C H A P T E R 3 Polynomial Functions
|
|
|
- Leon Cummings
- 5 years ago
- Views:
Transcription
1 C H A P T E R Polnomial Functions Section. Quadratic Functions and Models Section. Polnomial Functions of Higher Degree Section. Polnomial and Snthetic Division Section. Zeros of Polnomial Functions Section. Mathematical Modeling and Variation Review Eercises Problem Solving Practice Test
2 C H A P T E R Polnomial Functions Section. Quadratic Functions and Models You should know the following facts about parabolas. f a b c, a 0, is a quadratic function, and its graph is a parabola. If a > 0, the parabola opens upward and the verte is the point with the minimum -value. If a < 0, the parabola opens downward and the verte is the point with the maimum -value. The verte is ba, f ba. To find the -intercepts (if an), solve a b c 0. The standard form of the equation of a parabola is f a h k where a 0. (a) The verte is h, k. (b) The ais is the vertical line h. Vocabular Check. nonnegative integer; real. quadratic; parabola. ais or ais of smmetr. positive; minimum. negative; maimum. f opens upward and has verte, 0.. f opens upward and has verte, 0. Matches graph (g). Matches graph (c).. f opens upward and has verte 0,.. f opens downward and has verte 0,. Matches graph (b). Matches graph (h).. f opens downward. f opens upward and has verte and has verte,. Matches graph (f).,. Matches graph (a). 7. f opens downward and has 8. f opens downward and has verte, 0. verte,. Matches graph (e). Matches graph (d). 9
3 Section. Quadratic Functions and Models (a) (b) 8 Vertical shrink Vertical shrink and reflection in the -ais (c) (d) Vertical stretch Vertical stretch and reflection in the -ais 0. (a) (b) Vertical translation one unit upward Vertical translation one unit downward (c) (d) Vertical translation three units upward Vertical translation three units downward
4 98 Chapter Polnomial Functions. (a) (b) Horizontal translation one unit to the right Horizontal shrink and a vertical translation one unit upward (c) (d) Horizontal stretch and a vertical translation three units downward Horizontal translation three units to the left. (a) (b) (c) Horizontal translation two units to right, vertical shrink each -value is multiplied b, reflection in the -ais, and vertical translation one unit upward Horizontal translation one unit to the right, horizontal stretch (each -value is multiplied b ), and vertical translation three units downward (d) Horizontal translation two units to left, vertical shrink each -value is multiplied b, reflection in -ais, and vertical translation one unit downward Horizontal translation one unit to left, horizontal shrink each -value is multiplied b, and vertical translation four units upward
5 Section. Quadratic Functions and Models 99. f. h Verte: 0, Verte: 0, Ais of smmetr: 0 or the -ais Ais of smmetr: 0 Find -intercepts: Find -intercepts: ± -intercepts:, 0,, 0 ± -intercepts: ±, f 0. f Verte: 0, Verte: 0, Ais of smmetr: 0 or the -ais Ais of smmetr: 0 Find -intercepts: 0 8 -intercepts: ±8 ±, 0,, 0 Find -intercepts: 0 ±8 -intercepts: ±8, f 8. Verte:, Ais of smmetr: Find -intercepts: 0 ± ± 0 8 -intercepts:, 0,, f Verte:, Ais of smmetr: Find -intercepts: 0 Not possible for real No -intercepts h 8 0. g Verte:, 0 Ais of smmetr: -intercept:, Verte:, 0 Ais of smmetr: -intercept:, 0
6 00 Chapter Polnomial Functions. f. f 9 9 Verte:, Verte:, Ais of smmetr: Ais of smmetr: Find -intercepts: Find -intercepts: 0 0 Not a real number No -intercepts ± ±9 ± -intercepts: ±, 0. f. f Verte:, Verte:, Ais of smmetr: Ais of smmetr: Find -intercepts: Find -intercepts: ± 0 ± ± ± -intercepts: ±, 0 -intercepts:, 0,, 0
7 Section. Quadratic Functions and Models 0. h. f 0 Verte:, Ais of smmetr: Verte:, 7 8 Find -intercepts: 0 Not a real number ± No -intercepts Ais of smmetr: Find -intercepts: Not a real number No -intercepts 0 ± f 8. f Verte:, Ais of smmetr: Verte: 9 9, Find -intercepts: Ais of smmetr: or -intercepts:, 0,, Find -intercepts: intercepts:, 0,, f 0. f 0 Verte:, Ais of smmetr: -intercepts:, 0,, Verte: 0 0, 0 0 Ais of smmetr: -intercepts:, 0,, 0 80
8 0 Chapter Polnomial Functions. g 8. Verte:, Ais of smmetr: -intercepts: ±, 0 8 f 0 0 Verte:, 0 0 Ais of smmetr: -intercepts: ±, 0. f 8. Verte:, Ais of smmetr: -intercepts: ±, 0 f Verte: 9, Ais of smmetr: No -intercepts g. Verte:, Ais of smmetr: -intercepts: ±, 0 8 f 9 7 Verte:, 0 Ais of smmetr: -intercepts: ±, 0 0 0,, 7., 0 is the verte. 8. 0, is the verte. a 0 a f a 0 a Since the graph passes through the point we have: Since the graph passes through, 0, a0 0 a a a. So,. 9., is the verte. 0., is the verte. a f a Since the graph passes through the point, 0, we have: 0 a Since the graph passes through 0,, a0 a a a a a. So,.
9 Section. Quadratic Functions and Models 0., is the verte.., 0 is the verte. a f a 0 a Since the graph passes through the point, 0, we have: Since the graph passes through,, 0 a a a a. So,.., is the verte.., is the verte. f a f a Since the graph passes through the point 0, 9, we have: 9 a0 Since the graph passes through,, a a a a a f a. So, f.., is the verte.., is the verte. f a Since the graph passes through the point,, we have: a a a f f a Since the graph passes through 0,, a0 a a a. So, f. 7., is the verte. 8., is the verte. f a f a Since the graph passes through the point 7,, we have: a7 Since the graph passes through, 0, 0 a a a 0 a f a. So, f., 9., is the verte. 0. is the verte. f a Since the graph passes through the point we have: 0 a 9 a a 9 f 9, 0, f a Since the graph passes through,, a 8 a 9 8 a 9 8 a. So, f 9 8.
10 0 Chapter Polnomial Functions., 0 is the verte.., is the verte. f a f a Since the graph passes through the point we have: a 7 a f 7,, Since the graph passes through a 0 00a 9 00a 0 a. 0,, So, f intercepts: ±, 0 -intercept:, 0 ± intercepts:, 0,, 0 0 -intercepts:, 0,, 0 or f 8. f 0 -intercepts: 0, 0, (,0 -intercepts: 0, 0,, ) 0 0 or 0 0 The -intercepts and the solutions of f 0 are the same. 0 The -intercepts and the solutions of f 0 are the same. 9. f f intercepts:, 0,, 0 -intercepts:, 0, 0, ) or 0 The -intercepts and the solutions of f 0 are the same The -intercepts and the solutions of f 0 are the same.
11 Section. Quadratic Functions and Models 0. f intercepts:, 0,, ) or The -intercepts and the solutions of 0 0 f 0 are the same. f -intercepts: 7, 0,, The -intercepts and the solutions of f 0 are the same.. f intercepts:, 0, 7, or 7 0 The -intercepts and the solutions of f 0 are the same. f 7 0 -intercepts:, 0,, The -intercepts and the solutions of f 0 are the same.. f opens upward. g opens downward Note: f a has -intercepts, 0 and, 0 for all real numbers a 0. f, opens upward g f, opens downward g Note: f a has -intercepts, 0 and, 0 for all real numbers a f 0 0 opens upward 8. 0 g 00 opens downward 0 Note: f a 0 0 a 0 has -intercepts 0, 0 and 0, 0 for all real numbers a 0. f 8, opens upward g f, opens downward g Note: f a 8 has -intercepts, 0 and 8, 0 for all real numbers a f opens upward 70. f g 7 Note: 7 7 opens downward f a has -intercepts, 0 and, 0 for all real numbers a 0. 0, opens upward g f, opens downward g 0 Note: f a has -intercepts, 0 and, 0 for all real numbers a 0.
12 0 Chapter Polnomial Functions 7. Let the first number and the second number. Then the sum is Let first number and second number. Then, S, S. The product is P S. The product is P 0 0. P S P 0 S S S 0 The maimum value of the product occurs at the verte of P and is 0. This happens when. S S S The maimum value of the product occurs at the verte of P and is S. This happens when S. 7. Let the first number and the second number. 7. Let the first number and the second number. Then the sum is Then the sum is.. The product is P P. The product is P P. 7 The maimum value of the product occurs at the verte of P and is 7. This happens when and. Thus, the numbers are and. 7 The maimum value of the product occurs at the verte of P and is 7. This happens when and 7. Thus, the numbers are and (a) (b) A 00 This area is maimum when feet and 00 feet (c) A This area is maimum when feet and 00 feet. CONTINUED
13 Section. Quadratic Functions and Models CONTINUED (d) A (e) The are all identical. feet and feet The maimum area occurs at the verte and is 000 square feet. This happens when feet and feet. The dimensions are 0 feet b feet. 7. (a) Radius of semicircular ends of track: r Distance around two semicircular parts of track: d r (b) Distance traveled around track in one lap: d (c) Area of rectangular region: A The area is maimum when 0 and The verte occurs at b 9. The maimum height is feet. 9 a (a) The ball height when it is punted is the -intercept feet (b) The verte occurs at The maimum height is b a 9 0 f , 9 7 feet 0.0 feet. CONTINUED
14 08 Chapter Polnomial Functions 78. CONTINUED (c) The length of the punt is the positive -intercept ± or 8. The punt is approimatel 8. ft..8 ± C The verte occurs at b a The cost is minimum when 0 fitures. C 00, The verte occurs at The cost is minimum when units. 8. P , P The verte occurs at The profit is maimum when b a 0 0, ,000 units. The verte occurs at b 0 0. a 0. Because is in hundreds of dollars, dollars is the amount spent on advertising that gives maimum profit. 8. Rp p 00p 8. (a) R0 $,000 thousand R $,7 thousand R0 $,00 thousand (b) The revenue is a maimum at the verte b 00 a R,00 The unit price that will ield a maimum revenue of $,00 thousand is $. R p p 0p (a) R$ $ 0$ $08 R$ $ 0$ $8 R$8 $8 0$8 $ (b) The verte occurs at p b a 0 $. Revenue is maimum when price $. per pet. The maimum revenue is f$. $. 0$. $ C 99.8t.t, 0 t (a) (c) C0 0 Annuall: Dail: ,8,090 8,08,90 cigarettes 8879 cigarettes (b) Verte 0, 99 The verte occurs when 99 which is the maimum average annual consumption. The warnings ma not have had an immediate effect, but over time the and other findings about the health risks and the increased cost of cigarettes have had an effect.
15 Section. Quadratic Functions and Models (a) and (c) (a) 0 00 (b) (d) 99 (e) Verte occurs at (f) Minimum occurs at ear b a There will be approimatel,8,000 hairdressers and cosmetologists in 008. (b) 0.00s 0.00s a, b, c 0,09 s ± 0,09 s s s 0,09 0 ± 80,7 s 7., 9. s s 9 0,000 The maimum speed if power is not to eceed 0 horsepower is 9. miles per hour. 88. (a) and (c) (b) (d) The maimum of the graph is at., or about. mi/h. Algebraicall, the maimum occurs at b a mi/h. 89. True. The equation 0 has no real solution, 90. True. The verte of f is and the verte of so the graph has no -intercepts. is, 7,. g 9. f a b c a b a c a b a b a a b a b a c a b a f b a a b a b b a c b b a a c b b b ac a a c ac b a ac b a So, the verte occurs at b ac b, a a b a, f a b.
16 0 Chapter Polnomial Functions 9. Conditions (a) and (d) are preferable because profits would be increasing. 9. Yes. A graph of a quadratic equation whose verte is has onl one -intercept. 0, 0 9. If f a b c has two real zeros, then b the Quadratic Formula the are b ± b ac. a The average of the zeros of f is b b ac a b b ac a This is the -coordinate of the verte of the graph. b a b a. 9., and, 9. m 7,, m and m 98. The slope of the perpendicular line through 0, is m and the -intercept is b. m For a parallel line, m. So, for 8,, the line is 8 0. For Eercises 99 0, let f, and g f g f g 00. g f fg 7 f 7 g f g f g fg f g f 0 g f 0 g0 g 8 7
CHAPTER 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.
CHAPTER 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............
Review 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.
Quadratic Functions and Models
Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Use quadratic functions
STANDARD 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
Polynomial and Rational Functions
Polnomial and Rational Functions. Quadratic Functions and Models. Polnomial Functions of Higher Degree. Polnomial and Snthetic Division. Comple Numbers.5 Zeros of Polnomial Functions.6 Rational Functions.7
Name Class Date. Quadratic Functions and Transformations. 4 6 x
- Quadratic Functions and Transformations For Eercises, choose the correct letter.. What is the verte of the function 53()? D (, ) (, ) (, ) (, ). Which is the graph of the function f ()5(3) 5? F 6 6 O
CHAPTER 8 Quadratic Equations, Functions, and Inequalities
CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solving Quadratic Equations: Factoring and Special Forms..................... 7 Section. Completing the Square................... 9 Section.
2Polynomial and. Rational Functions
Polnomial and Rational Functions A ballista was used in ancient times as a portable rock-throwing machine. Its primar function was to destro the siege weaponr of opposing forces. Skilled artiller men aimed
Chapter 2 Polynomial, Power, and Rational Functions
Section. Linear and Quadratic Functions and Modeling 6 Chapter Polnomial, Power, and Rational Functions Section. Linear and Quadratic Functions and Modeling Eploration. $000 per ear.. The equation will
CHAPTER 1 Functions and Their Graphs
PART I CHAPTER Functions and Their Graphs Section. Lines in the Plane....................... Section. Functions........................... Section. Graphs of Functions..................... Section. Shifting,
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
CHAPTER 1 Functions, Graphs, and Limits
CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula.......... Section. Graphs of Equations........................ 8 Section. Lines in the Plane and Slope....................
SHORT 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) - -
Writing Quadratic Functions in Standard Form
Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval
Maintaining Mathematical Proficiency
Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +
UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS
Answer Ke Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questions. For the quadratic function shown below, the coordinates of its verte are, (), 7 6,, 6 The verte is
For Thought. 3.1 Exercises 142 CHAPTER 3 POLYNOMIAL AND RATIONAL FUNCTIONS. 1. False, the range of y = x 2 is [0, ).
CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS For Thought. False, the range of = is [0, ).. False, the verte is the point (, ). -5 -. True. True 5. True, since b a = 6 =. 6. True, the -intercept of = ( + )
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
CHAPTER 1 Functions, Graphs, and Limits
CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula... Section. Graphs of Equations...8 Section. Lines in the Plane and Slope... Mid-Chapter Quiz Solutions... Section.
= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
Skills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Give an eample of each term.. quadratic function 9 0. vertical motion equation s
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.
Ready 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
CHAPTER P Preparation for Calculus
CHAPTER P Preparation for Calculus Section P. Graphs and Models...................... Section P. Linear Models and Rates of Change............ Section P. Functions and Their Graphs................. Section
TEST REVIEW QUADRATICS EQUATIONS Name: 2. Which of the following statements is true about the graph of the function?
Chapter MATHEMATICS 00 TEST REVIEW QUADRATICS EQUATIONS Name:. Which equation does not represent a quadratic function?. Which of the following statements is true about the graph of the function? it has
SECTION 3.1: Quadratic Functions
SECTION 3.: Quadratic Functions Objectives Graph and Analyze Quadratic Functions in Standard and Verte Form Identify the Verte, Ais of Symmetry, and Intercepts of a Quadratic Function Find the Maimum or
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,
Chapter 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
Mini-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.
NAME DATE PERIOD. Study Guide and Intervention. Transformations of Quadratic Graphs
NAME DATE PERID Stud Guide and Intervention Write Quadratic Equations in Verte Form A quadratic function is easier to graph when it is in verte form. You can write a quadratic function of the form = a
Unit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
CHAPTER 3 Exponential and Logarithmic Functions
CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs......... Section. Properties of Logarithms..................
Math 1050 REVIEW for Exam 1. Use synthetic division to find the quotient and the remainder. 1) x3 - x2 + 6 is divided by x + 2
Math 0 REVIEW for Eam 1 Use snthetic division to find the quotient and the remainder. 1) 3-2 + 6 is divided b + 2 Use snthetic division to determine whether - c is a factor of the given polnomial. 2) 3-32
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..
y x+ 2. A rectangular frame for a painting has a perimeter of 82 inches. If the length of the frame is 25 inches, find the width of the frame.
. Simplif the complex fraction x 4 4 x. x 4 x x 4 x x 4 x x 4 x x E) 4 x. A rectangular frame for a painting has a perimeter of 8 inches. If the length of the frame is inches, find the width of the frame.
Chapter 10 Answers. Practice (0,0); maximum 2. (0,0); maximum 3. (0,0); minimum y = x 2, y = 3x 2, y = 5x 2 8. y 1
Chapter 0 Answers Practice 0-. (0,0); maimum. (0,0); maimum. (0,0); minimum. (0,0); minimum. (0,0); maimum. (0,0); minimum 7. =, =, =. =, =-, =-. =, =-, = 0. =, =, =-. =, =, =-7. =, =, =........ 7. 0 7...
C H A P T E R 9 Topics in Analytic Geometry
C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation
f(x) Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.
NAME DATE PERID 4-1 Practice Graphing Quadratic Functions Complete parts a c for each quadratic function. a. Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte. b.
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
Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas
Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve
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
LESSON #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
x x 5, x no real solutions. x a. 103 feet. b seconds
BRIDGE TO ALGEBRA B. 0. 9 3. 40 4. 5. 6 6. 9 5 6 7. 4 8. 3 9. 0 0. 7. 5,. 5, 3. no real solutions 4. 3 5 4 5. a. 03 feet b. 5.3 seconds 6. a. There are two times when the ball is si feet above the ground.
3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
b g if f x x x x x x
1. Find the product, if possible. AB, if A = 0 1, 4 1 0 4 4 3 15 B = 1 4 0 1 4 1 16 6 1 4 1 0 4 7 0 0 16 0 1 0 0 4 3 4 15. Find the equation of the parabola with vertex at the origin that passes through
CHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative
Instructor: 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
Path of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2.
- Quadratic Functions and Transformations Content Standards F.BF. Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative)
Maintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.
REVIEW 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.
Shape and Structure. Forms of Quadratic Functions. Lesson 2.1 Assignment
Lesson.1 Assignment Name Date Shape and Structure Forms of Quadratic Functions 1. Analze the graph of the quadratic function. a. The standard form of a quadratic function is f() 5 a 1 b 1 c. What possible
Math 1101 Chapter 3 Review. 1) f(x) = 2x2 + 2x - 4 A) Concave up B) Concave down. 2) f(x) = -2x2-2x + 2 A) Minimum B) Maximum. 3) f(x) = 0.
Math 11 Chapter 3 Review Determine if the graph of the function is concave up or concave down. 1) f() = + - Concave up B) Concave down Determine if the verte of the graph is a maimum point or a minimum
Law 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,
y = 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 (,
Graph Quadratic Functions in Standard Form
TEKS 4. 2A.4.A, 2A.4.B, 2A.6.B, 2A.8.A Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in
Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
Answers. Chapter Start Thinking Sample answer: y-intercept: 8 5. x x
. ( 7, ) 9. (, 9 ) 0. (, 7). no solution. (, 7). no solution. no solution. ( 7, ). infinitel man solutions 7. (, 7 ). infinitel man solutions 9. (, 9) 70. 9a + a + 7. b b + 9 7. c + 90c + 7. 9d d + 7.
MATH 115: Review for Chapter 3
MATH : Review for Chapter Can ou use the Zero-Product Propert to solve quadratic equations b factoring? () Solve each equation b factoring. 6 7 8 + = + ( ) = 8 7p ( p ) p ( p) = = c = c = + Can ou solve
3.1-Quadratic Functions & Inequalities
3.1-Quadratic Functions & Inequalities Quadratic Functions: Quadratic functions are polnomial functions of the form also be written in the form f ( ) a( h) k. f ( ) a b c. A quadratic function ma Verte
Exam 2 Review F15 O Brien. Exam 2 Review:
Eam Review:.. Directions: Completely rework Eam and then work the following problems with your book notes and homework closed. You may have your graphing calculator and some blank paper. The idea is to
Section 2.5: Graphs of Functions
Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree
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
Answers. Chapter Warm Up. Sample answer: The graph of h is a translation. 3 units right of the parent linear function.
Chapter. Start Thinking As the string V gets wider, the points on the string move closer to the -ais. This activit mimics a vertical shrink of a parabola... Warm Up.. Sample answer: The graph of f is a
d. 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,
a [A] +Algebra 2/Trig Final Exam Review Fall Semester x [E] None of these [C] 512 [A] [B] 1) Simplify: [D] x z [E] None of these 2) Simplify: [A]
![a [A] +Algebra 2/Trig Final Exam Review Fall Semester x [E] None of these [C] 512 [A] [B] 1) Simplify: [D] x z [E] None of these 2) Simplify: [A]](/thumbs/95/126290868.jpg "a [A] +Algebra 2/Trig Final Exam Review Fall Semester x [E] None of these [C] 512 [A] [B] 1) Simplify: [D] x z [E] None of these 2) Simplify: [A]")
) Simplif: z z z 6 6 z 6 z 6 ) Simplif: 9 9 0 ) Simplif: a a a 0 a a ) Simplif: 0 0 ) Simplif: 9 9 6) Evaluate: / 6 6 6 ) Rationalize: ) Rationalize: 6 6 0 6 9) Which of the following are polnomials? None
Mathematics 2201 Midterm Exam Review
Mathematics 0 Midterm Eam Review Chapter : Radicals Chapter 6: Quadratic Functions Chapter 7: Quadratic Equations. Evaluate: 6 8 (A) (B) (C) (D). Epress as an entire radical. (A) (B) (C) (D). What is the
Lesson 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)
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
Math 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 1 chapter si practice eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which equation matches the given calculator-generated graph and description?
9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson
Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric
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
MAT 1033C -- Martin-Gay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam
MAT 33C -- Martin-Ga Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Final Exam Review Part 2 #1 Page 1 / 21
Final Eam Review Part #1 Intermediate Algebra / MAT 135 Spring 017 Master ( Master Templates) Student Name/ID: v 1. Solve for, where is a real number. v v + 1 + =. Solve for, where is a real number. +
KCATM 2013 Algebra Team Test. E) No Solution. C x By. E) None of the Above are correct C) 9,19
KCTM 03 lgebra Team Test School ) Solve the inequality: 6 3 4 5 5 3,,3 3, 3 E) No Solution Both and B are correct. ) Solve for : By C C B y C By B C y C By E) None of the bove are correct 3) Which of the
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. x )
Midterm Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Decide whether or not the arrow diagram defines a function. 1) Domain Range 1) Determine
Vocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient.
CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. Term Page Definition Clarifing
Chapter 2: Quadratic and Other Special Functions. Exercises 2.1. x 2 11x 10 0 x 2 10x x ( x 10)(x 1) 0 x 10 0 or x 1 0
Mathematical Applications for the Management Life and Social Sciences 11th Edition Harshbarger SOLUTIONS MANUAL Full clear download at: https://testbankreal.com/download/mathematical-applications-managementlife-social-sciences-11th-edition-harshbarger-solutions-manual/
Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.
Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of
Review for Intermediate Algebra (MATD 0390) Final Exam Oct 2009
Review for Intermediate Algebra (MATD 090) Final Eam Oct 009 Students are epected to know all relevant formulas, including: All special factoring formulas Equation of a circle All formulas for linear equations
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Linear equations 1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Find the slope of the line passing through the points (, -3) and (2, -1). 1)
QUEEN ELIZABETH REGIONAL HIGH SCHOOL MATHEMATICS 2201 MIDTERM EXAM JANUARY 2015 PART A: MULTIPLE CHOICE ANSWER SHEET
QUEEN ELIZABETH REGIONAL HIGH SCHOOL MATHEMATICS 01 MIDTERM EXAM JANUARY 01 PART A: MULTIPLE CHOICE NAME: ANSWER SHEET 1. 11. 1.. 1... 1... 1... 1... 1.. 7. 17. 7. 8. 18. 8. 9. 19. 9. 10. 0. 0. QUADRATIC
Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which
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.
Chapter 1 Graph of Functions
Graph of Functions Chapter Graph of Functions. Rectangular Coordinate Sstem and Plotting points The Coordinate Plane Quadrant II Quadrant I (0,0) Quadrant III Quadrant IV Figure. The aes divide the plane
Math 141 Review for Midterm
Math 141 Review for Midterm There will be two parts to this test. Part 1 will have graph sketching, and no calculator is allowed. Part will have everthing else, and a calculator and/or graphing calculator
5. 2. The solution set is 7 6 i, 7 x. Since b = 20, add
Chapter : Quadratic Equations and Functions Chapter Review Eercises... 5 8 6 8 The solution set is 8, 8. 5 5 5 5 5 5 The solution set is 5,5. Rationalize the denominator. 6 The solution set is. 8 8 9 6
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
Algebra 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
CHAPTER 3 Graphs and Functions
CHAPTER Graphs and Functions Section. The Rectangular Coordinate Sstem............ Section. Graphs of Equations..................... 7 Section. Slope and Graphs of Linear Equations........... 7 Section.
4-1 Study Guide and Intervention
NAME DATE PERID 4-1 Study Guide and Intervention Graph Quadratic Functions Quadratic Function A function defined by an equation of the form = a 2 + b + c, where a 0 Graph of a Quadratic Function A parabola
Honors Algebra 2 ~ Spring 2014 Name 1 Unit 3: Quadratic Functions and Equations
Honors Algebra ~ Spring Name Unit : Quadratic Functions and Equations NC Objectives Covered:. Define and compute with comple numbers. Operate with algebraic epressions (polnomial, rational, comple fractions)
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).
PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.
MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for
The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.
Mathematics 10 Page 1 of 8 Quadratic Relations in Vertex Form The expression y ax p q defines a quadratic relation in form. The coordinates of the of the corresponding parabola are p, q. If a > 0, the
4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the
Mathematics 2201 Midterm Exam Review
Mathematics 0 Midterm Eam Review Chapter : Radicals Chapter : Quadratic Functions Chapter 7: Quadratic Equations. Evaluate: 8 (A) (B) (C) (D). Epress as an entire radical. (A) (B) (C) (D). What is the
MATH 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
2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.
Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =