3.1 Graphing Quadratic Functions. Quadratic functions are of the form.
|
|
- Virginia Stafford
- 5 years ago
- Views:
Transcription
1 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. Here, the parabola has the verte at. *Graph each of the following parabolas: A note about Aleks: Sometimes Aleks will ask ou to plot 5 points when graphing a parabola. Obviousl, one will be the verte. Usuall ou can use the leading coefficient of the parabola to help ou determine a second and third point easil. Remember to use the smmetr of the parabola to help find addtional points. Eample: 41
2 *For each of the following functions, complete the square to write each in the form. State the verte. Also find the -intercepts of each (ou ma wish to factor or use the quadratic formula)
3 B. Quadratic Functions The Verte Formula The quadratic function has its verte at. *Use the verte formula to find the verte of each of the following functions: 9. A note about vocabular of quadratic functions: If is an -intercept of the graph of it can also be said that is a solution or of the equation Further, and is a of can be called a of 3.1 #47 A projectile is thrown upward with an initial velocit of 176 ft/sec. After t seconds, its height above the ground is given b the function. a. Find the projectile's height above the ground after 2 seconds. 10. b. Sketch the graph modeling the projectile's height. c. What is the projectile's maimum height? What is the value of t at this height? 11. d. How man seconds after it is thrown will the projectile strike the ground? 43
4 ALEKS Problem Find a quadratic function f whose graph is shown below. 44
5 3.2 Long & Snthetic Division 3. Divide: Long division alwas works; snthetic division onl works when dividing b factors (those without eponents). 1. Divide: Snthetic division can also be used to evaluate polnomials: 4. If, find in two was. 2. Divide using snthetic division: 45
6 5. Use snthetic division to determine whether is a factor of 6. Write a polnomial of least degree having the given zeroes (roots, solutions): a. b. 3.2 #66 Given that is a zero of, write in a completel factored form. c. d. 3.2 #70 Given that is a zero of completel factored form., write in a 3.2 #72 Factor: 46
7 3.2 #77 Factor: 3.3 Factoring Higher-Degreed Polnomials The Rational Zero Theorem: If has rational zeroes, then the are of the form where a is the set of factors of the leading coefficient, and b is the set of factors of the constant term. *List all possible rational zeroes of Theorem: If the sum of the coefficients is, then is a zero (and is a factor). If after changing the signs of the coefficients of the odd-degreed terms, the sum of the "new" coefficients is zero, then is a zero. Returning to our previous problem, factor: Factor: We need some new tools to break this down further... 47
8 The Intermediate Value Theorem: If is a polnomial and, and and, then there eists some number "c" in such that 3.3 #44 Factor. Find the zeroes. picture: 3.3 #30 Use the Intermediate Value Theorem to determine whether a zero eists for on the interval. 3.3 #49 Factor. Find the zeroes. 48
9 Descartes' Rule of Signs: Given the polnomial, the number of positive real zeroes is equal to the number of variations in sign of, or less that that b subtracting twos. The number of negative real zeroes is equal to the number of variations in sign of, or less that that b subtracting twos. 2. Theorem: If is a polnomial with real coefficients, then if there are an comple zeroes, the will be comple conjugate pairs. Possible positive zeroes Possible negative zeroes Possible comple zeroes Total number of zeroes *Use Descartes' Rule of Signs to count the number of possible positive, negative and comple zeroes (roots) of each: 1. Upper and Lower Bounds Propert: Given is a polnomial with real coefficients. Possible positive zeroes Possible negative zeroes Possible comple zeroes Total number of zeroes 1. If is divided b using snthetic division and all coefficients in the quotient row are either positive or zero, then is an upper bound on the zeroes of P. 2. If is divided b using snthetic division and all coefficients in the quotient row alternate in sign, then is a lower bound on the zeroes of P. *For both 1 and 2, zero coefficients can be either positive or negative as needed. 49
10 3.3 #89 Use the Rational Zero Theorem and Descartes' Rule of Signs along with the tests for 1 and to find all the zeroes of 3.3 #96 Use the Rational Zero Theorem and Descartes' Rule of Signs along with the tests for 1 and to find all the zeroes of 50
11 ALEKS Problem (requires the use of the Aleks graphing calculator) A 2 foot thick slice is cut off the top of a cube, resulting in a rectangular bo that has volume. Use the ALEKS graphing calculator to find the side length of the original cube. Round our answer to two decimal places. ALEKS Problem Suppose that the polnomial has real coefficients with Suppose also that has the following zeroes: 2, 3, Using this information, answer the following questions: a. What is another zero of? b. At most, how man real zeroes of are there? c. At most, how man imaginar zeroes of are there? ALEKS Problem (requires the use of the Aleks graphing calculator) The width of a rectangular bo is 2 times its length, and its height is 4 ft more than its length. The volume of the bo is. Use the ALEKS graphing calculator to find the length of the bo. Round our answer to two decimal places. ALEKS Problem Find all other zeroes of the polnomial, given that is a zero. 51
12 3.4 Graphing Polnomial Functions Theorem: A polnomial of degree has at most vertices (or less than that b subtracting twos). There are two tpes of -intercepts: Cut and Bounce: A "bounce" -intercept occurs when the zero has an multiplicit; a "cut" occurs when the zero has an multiplicit. 52
13 Sketch: ALEKS Problem Below is the graph of a polnomial function with real coefficients. Use the graph to answer the questons about. All local etrema of are shown in the graph. Sketch: Hint: The graph contains the following points: a. The function is increasing over which intervals? b. The function has minima at which - values? c. What is the sign of the leading coefficient of? d. Which of the following is a possibilit for the degree of? Check (circle) all that appl
14 54
15 3.5, 3.6 Graphing Rational Functions Rational functions are of the form where and have no Graph and then write a piecewise-defined fucntion which would make into a continuous function. factors in common. If and DO have factors in common, ou get removable discontinuities in our graph. For eample: Three Tpes of Discontinuities: removable discontinuities We can "fill the gap" b creating a piecewisedefined function: nonremovable (gap) discontinuities nonremovable (asmptotic) discontinuities 55
16 Steps for Graphing Rational Functions: Graph: 1. Set the numerator equal to zero to find the -intercept. 2. Set the denominator equal to zero to determine locations of an vertical asmptotes. 3. Look at the degrees of the numerator and denominator... a. If the degree of the denominator is higher, then there is a horizontal asmptote at, the -ais. b. If the degree of the numerator is higher, then do long division to determine the diagonal (oblique) or parabolic asmptote. c. If the degree of the numerator and the denominator is the same, then there is a horizontal asmptote at the ratio of the leading coefficients. Graph: 4. Use an - chart to plot additonal points. Etra note: If the denominator contains an even eponent, then the vertical asmptote which it corresponds to has curves on both sides that approach the same infinit. If each vertical asmptote referred to b the denominator has an odd multiplicit, then the curve approaches opposite infinities on either side of the vertical asmptote. 56
17 Graph: Graph: 57
18 Graph: Graph: 58
19 3.7 Polnomial and Rational Inequalities 3.7 #42 Solve: Recall... When is? 3.7 #10 where is 3.7 #54 Solve: 3.7 #14 Solve: 3.7 #20 Solve: 59
20 Find the domain of Solve: Solve: 60
21 3.8 Variation Direct Variation Solve the following: 6. m varies directl as the square of. If m=200 when =20, find m when =32. Inverse Variation "k" is called the * Write a variation model for each: Aleks Problems: 1. W varies directl as Z. 2. m varies inversel as t. 3. w varies jointl as p and f. Find the value of k for each m varies directl as and when is 8, m is T is inversel proportional to and when is 40, T is
22 Chapter 3 Review All Aleks Problems 1. 62
23 2. For the polnomial below, 3 is a zero. Epress g() as a product of linear factors. g 3 2 ( ) Use the rational zero theorem to list all possible rational zeros of the following. h( ) Find the equation of the quadratic function f whose graph is shown below. 5. Write an equation that epresses the following relationship. Use k as the constant of proportionalit. P varies jointl with the square of d and the cube of u. 63
24 6. The function below has at least one rational zero. Use this fact to find all zeros of the function. 7. Graph the rational function f( ) f 3 2 ( ) On a given planet, the weight of an object varies directl with the mass of the object. Suppose that an object whose mass is 9 kg weighs 90 N. Calculate the mass of another object that weighs 40 N. 64
25 9. Graph the parabola Plot the verte and four additional points, two on each side of the verte Graph the parabola Plot the verte and four additional points, two on each side of the verte. 65
26 11. For the polnomial below, 1 is a zero. Epress h() as a product of linear factors. h 3 2 ( ) Sketch 2 1 f( ) Sketch: Use Descartes' Rule of Signs to determine the number of possible positive real zeros and negative real zeros of f ( ) Solve
27 4.1 Functions and Thier Inverses A function is if it passes both a vertical and horizontal line test. If a function is one-to-one, then it is (it has an inverse which is also a function). *Find the inverse of each function and then graph each on the same graph: 1. To find the inverse of a function from its equation, switch the and, and then solve for the "new". To find the inverse of a function from its graph, reflect the graph across the line. If is on, then is on the graph of its inverse. *Given the graph of, graph its inverse
28 3. Aleks Problem: The one-to-one functions and are defined as follows: Find the following: To prove algebraicall that two functions are inverses of each other, show that and. Aleks Problem: The one-to-one functions defined b. Find, the inverse of. Then, give the domain and range of using interval notation. is 4.1 #57 Prove that and are inverses of each other. 68
29 4.2, 4.3 Eponential & Logarithmic Functions Graph is an eponential equation. Graph: Calculators: Eponents and Logarithms Eponential kes: Convert each from logarithmic form to eponential form (or vice versa): Logarithmic kes: Logarithmic Form Eponential Form Definition: As On our calculator, find the following values:
30 5. Aleks Problem: Graph the function: 6. Change-of-Base Formula: Solve each equation: Evaluate each of the following:
31 4.4 Solving Ep. & Log. Equations 4.2 #57 Solve: Properties of Logarithms: #67 Solve: 7. *Write each as separate, simplified logarithms: #57 Solve: 71
32 Solve: 4.4 #73 Solve: Solve: 4.4 #73 Solve: 72
33 4.4 #83 Solve: Aleks Problem: Consider the equation. Find the value of. Round our answer to 3 decimal places 4.4 #22 Solve: 73
34 4.5 Applications of Ep. & Log Equations Compound interest: For how long should $1000 be invested at 1.1% compounded dail for the mone to double? $1000 is invested at 1.2% for 10 ears. Find the value of the account after 10 ears if the interest is compounded... a. quarterl. b. monthl. c. dail. d. continuousl. 74
35 If $1500 is invested at 2.25%, compounded monthl, how long would it take to grow into $1800? How much should be invested at 1.1%, compounded dail, to grow to $2000 in 6 months? Annuit Formula (optional, if time) If $100 is invested ever month into an account which earns 8.5% interest, compounded monthl, for 45 ears, what would the balance of the account be at the end of the 45 ears? For how long should $10,000 be invested at 2.5% compounded dail in order for the mone to triple? nt Pn r Use A 1 1, where P is the r n amount deposited n times per ear. 75
36 Chapters 3 & 4 Mied Review 1. Solve: 3. Graph: 2. Given that is a zero, find the other zeroes of. 76
37 4. Graph: 6. Graph 7. Find all the zeroes of 5. Graph: 77
38 8. Sketch: 11. Solve: 9. Solve: 12. Solve: 10. Solve: 78
39 13. Solve: 15. For continuousl compounding interest, at what interest rate should $500 be invested so that is grows to $750 in 8 ears? 16. Find the verte: 14. Solve: 17. Graph 79
40 18. For how long should $800 be invested at 4.3%, compounded dail, in order for it to grow to $2000? 20. Use the Rational Zero Theorem and Descartes' Rule of Signs along with the tests for 1 and to find all the zeroes of 19. Find and graph the inverse of 80
41 21. Graph f( ) For f( ), find f 8 5 domain and range of f 1 ( ). 1 ( ), and the 22. Write a piece-wise defined function to make f() (from the previous problem) continuous. 81
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 informationPolynomial 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 informationC)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 informationCollege 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 informationSHORT 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 informationCHAPTER 2 Polynomial and Rational Functions
CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question
Midterm Review 0 Precalculu Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question ) A graph of a function g is shown below. Find g(0). (-, ) (-, 0) - -
More informationMath 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 informationName 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 informationExam practice Disclaimer. The actual test does not mirror this practice. This is meant as a means to help you understand the material.
Eam 3 24 practice Disclaimer. The actual test does not mirror this practice. This is meant as a means to help ou understand the material. Graph the function. 1) f() = 2 2 + 4 + 3 1) Sketch the graph of
More informationInstructor: Imelda Valencia Course: A3 Honors Pre Calculus
Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression
More informationTest # 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 informationLaw of Sines, Law of Cosines, Heron s Formula:
PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,
More informationLearning 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 informationREVIEW. 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 informationPre-Calculus B Semester 1 Review Packet December 2015
Pre-Calculus B Semester Review Packet December 05 Name DISCLAIMER The memor on all calculators will be cleared the da of the final. If ou have programs on our calculator that ou would like to keep, please
More informationa 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 informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
0 Final Practice Disclaimer: The actual eam differs. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Epress the number in scientific notation. 1) 0.000001517
More informationAlgebra 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 informationMATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)
MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the
More informationSelf- assessment 1010 (Intermediate Algebra)
Self- assessment (Intermediate Algebra) If ou can work these problems using a scientific calculator, ou should have sufficient knowledge to demonstrate master of Intermediate Algebra and to succeed in
More informationMath 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 informationGlossary. 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 information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationChapter 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 informationFirst Semester Final Review NON-Graphing Calculator
Algebra First Semester Final Review NON-Graphing Calculator Name:. 1. Find the slope of the line passing through the points ( 5, ) and ( 3, 7).. Find the slope-intercept equation of the line passing through
More informationHonors Pre-Calculus. Multiple Choice 1. An expression is given. Evaluate it at the given value
Honors Pre-Calculus Multiple Choice. An epression is given. Evaluate it at the given value, (A) (B) 9 (C) 9 (D) (E). Simplif the epression. (A) + (B) (C) (D) (E) 7. Simplif the epression. (A) (B) (C) (D)
More informationChapter 9 Vocabulary Check
9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section 9.. 67. f = + 68. f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population
More informationa 2 x y 1 x 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 informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationTest #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 informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationMAT 116 Final Exam Review
MAT 116 Final Eam Review 1. Determine the equation for the line described, writing the answer in slope-intercept form. The line is parallel to = 8 and passes through ( 1,9). The line is perpendicular to
More informationReady 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 informationChapter 8 Vocabulary Check
28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationAlgebra 2 Honors Summer Packet 2018
Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear
More informationSolutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 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
More informationAnswers 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 informationCHAPTER 3 Polynomial Functions
CHAPTER Polnomial Functions Section. Quadratic Functions and Models............. 7 Section. Polnomial Functions of Higher Degree......... 7 Section. Polnomial and Snthetic Division............ Section.
More informationMini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
More information20 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 information5. Perform the indicated operation and simplify each of the following expressions:
Precalculus Worksheet.5 1. What is - 1? Just because we refer to solutions as imaginar does not mean that the solutions are meaningless. Fields such as quantum mechanics and electromagnetism depend on
More informationChapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller
Chapter 4 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 4 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal
More informationMath 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 information2.1 Evaluate and Graph Polynomial
2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of
More informationreview 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 informationMath 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+ = + + = 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 informationAP 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 informationMth 95 Module 4 Chapter 8 Spring Review - Solving quadratic equations using the quadratic formula
Mth 95 Module 4 Chapter 8 Spring 04 Review - Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b
More informationTest # 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 information2, g(x) = 4x ) x - 12
Math 11000 Practice Final Eam Page 1 1) Find f(-1) when f() = 3 - - 8 + 3 1) ) Given f() = + - 3-1 and g() =, find the sum, (f + g)(). ) + 3) Given f() = -63 + 9 + and g() = -53 + - 5, find the difference,
More informationName 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 informationHCC-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 informationAP 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 information6.4 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 informationMATH 91 Final Study Package Name
MATH 91 Final Stud Package Name Solve the sstem b the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to epress the solution set. 1) - = 1 1)
More informationThe semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. n times per year: 1
ALGEBRA B Semester Eam Review The semester B eamination for Algebra will consist of two parts. Part 1 will be selected response. Part will be short answer. Students ma use a calculator. If a calculator
More informationALGEBRA 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 informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationSummary, Review, and Test
Summar, Review, and Test 79 56. Galileo s telescope brought about revolutionar changes in astronom. A comparable leap in our abilit to observe the universe took place as a result of the Hubble Space Telescope.
More informationFind the distance between the pair of points. 2) (7, -7) and (3, -5) A) 12 3 units B) 2 5 units C) 6 units D) 12 units B) 8 C) 63 2
Sample Departmental Final - Math 9 Write the first five terms of the sequence whose general term is given. 1) a n = n 2 - n 0, 2,, 12, 20 B) 2,, 12, 20, 30 C) 0, 3, 8, 1, 2 D) 1,, 9, 1, 2 Find the distance
More informationreview 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 informationPractice A ( 1, 3 ( 0, 1. Match the function with its graph. 3 x. Explain how the graph of g can be obtained from the graph of f. 5 x.
8. Practice A For use with pages 65 7 Match the function with its graph.. f. f.. f 5. f 6. f f Lesson 8. A. B. C. (, 6) (0, ) (, ) (0, ) ( 0, ) (, ) D. E. F. (0, ) (, 6) ( 0, ) (, ) (, ) (0, ) Eplain how
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 2 Stud Guide-Chapters 8 and 9 Name Date: Time: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all square roots of the number. ) 600 9,
More informationSTANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The domain of a quadratic function is the set of all real numbers.
EXERCISE 2-3 Things to remember: 1. QUADRATIC FUNCTION If a, b, and c are real numbers with a 0, then the function f() = a 2 + b + c STANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100
Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation
More informationAdvanced Algebra Scope and Sequence First Semester. Second Semester
Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs
More informationChapters 8 & 9 Review for Final
Math 203 - Intermediate Algebra Professor Valdez Chapters 8 & 9 Review for Final SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the formula for
More informationUse the slope-intercept form to graph the equation. 8) 6x + y = 0
03 Review Solve the inequalit. Graph the solution set and write it in interval notation. 1) -2(4-9) < - + 2 Use the slope-intercept form to graph the equation. 8) 6 + = 0 Objective: (2.8) Solve Linear
More information3.1 Exponential Functions and Their Graphs
.1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationGraphing Calculator Computations 2
Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationREVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationReview Exercises for Chapter 2
Review Eercises for Chapter 7 Review Eercises for Chapter. (a) Vertical stretch Vertical stretch and a reflection in the -ais Vertical shift two units upward (a) Horizontal shift two units to the left.
More information(2.5) 1. Solve the following compound inequality and graph the solution set.
Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.
More informationAlgebra 2 CPA Summer Assignment 2018
Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou
More informationAlgebra 2 Semester Exam Review
Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More informationFINAL 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 informationLesson Goals. Unit 5 Exponential/Logarithmic Functions Exponential Functions (Unit 5.1) Exponential Functions. Exponential Growth: f (x) = ab x, b > 1
Unit 5 Eponential/Logarithmic Functions Eponential Functions Unit 5.1) William Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Recognize and
More informationLogarithms. Bacteria like Staph aureus are very common.
UNIT 10 Eponentials and Logarithms Bacteria like Staph aureus are ver common. Copright 009, K1 Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,
More informationf(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 informationChapter 11 Exponential and Logarithmic Function
Chapter Eponential and Logarithmic Function - Page 69.. Real Eponents. a m a n a mn. (a m ) n a mn. a b m a b m m, when b 0 Graphing Calculator Eploration Page 700 Check for Understanding. The quantities
More informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationHonors Algebra 2: Semester 1 Review
Name Block Date Honors Algebra : Semester 1 Review NON-CALCULATOR 6-5 1. Given the functions f ( ) 5 11 1, g( ) 6 ( f h)( ) b) ( g f )( ), and h ( ) 4, find each function. g c) (g h)( ) d) ( ) f -1, 4-7,
More informationd. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36
Vertices: (.8, 5.), (.37, 3.563), (.6, 0.980), (5.373, 6.66), (.8, 7.88), (.95,.) Graph the equation for an value of P (the second graph shows the circle with P 5) and imagine increasing the value of P,
More informationFinal Exam Review Spring a. Is this a quadratic? 2 a. Is this a quadratic? b. EXPLAIN why or why not. b. EXPLAIN why or why not!!
Final Exam Review Spring 01-013 Name Module 4 Fill in the charts below. x x -6 0 Change in 0 0 Change in -3 1 1-1 4 5 0 9 3 10 16 4 17 5 5 5 6 6 36 6 37 1 Is this a quadratic? Is this a quadratic? b. EXPLAIN
More informationReady To Go On? Skills Intervention 7-1 Exponential Functions, Growth, and Decay
7A Find these vocabular words in Lesson 7-1 and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention 7-1 Eponential Functions, Growth, and Deca eponential growth eponential deca asmptote
More informationThe formulas below will be provided in the examination booklet. Compound Interest: r n. Continuously: n times per year: 1
HONORS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part will be selected response on which a calculator will not be allowe Part will be short answer
More informationy = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
More informationA. 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 informationMath125 Exam 5 Review Name. Do the following as indicated.
Math Eam Review Name Do the following as indicated. For the given functions f and g, find the requested function. ) f() = - 6; g() = 9 Find (f - g)(). ) ) f() = 33 + ; g() = - Find (f g)(). 3) f() = ;
More informationReview 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