps, but strongly recommended if you want to TEACH Precalculus in High School!

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You can procure these books

a) by approaching friendly High School teachers who might have copies of PreCalculus books to spare for a couple of months

b) by borrowing them from friends who recently did Math courses in college

c) by purchasing them (the best option as they're terrific for Reference purposes!) from a college bookstore (expensive!) OR online (used, hence, affordable!) at Amazon.com or half.com.

d) by borrowing them from your local Public Library. This would entail renewing the books repeatedly, for which wooing date-less librarians wearing large glasses would help...

e) by stealing them from college bookstores, which are, anyway, the face of evil behemoth corporations...

In general, I would strongly recommend having at least TWO of the above Algebra texts!

I've known chaps that do some sort of 'online research' for certain topics. Typical remark: "I was on the internet looking for stuff about parabolas (or vectors)!" I can't fathom what in the blazes these fellows are blathering about, as, to me, it all seems such a beastly WASTE OF ONE'S TIME: consult a bloody %&$#@%$# book!! (For starters, you'd pull up 8769876098 trillion gazillion quintillion mazillion billion jillion number of pages on parabolas! Oh, did I forget to say mahallion?)

Hankering for more CSET Math Qs? Click on Need More Practice Questions? To purchase a vast database of Qs that I've compiled from my experience as a test taker and CSET Single Subject Math instructor at Cal State, San Bernardino, email me at innovationguy@yahoo.com. The CSET Qs on this site are taken from that collection of questions!

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SUBTEST III: Calculus & Trigonometry Prep Resources

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What books do I recommend for the Subtest III: Calculus/Trigonometry portion of the Single Subject Math CSET?

Both Calculus and Trigonometry require their own preparation materials. The History of Math component is only a small part of the test and I don't believe you can 'prepare' for it in the conventional sense!

For a fairly exhaustive and descriptive study content for Subtest III, consult the link Subtest III: Calculus and Trigonometry Syllabus.

For Calculus, I earnestly recommend any - one or more! - of the following books:

a) Calculus and Analytical Geometry

By: Thomas/Finney
Edition: 9/10th
Pub: Addison Wesley;
Comments: This book is simply a classic, there's nothing more to be said! The authors (G B Thomas and Ross L Finney) have written books with IDENTICAL titles INDIVIDUALLY too, but all these books are of the same extraordinary calibre!! Also, this book is available as one book or as two (Vol 1 and Vol. 2): in the latter case, Vol 1 is more than adequate for the CSET.

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b) Calculus (4th Edition) OR
Calculus: Concepts and Contexts (2nd Edition)

By: James Stewart
Pub: Brooks/Cole
Comments: Another excellent set of books, very simply and ably written by someone who has also co-authored one of the Algebra books I've strongly recommended.

c) Calculus

By: Larson/Hostetler/Edwards
Edition: 5th/6th
Pub: Heath
Comments: Comparable to the others! Very user-friendly, and a standard text for AP Calculus in High Schools.

d) Calculus

By: Smith/Minton
Edition: 2nd
Pub: McGrawHill
Comments: A book written in a singularly detailed manner: these chaps take nothing for granted, and there are small, insightful observations about problem-solving, offering tips, pointers and pitfalls at all the right places, which enormously endeared itself to me!

For Trigonometry, any of the books I've recommended for Subtest I: Algebra would be sufficient. (Examine Subtest I: Algebra Prep Resources).

In the Algebra books, there are usually 3 chapters devoted to Trigonometry, and all three merit your attention and mastery!

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Otherwise, there are dedicated books on the subject that are excellent, for instance:

Analytical Trigonometry with Applications

By: Barnett/Ziegler
Edition: 6th
Pub: PWS Publishing

Remember: You need not purchase the current edition of any of the above textbooks! Changes from one edition to the next is usually incremental , and it's a bloody beastly scam of rapacious book publishers to insure continuous sales by supposedly updating their books!

Hankering for more CSET Math Qs? Click on Need More Practice Questions? To purchase a vast database of Qs that I've compiled from my experience as a test taker and CSET Single Subject Math instructor at Cal State, San Bernardino, email me at innovationguy@yahoo.com. The CSET Qs on this site are taken from that collection of questions!

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General Information about the CSET

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(This is based on information on the CSET website & is interspersed with my comments and opinions!)

1. What is the CSET?

Teacher candidates in California are required to demonstrate competence in the subject matter they will be authorized to teach. The California Commission on Teacher Credentialing (CCTC) provides candidates with two options for satisfying this requirement.

Candidates can either

(a) complete a CCTC-approved subject matter preparation program (ie. relevant coursework at a College/University) or

(b) pass the appropriate CCTC-adopted subject matter test(s).

The California Subject Examinations for Teachers (CSET®) is a new testing program serving as the examination option for the subject matter competence requirement.

Official information about CSET can be obtained at:

http://www.cset.nesinc.com/

2. What materials are officially available for CSET preparation?

A test guide, providing information about the tests, is available via the CSET Web site. Each test guide includes the content to be covered, information on the test structure, scoring scales, sample test questions, sample responses to constructed-response questions, and an annotated list of suggested resources that candidates can use to study for the test.

Caution! There's a general consensus amongst teachers that've taken the CSET that the pattern of Qs on the official Test Guides are quite different from those on the Test!! I have myself found the Test Guides to be not very representative of the Test itself!

This is not entirely surprising as the Guides offer only a sample of the diverse concepts and skills required. So skim through it only to get a very general idea!

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3. How much time do I have to complete each subtest of the CSET?

Each testing session is five hours in length. Examinees can choose to take up to three Subtests in one testing session. However, individual subtests are not timed so that candidates can allocate their time flexibly from one Subtest to another.

4. What are the requirements about the CSET Single Subject Math test?

There are 3 Subtests one needs pass to meet the CSET Single Subject Mathematics teaching credential requirement.

Subtest I: Algebra / Number Theory
Subtest II: Geometry / Probability & Statistics
Subtest III: Calculus / Trigonometry / History of Mathematics

Further, there is a Foundational Mathematics Teaching Credential for which one needs to pass only Subtest I and II. This authorizes teaching only in limited Math content areas: General Mathematics, Algebra, Geometry, Probability and Statistics and Consumer Math.

In other words, the Foundational Credential allows one teach the following classes: Algebra I, Geometry, Business Math, Algebra II. Certain school districts prohibit teaching Algebra II since it requires Trigonometry, which is a part of Subtest III.

Most school districts require only a Foundational Credential for employment purposes!

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5. What is the format of the tests?

All 3 Subtests are similarly structured:

30 Multiple Choice Qs + 4 Constructed (or Free) Response Qs (Written/Essay Type)

The Free Response Qs are graded according to the demonstration of the following criteria:

Purpose
Subject Matter Knowledge
Support
Depth and Breadth of Understanding

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Checklist of Skills for Subtest I

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What are the checklist of skills one must possess for Subtest I?

BASIC SKILLS: you must:

• Know distance formula.
• Midpoint formula
• Know symmetricity of graphs with respect to x-axis, y-axis, origin, y = x, line y = -x.
• Know the points slope form and slope intercept form of equation of lines.
• Know the condition for parallel and perpendicular lines using slopes.
• Prove the theorem on parallel and perpendicular lines using slopes.
• Know the definition of a function and relation.
• Know the Vertical Line Test for determining if a set of points belongs to a function.
• Be able to determine the domain and range of a function algebraically and graphically.
• Be able to factorize expressions using (a ± b)², (a ± b)³, a² – b² , a³ ± b³
• Know how to complete the square of a quadratic expression.
• Determine if a function is even/odd.
• Be able to rationalize the denominator to simplify complex (i) and irrational (√) expressions
  

For QUADRATIC FUNCTIONS, you must:

• Be able to derive the quadratic formula
• Derive the sum of roots of a quadratic equation is –b/a and the product of roots is c/a.
• Know the relationship between the roots of a quadratic equation and its graph based on the discriminant, D = b² – 4ac, being positive, negative or equal to zero
• Be able to convert a quadratic function in standard form to parabola form by completing the square.
• Determine the vertex, axis of symmetry, roots/x-intercepts of a quadratic function/ parabola.
• Determine if a parabola opens up or down, and based on that, know if the function has a maximum or minimum.
• Be able to solve variations of quadratic equations using substitution: for example, ax⁴ + bx² + c = 0, √(ax⁴ + bx² + c) = 0, ax² + b/x² + c = 0, √(ax + b) + √ (cx + d) = 0, etc. and find the maximum/ minimum of such functions.
• Be able to derive the coordinates of the vertex of a quadratic function in standard form.
• Model word problems related to quadratic functions and determine maximum/ minimum values.
• Determine roots/ x intercepts through quadratic formula/ completing the square.
• Know properties of complex numbers: addition, subtraction, multiplication, and division.
• Be able to represent the sum, difference, product and quotient of complex numbers in a + bi form.
• Know that irrational and complex roots of functions occur as conjugates.
• Be able to graph and interpret inequalities relating to quadratic functions: know for which intervals the graph lies above/ below the x-axis
  

For POLYNOMIAL FUNCTIONS, you must:

• Be able to determine the number of turning points for a function.
• Determine the end behavior of the function.
• Determine by visual inspection if a graph represents a certain polynomial function.
• Know characteristics of even/ odd functions.
• Know shapes of the most important BASE GRAPHS:

1. f(x) = √x
2. f(x) = x²
3. f(x) = xⁿ, for n = odd/ even
4. f(x) = 1/x
5. f(x) = 1/xⁿ, for n = odd/ even
6. f(x) = ³√x
7. f(x) = ⁿ√x, for n = odd/ even
8. f(x) = |x |

• Be able to transform the above base graphs for the following cases:

1. Given f(x), graphing f(x + c) or f(x – c)
2. Given f(x), graphing f(x) + c or f(x) – c
3. Given f(x), graphing c f(x), for c > 1 and c <>
4. Given f(x), graphing f(-x) and –f(x)

• Be able to graph polynomial functions in factored form

1. X-intercepts
2. Y-intercepts
3. End behavior of function
4. Intervals where graph is above/ below the x-axis by using a sign table.

• Be able to divide two polynomial functions using long division
• Be able to divide a polynomial and a linear (binomial) expression using Synthetic Division, and determine the Quotient and Remainder
• Know the proofs of Remainder Theorem and Factor Theorem
• Use the Factor Theorem and Remainder Theorem to determine if a monomial is a factor of the polynomial function.
• Be able to prove the Rational Roots Theorem.
• Apply the Rational Roots Theorem to determine the possible roots of a polynomial.
• Apply the Descartes Rule of Signs to determine the number of positive, negative and imaginary roots of a polynomial.
• Determine the polynomial function given its roots and their multiplicity
• Be able to calculate the roots, real and imaginary, of a polynomial.
• Be able to graph and interpret inequalities relating to polynomial functions: know for which intervals the graph lies above/ below the x-axis.
  

For RATIONAL FUNCTIONS, f(x) = P(x)/ Q(x), you must:

• Be able to graph various forms of rational functions by determining its:

1. x intercepts (solve: numerator = 0)
2. y intercepts (substitute: x = 0 into the function)
3. vertical asymptotes (solve: denominator = 0)
4. horizontal asymptotes (imagine x to be a large number, M, and simplify.)
5. slant asymptotes, if any, if the numerator has degree 1 more than the denominator

For INVERSE OF FUNCTIONS you must

• Be able to determine the inverse of functions (swap x and y in the original function and solve for y)
• Know properties of inverse functions
• Know the behavior of the inverse function, when given the graph of the original function (ie. the inverse function and the original function are symmetrical about the line y =x).
• Know the relationship of the domain and range of inverse functions and the original function.
• Know the horizontal line test for determining if a function has an inverse.
• Know what a one-to-one function is.

For COMPOSITE FUNCTIONS, f(x) = h(g(x)); h(x)/g(x); h(x)g(x), etc. you must

• Be able to determine the domain and range.
• Find values of composite functions for given values of x (x = a…)
  

For LOGARITHMIC AND EXPONENTIAL FUNCTIONS you must

• Know the general form (algebraically and graphically) of logarithmic (ie. log x) and exponential (ie. a^x or e^x) functions.
• Know the relationship between logarithmic (log₁₀ x or log _ex) and exponential (a^x or e^x) functions, and transform exponential to logarithmic functions and vice versa.
• Be able to calculate the inverse of logarithmic and exponential functions.
• Graph logarithmic and exponential functions using elementary transformations
• Be able to determine the domain and range of transformed logarithmic and exponential functions.
• Know the properties of logarithms and exponents, and apply properties to simplify expressions
• Solve equations and inequalities with logs and exponents using definitions and properties of logs and exponents
• Be able to determine extraneous solutions to equations/inequalities
• Know the formula for compound interest for the case of continuous compounding
• Be familiar with exponential growth and decay situations, and related doubling time and half-life problems
  

For VECTORS you must:

• Be able to determine the magnitude and direction of a vector
• Know the representation of vectors in standard (ai + bj) and component form
• Know basic vector operations
• Be able to determine unit vectors in a given direction
• Be able to calculate the direction angles of vectors
• Apply elementary vector properties to solve real-world problems
• Be able to calculate the angle between 2 vectors using dot product of vectors, and determine if 2 vectors are parallel or perpendicular
  

For SYSTEMS OF EQUATIONS, you must:

• Be able to solve systems of equations (lines, circles, ellipses, parabolas, hyperbolas) simultaneously to determine points of intersection.
• Solve systems of inequalities and be able to graph the shaded region representing all possible values of (x, y).
• Be able to determine the maximum and minimum values of an objective function of a linear programming problem/ situation.
• Be able to solve word problems involving a linear programming situation: choose appropriate variables, determine constraints and objective function, plot lines representing constraints, determine corner points for shaded polygon and calculate maximum/ minimum values of the objective function.
• Be able to solve systems of linear equations by finding the reduced echelon form of a matrix by performing row transformations
• Be able to classify systems of linear equations as consistent/ inconsistent/ possessing infinite solutions.
  

For MATRICES and DETERMINANTS, you must:

• Know basic properties of matrices
• Know the criteria for multiplying matrices
• Be able to multiply matrices
• Know properties of matrix multiplication.
• Be able to find the inverse of a matrix using determinants.
• Be able to calculate the inverse of a matrix by reducing it to echelon form.
• Be able to solve systems of linear equations by calculating the inverse of a matrix.
• Be able to find the value of a 2 X 2 and 3 X 3 determinants.
• Know properties of determinants pertaining to row and column transformations.
• Know Cramer’s Rule for solving linear equations using determinants.
  

For SEQUENCES, SERIES, PERMUTATIONS AND COMBINATIONS you must:

• Be able to determine the general term of an arithmetic/ geometric series using the common difference/ ratio.
• Be able to calculate the sum of n terms of an arithmetic/ geometric series.
• Be able to calculate the sum of an infinite geometric series.
• Know the principle of Mathematical Induction.
• Be able to apply the Principle of Mathematic Induction to prove elementary propositions/ conjectures.
• Know the (r + 1)^th term in the expansion of (a + b)ⁿ using the binomial theorem.
• Expand binomial expressions using the binomial theorem and calculate specific terms.
• Be familiar with Pascal’s Triangle.
• Know the fundamental counting principle.
• Be able to calculate the number of different permutations & combinations of r elements that can be obtained from a set of n elements.

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