# Exam #4 Topics/Review

Listed below are the topics and exercises to review for the 4th midterm exam on Wednesday Dec 13.

Note that the MAT1275 Final Exam Review sheet is available as a pdf inÂ Files, and is also available in the Final Exam Review post.

This exam will also include one algebra topic from earlier in the semester:

• Solving systems involving nonlinear equations
• Final Exam Review: #7
• Exam #2: question #5
• Algebra textbook, Sec 9.4: Example 1(b) (p777), Example 4 (p779)

# Final Exam Review – Study Guide

Our final exam will be Monday, Dec 18 (at the usual class time, 10-11:40am).

The questions on the final exam will be very similar in format to the exercises on the Math Departmentâ€™s Final Exam Review sheet:

MAT1275 Final Exam Review

(A hardcopy of the Final Exam Review sheet was handed out in class, and the pdf is also available online at http://www.citytech.cuny.edu/mathematics/docs/review/MAT1275FinalRview.pdfÂ or on OpenLab: https://openlab.citytech.cuny.edu/groups/mat1275en-ganguli-fall2017/files/).

Note that many of the exercises on our midterm exams and quizzes were very similar in format to these exercises. Â The best way to review for the final is to work through the Final Exam Review sheet, and study the related midterm exam solutions and quiz solutions as you do so.Â  All the midterm and quiz solutions are on OpenLab–download them!

Below is a guide to the exercises on the Final Exam Review sheet, with similar exam questions, WebWork exercises, and textbook examples listed for further review.

#1: Solving Quadratic Equations (using the quadratic formula)

• Quiz #3 (question #2)
• Quiz #4 (question #3)
• Exam #1 (question #3)
• WebWork: â€śQuadraticFormulaâ€ť set

#2: Graphing Quadratic Functions (by completing the square to identify the vertex)

• â€śGraphing Quadraticsâ€ť class handout (pdf available on OpenLab)
• Quiz #4
• Exam #2 (question #3)
• Exam #3 (question #1)

#3: Simplifying Complex Fractions (using least common denominators)

• Exam #2 (question #2)
• Exam #3 (question #6)
• WebWork: “ComplexFractions-Method1” set, Problem 6

#4: Graphing Circles (by completing the squares to identify the center and radius)

• Quiz #5 (question #1)
• Exam #2 (question #4)
• WebWork: â€śCirclesâ€ť set, Problems 1-4

#5: Complex Numbers (multiplying via â€śFOILâ€ť; dividing using the complex conjugate)

• Exam #2 (question #2)
• Quiz #8 (question #3)
• WebWork: “ComplexNumbers” set, Problems 5-8

#6: Solving a system of 3 linear equations in 3 variables

• Quiz #1
• Exam #1 (question #5)
• Algebra textbook, Sec 3.6: Example 1 (pp280-281)

#7: Solving systems involving nonlinear equations

• Exam #2 (question #5)
• Algebra textbook, Sec 9.4: Example 1(b) (p777), Example 4 (p779)

#8: Finding the values of the trig functions based on the value of one trig function (and information about the sign of one other trig function):

• Exam #3 (question #5)
• WebWork: â€śCoordinatePlaneTrigâ€ť set, Problems 4 & 5
• Trigonometry textbook, Sec 1.3: Example #5 (p26)

#9: Using the Law of Sines or Law of Cosines

• Exam #4 (TBD)
• Trig textbook, Sec 7.1: Example 1 (pp317-318)
• Sec 7.2: Examples 2 & 3 (pp331-332)

#10: Verifying Trig Identitiesâ€“use the fundamental trig identities and algebra

• Exam #4 (TBD)
• Trig textbook, Sec 1.4: Examples 2-4 (pp32-33)
• Trig textbook, Sec 5.1: Examples 1 & 2 (pp212-213)

#11: Solving Trig Equations (using the unit circle)

• Exam #4 (TBD)
• Quiz #8 (question #2)
• WebWork: â€śSolvingTrigEquationsâ€ť set (Problems 1-7)

#12: Solving an exponential equation (using logarithms):

• Exam #3 (question #3)
• Quiz #7 (question #1)
• Algebra textbook, Sec 8.7: Example 7 (p722)

#13: Evaluating logarithms (using the definition of logarithms)

• Exam #3 (question #2)
• Quiz #6 (questions #1(c) & #1(d))
• WebWork: â€śLogarithmicFunctionsâ€ť (Problems 6-10)
• Algebra textbook, Sec 8.4: Examples 1 & 2 (pp682-683)

# Exam #3 Review

Listed below are the topics and exercises to review for the 3rd midterm exam on Wednesday Nov 22.

Note that the MAT1275 Final Exam Review is available as a pdf inÂ Files, and is also available on the Math Department website.

• logarithms
• Quiz #6
• MAT1275 Final Exam Review: #13
• WebWork: LogarithmicFunctions, Problems 6-10
• Algebra textbook: Sec 8.4, pp683-685 (Examples 2-4)
• solving exponential equations

• Quiz #7
• MAT1275 Final Exam Review: #12
• WebWork: ExponentialEquations-Calc, Problems 1,3
• Algebra textbook: Sec 8.7, pp722-723 (Example 7)
• trigonometry ratios in right triangles

• Quiz #7
• WebWork: TrigonometryRatios, Problems 4-6; SolvingRightTriangles, Problems 1-4
• Trigonometry textbook: Section 2.2, pages 54-56; Section 2.3, pages 63-66
• trigonometry in the coordinate plane
• MAT1275 Final Exam Review: #8
• WebWork: CoordinatePlaneTrig, Problems 1-5
• Trigonometry textbook: Section 1.3, pages 22-27

This exam will also include a couple topics from earlier exams:

• graphing quadratic functions:
• specifically, graphing a quadratic function of the form y = -x^2 + bx + c (so a parabola which opens downwards)
• MAT1275 Final Exam Review: #2(c)
• Algebra textbook,
• simplifying complex (i.e., nested) fractions
• Quiz #5, question #2
• Exam #2: question #1
• MAT1275 Final Exam Review: #3

# Exam #2: Review Exercises

Listed below are the topics and exercises to review for the exam this Friday (Oct 27).

Note that the MAT1275 Final Exam Review is available as a pdf inÂ Files, and is also available on the Math Department website.

• graphing quadratic functions:
• review the 7-page “Graphing Quadratic Functions” handout/HW assignment (pdf available inÂ Files), especially Question 5
• Quiz #4 (solutions posted in Files)
• “ShiftingParabolas” WebWork: #5 & #6 (review #1-4 if necessary)
• “ParabolaVertices” WebWork: #1 & #2
• MAT1275 Final Exam Review: #2
• equations and graphs of circles:
• “Circles” WebWork: #1-4 (also try graphing the given circles)
• Quiz #5, question #1 (solutions will be posted to Files)
• MAT1275 Final Exam Review: #4
• adding rational expressions and complex fractions:
• “AddRationalExpressions2” WebWork: all
• “ComplexFractions-Method1” WebWork: complete the exercises (due Wed)
• Quiz #5, question #2 (solutions will be posted to Files)
• MAT1275 Final Exam Review: #3
• nonlinear systems:
• “NonLinearSystems” WebWork: #2, 4, 5
• MAT1275 Final Exam Review: #7

# Exam #1: Review Exercises

Here are the topics and exercises to review for the exam this Wednesday (Sept 27), listed according to WebWork/homework assignments:

• LinesReview: #2-4 (slopes and equations of lines)
• LinearSystems: #1, 2, 4-7 (graphing lines, solving 2×2 linear systems)
• Quiz #1: 3×3 linear system (solution is posted on OpenLab Files)
• see also textbook pp280-281 (also posted on OpenLab Files)
• DifferenceOfSquares (#1-3)
• HW#2 (factoring quadratic polynomials & solving quadratic equations by factoring)
• will be handed back at beginning of class Wed so that we can discuss; solutions will be posted to OpenLab later today
• review GCF-Factoring if needed (in particular factoring by grouping, since that’s needed for the ac-method)
• SquareRootProperty: #3-9
• solving quadratic equations by completing the square and using the square root property
• ComplexNumbers (due Tuesday, 9/26 – complete this as review for the exam)

# Video Playlists: Complex Numbers

I have mentioned in class that online videos are a good way to review material from class and to go through additional example, and I’ve previously posted a few videos on here from PatrickJMT (you can find all video posts atÂ https://openlab.citytech.cuny.edu/mat1275en-ganguli-fall2017/category/video/).

This week we introduced imaginary and complex numbers. You can review that material via the following two video playlists:

The first one is via a website called “Math 1275 Student Video Resources” which was compiled by another CityTech math professor. It contains links to Khan Academy videos, organized according to the MAT1275 syllabus. Here is the page for Complex Numbers:

https://openlab.citytech.cuny.edu/math1275videolibrary/complex-numbers/

The 2nd one is a YouTube playlist of all 21 of PatrickJMT’s Complex Numbers videos:

# Derivation of the Quadratic Formula

Here’s a snapshot of the derivation of the quadratic formula from the blackboard on Friday:

(Note that my “blackboard management” wasn’t ideal: the derivation starts in the middle board and the continues on LHS board. The RHS board has a numerical example, which follows the same steps as the general derivation.)

Also, here is PatrickJMT’s 7min video going thru the derivation:

You can also read the derivation on the Wikipedia page for the quadratic formula, or in Sec 7.2 of the textbook (p592).

# PatrickJMT Videos: Factoring Differences of Squares

You may find online videos useful to review material from class and see additional examples. I particularly like the videos of a math teacher who goes by the name of “PatrickJMT.” Â He has posted hundreds of short videos on his website patrickjmt.com, including many which cover topics we will cover in the course (all the videos are also available on YouTube–his YT channel is here.)

Here are three videos which present examples of factoring differences of squares:

Factoring the Difference of Two Squares â€“ Ex 1

Factoring the Difference of Two Squares â€“ Ex 2

Factoring the Difference of Two Squares â€“ Ex 3