Solutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1

Size: px
Start display at page:

Download "Solutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1"

Transcription

1 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given so be careful. Work out the solutions before looking at this ke!. Simplif: The LCD is + so convert the first two terms into fractions with + in the denominator and simplif. This simplification should be true for an value of, ecept =, which is not in the domain of the epression. So, we can check the simplification b replacing with, sa, in the original and simplified epression. If we end up with the same value then we are confident that the simplification is correct (this does not prove it is correct but increases our confidence that it is). To do such a check don t use 0 or since those numbers have special properties that could mess up the check. Let's tr =. Original: Simplified: Simplif: a a a a Multipl each term of the binomial b each term of the other polnomial and then simplif. ANS: e a a a a a a a a a a a a a a a a a a a a a a ANS: d

2 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page. The epression / / / z is equal to: The negative eponent of sends it to the denominator. That is, /. / For rational eponents, the numerator of the eponent is the power of the variable and the mn / n m denominator of the eponent is the inde of the root. That is,. / / / z / / z / z z ANS: c. Find the Least Common Multiple:,, The LCM is the product of the largest power of each individual factor. The individual factors are ( + ), ( ), ( + ) The product of the largest power of each of these is LCM. Divide b Do this just like a long division problem in arithmetic: ANS: a So, the quotient is and the remainder is. ANS: c

3 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page. Find the midpoint of (, ) and (, ). Use the midpoint formula. Remember, this formula is like the average of the coordinates, m and the average of the coordinates, m. m m So, the coordinates of the midpoint is,. 7. If a leg of a right triangle has length and the hpotenuse has length, then the other leg will have length: ANS: b Use the Pthagorean Theorem: a b c where a and b are the lengths of the sides of a right triangle and c is the length of the hpotenuse. b 9b b 7 b 7 b 7 Since b is a length it cannot be negative. So, there is one solution, 7. ANS: c

4 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page. 8. Consider the functions f and g Which from the following is equal to g f after simplification? Replace the in with the epression for f g g f g and simplif: ANS: c 9. If (, b) is a point on the graph of, what is b? Since we are given that (, b) is a point on the graph, we know that when =, = b. So, put these into the given equation and simplif: b b 0 0. The Average Rate of Change of the function f from = to = is: ANS: a The ARC formula is like the one for slope of a line. The ARC from c to for f() is: f f c ARC c 9 ANS: e

5 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page. Which of the following is true about the rational function R? Recall that for a rational function if the degree of the numerator is more than the degree of the denominator it will have an oblique asmptote. To find it, do the long division. 0 So, we have an oblique asmptote at. ANS: b logb. The value of logb a is: When ou see the quotient of two logs of the same base, think about the change of base log formula, log b a. This looks like the change of base in reverse. log a b Lets get rid of the eponents b putting them in front of the logs. Then use the change of base in reverse. logb logb log b log a ANS: a log a logba logba b

6 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page is: To find the inverse, echange and and solve for.. The inverse function of f f f We can check to see if this is the inverse b seeing if f f : f f f. Which of the given statements about eponentials and logs is true? First, recall that when converting from an eponential to a log, the base of the log is the same as the base of the eponential; ANS: d the log epression is equal to the eponent of the eponential (remember, a log is an eponent); the epression the eponential is equal to is the argument of the log. If ou cannot remember all that, then simpl take the log of both sides of the eponential equation and see what ou get (ou must use the same base for the log as the eponential). Choice a is true: e N is equivalent to lnn We can show this as follows: ln e N e lnn ln N ANS: a

7 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page 7. The domain of the function f log is: The base of a log must be positive. So, 0. The base of a log must not be. So, 0 Also, the domain of the log is positive real numbers. So, 0 We can combine these three conditions, >, 0, and > into a single statement < <, 0. Choice c sas < <, 0 but this is a tpo since it sas that <. The answer is supposed to be c. ANS: c f log Part : Free Response Questions. Here ou work out the problems and ou MUST show our work. You can earn partial credit on these so being clear in what ou are doing is important.. Simplif: Start simplifing at the lower right and work our wa up through the fraction. Now, we have. So, work on the net level: Now, we have do the last part: ANS:

8 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page 8 Check: Tr = and see if ou get the same value for the original and the simplified version. Original: Simplified: 0. f ln. There are several things to look for: Division b 0, negative inside a square root, negative argument for the logarithm, base of the logarithm being negative or 0 or. Division b 0:. Determine the domain of the function 0 0 So, Negative inside the square root: 0 Negative argument for the logarithm: 0 Base of the logarithm being negative or 0 or : ln log, where e so this is fine. e So, cannot be and it must be between and. The domain is < <. ANS: < <. Find the center and the radius of the circle 8 0. Use completing the square to write this in the standard form for a circle: 8 0 h k r 8 0 To complete the square in, be sure the coefficient of the term is. It is. If it were not, then ou would factor out the coefficient before tring to complete the square. Now, find half the coefficient of the -term and square it. The coefficient is so we have 9 Add and subtract this value from the -terms f ln

9 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page 9 Complete the square for in the same wa. The whole process looks like this: h k r So, the center is (, ) and the radius is. ANS: center = (, ), radius =. What function is finall graphed after the following transformations are applied to the graph of f. There is a great summar of transformations on page 99 of Sullivan. a. Reflect about the -ais. This is f f. So, we have g b. Compress horizontall b a factor of. This is f f a, a. So, we have h ( ) c. Shift left unit. This is f f h. So, we have j You are not asked for the graphs, but here the are anwa. ANS: j f g h () j

10 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page 0. Graph the rational function R 7 Characteristics of a function that help us to sketch a graph are the domain, -intercepts, -intercepts, vertical asmptotes, horizontal or oblique asmptotes, and end behavior of the graph. Domain. Factor the polnomials and note where the denominator is 0. R 7 9 We see that the denominator is 0 when = and when =. So the domain is,. Also note that there is a common factor of. Reducing b that factor gives us this: 9 R This means there will be a hole in the graph at =. Hole at = -intercepts: These are the values of when = 0. 9 R The right side of the equation is not factorable so use the quadratic formula to solve., b b ac a This has no real solutions so there are no -intercepts. No -intercepts

11 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page -intercept: This is the value of when = 0. 9 R 0 09 R intercept at (0, 9) 9 Vertical asmptotes: These occur at the zeros of the denominator of R. The denominator is +, so the line = is a vertical asmptote. Vert asm: = Horizontal or oblique asmptote: There are three possibilities. If n = degree of numerator and m = degree of denominator, then if n < m the -ais is a horizontal asmptote. The equation is = 0. a n = m there is a horizontal asmptote, n (the ratio of the coefficients of dominant terms). b n = m + there is an oblique asmptote. Find this b using long division. n > m + there are no horizontal or oblique asmptotes. 9 For R n = m + so we use long division to find the oblique asmptote n The quotient is + so the oblique asmptote is = +. Obl Asm: = + Does the graph cross the horizontal asmptote? That is, is there an such that R() =? R Since this is a contradiction, the graph does not cross the asmptote. No cross End behavior: Since we have an oblique asmptote at = +, the graph approaches this line as.

12 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Now, lets put all the information together to sketch the graph. Here are the intercepts and asmptotes: = = We onl have one point plotted, the -intercept. The graph must go through this point and it must approach the vertical and oblique asmptotes but not touch them. That tells us the upper part of the graph looks like this (remember the hole at = ). = = For the other part of the graph, we note that there are no -intercepts so the graph does not cross the -ais. Therefore, it cannot be above the line = +. Also, the graph must approach the asmptotes but not touch them. Therefore, the bottom part of the graph must look like the following: = =

13 Solutions to the Math 0 Sample Final Eam (from Spring 00) Page We can plot a few points in each region to see the are on the graph we have sketched. Let's use the zeros of R() of BOTH the numerator and denominator to break up the domain into intervals and use test values of within these intervals to get some points to plot. It will be easiest to do the calculations if we use the original R 7 Interval: (, ) (, ) (, ) Tr: Substitute: Simplif: Point: (, 7) (,.) (, 7.) = = + -. Solve log log. Use the product propert of logs to combine the logs into a single log and then convert to an eponential. log or 0 or There appear to be two answers but recall that the domain of the log is positive real numbers. If we put in for we get a negative argument so cannot be a part of the solution. The answer is =. Check: log log? log log? 0?Yes ANS: = f gwhen log f log g

Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1

Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1 College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,

More information

College Algebra Final, 7/2/10

College Algebra Final, 7/2/10 NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Final Eam Review MAC 1 Spring 0 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve and check the linear equation. 1) (- + ) - = -( - 7) {-

More information

Math 1051, Robertson, Exam 3 on Chapters 3 & 4 on Friday 12 November 2010 KEY page 1

Math 1051, Robertson, Exam 3 on Chapters 3 & 4 on Friday 12 November 2010 KEY page 1 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page. You earned points out of. Ans: f 6 Write the equation of a quadratic function whose graph has the following characteristics: It opens down;

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Final Eam Review MAC 1 Fall 011 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve and check the linear equation. 1) (- + ) - = -( - 7) A)

More information

Math 154 :: Elementary Algebra

Math 154 :: Elementary Algebra Math :: Elementar Algebra Section. Section. Section. Section. Section. Section. Math :: Elementar Algebra Section. Eponents. When multipling like-bases, ou can add the eponents to simplif the epression..

More information

Answers for the problems can be found at the end of this packet starting on Page 12.

Answers for the problems can be found at the end of this packet starting on Page 12. MAC 0 Review for Final Eam The eam will consists of problems similar to the ones below. When preparing, focus on understanding and general procedures (when available) rather than specific question. Answers

More information

Questions From Old Exams

Questions From Old Exams MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE Questions From Old Eams. Write the equation of a quadratic function whose graph has the following characteristics: It opens down; it is stretched

More information

Algebra/Pre-calc Review

Algebra/Pre-calc Review Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge

More information

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Math Review Packet #5 Algebra II (Part 2) Notes

Math Review Packet #5 Algebra II (Part 2) Notes SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of

More information

MATH 0312 FINAL EXAM REVIEW ITEMS

MATH 0312 FINAL EXAM REVIEW ITEMS MATH 012 FINAL EXAM REVIEW ITEMS Name The items on this review are representative of the items that ou might see on our course final eam. No formul sheets are allowed and calculators are not allowed on

More information

L43-Mon-12-Dec-2016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45 Page 27. L43-Mon-12-Dec-2016-Rev-Cpt-4-HW44-and-Rev-Cpt-5-for-Final-HW45

L43-Mon-12-Dec-2016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45 Page 27. L43-Mon-12-Dec-2016-Rev-Cpt-4-HW44-and-Rev-Cpt-5-for-Final-HW45 L43-Mon-1-Dec-016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45 Page 7 L43-Mon-1-Dec-016-Rev-Cpt-4-HW44-and-Rev-Cpt-5-for-Final-HW45 L43-Mon-1-Dec-016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45

More information

math FALL developmental mathematics sullivan 1e

math FALL developmental mathematics sullivan 1e TSIpractice eam review 1 131 180 plus 34 TSI questions for elementar and intermediate algebra m0300004301 aaa Name www.alvarezmathhelp.com math0300004301 FALL 01 100 interactmath developmental mathematics

More information

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

More information

Example 1: What do you know about the graph of the function

Example 1: What do you know about the graph of the function Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What

More information

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1 Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab

More information

Math 111 Lecture Notes

Math 111 Lecture Notes A rational function is of the form R() = p() q() where p and q are polnomial functions. The zeros of a rational function are the values of for which p() = 0, as the function s value is zero where the value

More information

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

More information

1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)

1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given

More information

Name. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Name. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. REVIEW Eam #3 : 3.2-3.6, 4.1-4.5, 5.1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the Leading Coefficient Test to determine the end behavior

More information

Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have

Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember

More information

Name Please print your name as it appears on the class roster.

Name Please print your name as it appears on the class roster. Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

More information

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

1. Simplify each expression and write all answers without negative exponents. for variable L.

1. Simplify each expression and write all answers without negative exponents. for variable L. MATH 0: PRACTICE FINAL Spring, 007 Chapter # :. Simplif each epression and write all answers without negative eponents. ( ab ) Ans. b 9 7a 6 Ans.. Solve each equation. 5( ) = 5 5 Ans. man solutions + 7

More information

3.1 Graphing Quadratic Functions. Quadratic functions are of the form.

3.1 Graphing Quadratic Functions. Quadratic functions are of the form. 3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.

More information

Math 1051 Moodle Quiz Solutions

Math 1051 Moodle Quiz Solutions Math 1 Moodle Quiz Solutions There is a one question Moodle quiz associated with most lectures. A quiz will open the day of the lecture and close at midnight on the day before the net lecture (e.g., a

More information

Test # 33 QUESTIONS MATH131 091700 COLLEGE ALGEBRA Name atfm131bli www.alvarezmathhelp.com website MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

More information

ALGEBRA 1 CP FINAL EXAM REVIEW

ALGEBRA 1 CP FINAL EXAM REVIEW ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8

More information

FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name

FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name 1) Find the SUM of the solutions of the equation. 82 + 0 = 16 Use the quadratic formula to solve the equation. (All solutions are real numbers.)

More information

McKinney High School AP Calculus Summer Packet

McKinney High School AP Calculus Summer Packet McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work

More information

review math0410 (1-174) and math 0320 ( ) aafinm mg

review math0410 (1-174) and math 0320 ( ) aafinm mg Eam Name review math04 (1-174) and math 0320 (17-243) 03201700aafinm0424300 mg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif. 1) 7 2-3 A)

More information

Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.

Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint. Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the

More information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots. Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to

More information

REVIEW. log e. log. 3 k. x 4. log ( x+ 3) log x= ,if x 2 y. . h

REVIEW. log e. log. 3 k. x 4. log ( x+ 3) log x= ,if x 2 y. . h Math REVIEW Part I: Problems Simplif (without the use of calculators) ln log 000 e 0 k = k = k 7 log ( ) 8 lo g (log ) Solve the following equations/inequalities Check when necessar 8 =0 9 0 + = log (

More information

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1 Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

More information

Finding Slope. Find the slopes of the lines passing through the following points. rise run

Finding Slope. Find the slopes of the lines passing through the following points. rise run Finding Slope Find the slopes of the lines passing through the following points. y y1 Formula for slope: m 1 m rise run Find the slopes of the lines passing through the following points. E #1: (7,0) and

More information

Test #4 33 QUESTIONS MATH1314 09281700 COLLEGE ALGEBRA Name atfm1314bli28 www.alvarezmathhelp.com website SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

More information

Northwest High School s Algebra 2/Honors Algebra 2

Northwest High School s Algebra 2/Honors Algebra 2 Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar

More information

RAMAPO&INDIAN*HILLS*SCHOOL*DISTRICT*

RAMAPO&INDIAN*HILLS*SCHOOL*DISTRICT* 1 * * Algebra 2 CP Summer Packet RAMAPO&INDIAN*HILLS*SCHOOL*DISTRICT* DearRamapo*IndianHillsStudent: Pleasefindattachedthesummerpacketforourupcomingmathcourse.Thepurposeof thesummerpacketistoprovideouwithanopportunittoreviewprerequisiteskillsand

More information

HCC-SE MATH DEPT. 1 Revised Fall 2008

HCC-SE MATH DEPT. 1 Revised Fall 2008 FINAL EXAM REVIEW ITEMS Math : College Algebra Find the -intercepts and an -intercepts. ) f() = + 7-0 ) = Name ) Select the equation that describes the graph. Solve the equation and epress the solution

More information

Section 5.1 Model Inverse and Joint Variation

Section 5.1 Model Inverse and Joint Variation 108 Section 5.1 Model Inverse and Joint Variation Remember a Direct Variation Equation y k has a y-intercept of (0, 0). Different Types of Variation Relationship Equation a) y varies directly with. y k

More information

6.4 graphs OF logarithmic FUnCTIOnS

6.4 graphs OF logarithmic FUnCTIOnS SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS

More information

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of

More information

A. Simplifying Polynomial Expressions

A. Simplifying Polynomial Expressions A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +

More information

HONORS PRE-CALCULAUS ACP Summer Math Packet

HONORS PRE-CALCULAUS ACP Summer Math Packet Name Date Section HONORS PRE-CALCULAUS ACP Summer Math Packet For all incoming Honors Pre-Calculus ACP students, the summer math packet will be on the school website. Students will need to print a cop

More information

review for math TSI 55 practice aafm m

review for math TSI 55 practice aafm m Eam TSI Name review for math TSI practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the

More information

Algebra Review. Unit 7 Polynomials

Algebra Review. Unit 7 Polynomials Algebra Review Below is a list of topics and practice problems you have covered so far this semester. You do not need to work out every question on the review. Skip around and work the types of questions

More information

Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general

Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate

More information

Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!

Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit! Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Practice for the Final Eam MAC 1 Sullivan Version 1 (2007) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the distance d(p1, P2) between the points

More information

Math Intermediate Algebra

Math Intermediate Algebra Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and

More information

Algebra II Midterm Exam Review Packet

Algebra II Midterm Exam Review Packet Algebra II Midterm Eam Review Packet Name: Hour: CHAPTER 1 Midterm Review Evaluate the power. 1.. 5 5. 6. 7 Find the value of each epression given the value of each variable. 5. 10 when 5 10 6. when 6

More information

Course 15 Numbers and Their Properties

Course 15 Numbers and Their Properties Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

More information

C)not a function. B) function domain: {-3, 2, 4, 6} range: {-7, 4, 2, -1}

C)not a function. B) function domain: {-3, 2, 4, 6} range: {-7, 4, 2, -1} Name Spring Semester Final Review (Dual) Precalculus MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the relation represents a function.

More information

AP Calculus AB Summer Assignment Mrs. Berkson

AP Calculus AB Summer Assignment Mrs. Berkson AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre- Calculus skills required for AP Calculus AB. Net ear we will be starting off the

More information

5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).

5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5). Rewrite using rational eponents. 2 1. 2. 5 5. 8 4 4. 4 5. Find the slope intercept equation of the line parallel to y = + 1 through the point (4, 5). 6. Use the limit definition to find the derivative

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=

More information

f(x) = 2x 2 + 2x - 4

f(x) = 2x 2 + 2x - 4 4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms

More information

MATH STUDENT BOOK. 12th Grade Unit 2

MATH STUDENT BOOK. 12th Grade Unit 2 MATH STUDENT BOOK 12th Grade Unit 2 Unit 2 FUNCTIONS MATH 1202 FUNCTIONS INTRODUCTION 3 1. LINEAR FUNCTIONS 5 GRAPHS OF LINEAR FUNCTIONS 5 EQUATIONS OF LINEAR FUNCTIONS 10 SELF TEST 1: LINEAR FUNCTIONS

More information

math0320 FALL interactmath sections developmental mathematics sullivan 1e

math0320 FALL interactmath sections developmental mathematics sullivan 1e Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan

More information

ZETA MATHS. Higher Mathematics Revision Checklist

ZETA MATHS. Higher Mathematics Revision Checklist ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions

More information

MATH 60 Review Problems for Final Exam

MATH 60 Review Problems for Final Exam MATH 60 Review Problems for Final Eam Scientific Calculators Onl - Graphing Calculators Not Allowed NO CLASS NOTES PERMITTED Evaluate the epression for the given values. m 1) m + 3 for m = 3 2) m 2 - n2

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted

More information

a 2 x y 1 y SOL AII.1a

a 2 x y 1 y SOL AII.1a SOL AII.a The student, given rational, radical, or polnomial epressions, will a) add, subtract, multipl, divide, and simplif rational algebraic epressions; Hints and Notes Rules for fractions: ) Alwas

More information

3.3 Limits and Infinity

3.3 Limits and Infinity Calculus Maimus. Limits Infinity Infinity is not a concrete number, but an abstract idea. It s not a destination, but a really long, never-ending journey. It s one of those mind-warping ideas that is difficult

More information

Unit 3 Notes Mathematical Methods

Unit 3 Notes Mathematical Methods Unit 3 Notes Mathematical Methods Foundational Knowledge Created b Triumph Tutoring Copright info Copright Triumph Tutoring 07 Triumph Tutoring Pt Ltd ABN 60 607 0 507 First published in 07 All rights

More information

Fox Lane High School Department of Mathematics

Fox Lane High School Department of Mathematics Fo Lane High School Department of Mathematics June 08 Hello Future AP Calculus AB Student! This is the summer assignment for all students taking AP Calculus AB net school year. It contains a set of problems

More information

AP Calculus AB Summer Assignment Mrs. Berkson

AP Calculus AB Summer Assignment Mrs. Berkson AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre- Calculus skills required for AP Calculus AB. Net ear we will be starting off the

More information

3x 2. x ))))) and sketch the graph, labelling everything.

3x 2. x ))))) and sketch the graph, labelling everything. Fall 2006 Practice Math 102 Final Eam 1 1. Sketch the graph of f() =. What is the domain of f? [0, ) Use transformations to sketch the graph of g() = 2. What is the domain of g? 1 1 2. a. Given f() = )))))

More information

EOC Review. Algebra I

EOC Review. Algebra I EOC Review Algebra I Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations.. 5 6 +.

More information

review for math TSI 182 practice aafm m

review for math TSI 182 practice aafm m Eam TSI 182 Name review for math TSI 182 practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif.

More information

c) domain {x R, x 3}, range {y R}

c) domain {x R, x 3}, range {y R} Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..

More information

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do

More information

Name Date. Show all work! Exact answers only unless the problem asks for an approximation.

Name Date. Show all work! Exact answers only unless the problem asks for an approximation. Advanced Calculus & AP Calculus AB Summer Assignment Name Date Show all work! Eact answers only unless the problem asks for an approimation. These are important topics from previous courses that you must

More information

Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs

Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs Sample Questions to the Final Eam in Math 1111 Chapter Section.1: Basics of Functions and Their Graphs 1. Find the range of the function: y 16. a.[-4,4] b.(, 4],[4, ) c.[0, ) d.(, ) e.. Find the domain

More information

Performing well in calculus is impossible without a solid algebra foundation. Many calculus

Performing well in calculus is impossible without a solid algebra foundation. Many calculus Chapter Algebra Review Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps

More information

Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions. Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and

More information

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology. Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph

More information

Ready To Go On? Skills Intervention 6-1 Polynomials

Ready To Go On? Skills Intervention 6-1 Polynomials 6A Read To Go On? Skills Intervention 6- Polnomials Find these vocabular words in Lesson 6- and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading

More information

PreCalculus. Ocean Township High School Mathematics Department

PreCalculus. Ocean Township High School Mathematics Department PreCalculus Summer Assignment Name Period Date Ocean Township High School Mathematics Department These are important topics from previous courses that ou must be comfortable doing before ou can be successful

More information

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph

More information

WBHS Algebra 2 - Final Exam

WBHS Algebra 2 - Final Exam Class: _ Date: _ WBHS Algebra 2 - Final Eam Multiple Choice Identify the choice that best completes the statement or answers the question. Describe the pattern in the sequence. Find the net three terms.

More information

Math 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions

Math 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions 1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains

More information

Lesson #9 Simplifying Rational Expressions

Lesson #9 Simplifying Rational Expressions Lesson #9 Simplifying Rational Epressions A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Factor the following epressions: A. 7 4 B. y C. y 49y Simplify: 5 5 = 4

More information

20 points Completion 20 points - Accuracy

20 points Completion 20 points - Accuracy Algebra II Final Eam REVIEW 015 Name 0 points Completion 0 points - Accurac The eam review will be graded on completion (0 points) and randoml selected problems answered correctl with accurate work shown

More information

Eam Name algebra final eam review147 aam032020181t4highschool www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation.

More information

Which Mathematics Course Should You Take? August 22, 2018 Which mathematics course you should take depends on your current mathematics skill level

Which Mathematics Course Should You Take? August 22, 2018 Which mathematics course you should take depends on your current mathematics skill level Which Mathematics Course Should You Take? August, 018 Which mathematics course you should take depends on your current mathematics skill level and your intended major. This is a conversation you should

More information

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation? Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

Visit us at: for a wealth of information about college mathematics placement testing!

Visit us at:   for a wealth of information about college mathematics placement testing! North Carolina Early Mathematics Placement Testing Program, 9--4. Multiply: A. 9 B. C. 9 9 9 D. 9 E. 9 Solution and Answer to Question # will be provided net Monday, 9-8-4 North Carolina Early Mathematics

More information

Section 1.2 A Catalog of Essential Functions

Section 1.2 A Catalog of Essential Functions Page 1 of 6 Section 1. A Catalog of Essential Functions Linear Models: All linear equations have the form y = m + b. rise change in horizontal The letter m is the slope of the line, or. It can be positive,

More information

Diagnostic Tests Study Guide

Diagnostic Tests Study Guide California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra

More information

1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10

1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10 CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series.

More information

N x. You should know how to decompose a rational function into partial fractions.

N x. You should know how to decompose a rational function into partial fractions. Section 7. Partial Fractions 0. 0 7 0 0 0 0 Solution:, 0 Equation Equation Eq. Eq. 07. nswers will var. Section 7. Partial Fractions N You should know how to decompose a rational function into partial

More information

Test # 2 Review Sections (2.4,2.5,2.6, & ch. 3) Math 1314 Name

Test # 2 Review Sections (2.4,2.5,2.6, & ch. 3) Math 1314 Name Test # Review Sections (.,.,., & ch. 3) Math 131 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the equation of the line. 1) -intercept,

More information