# 1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.

Size: px
Start display at page:

Download "1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names."

Transcription

1 Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1-. Think of these as parent functions too. 1 a) f ( ) b) f( ) Why is this a power function? Why is this a power function? Graph this function (label 5 points) Graph this function (label pts & asymptotes) y y Domain: Domain: Range: Range: Symmetry: Symmetry: Boundedness: Boundedness: Asymptotes: Asymptotes: Discontinuities: Discontinuities: Increasing/Decreasing: Increasing/Decreasing: Etrema: Etrema:

2 a. Graphs of Power Functions: The key to power function graphs is the eponent in y. If a is a positive even number or a positive odd number, you already know what these functions look like from your parent functions. a) Use your knowledge of parent functions to fill in the 1 st column of the chart below with a basic sketch. a 1 a 0 0 a 1 Odd Even b) Use your graphing calculator to graph the following functions and fill in the nd column in your chart. 4 5 y, y, y, y,... c) Use your graphing calculator to graph the following functions and fill in the nd column in your chart y, y, y, y,... d) Now go back to the functions given in question and rewrite them in Which part of the chart above do they match? y a form. 4. Match the equations to one of the curves labeled in the figure below y=- 4 a b c y= d y= 1 4 y=- 5 y =- - y= 1.7 g j i h

3 5. For mammals and other warm-blooded animal to stay warm requires quite a bit of energy. Temperature loss is related to surface area, which is related to body weight, and temperature gain is related to circulation, which is related to pulse rate. In the final analysis, scientists have concluded that the pulse rate r of mamals is a power function of their body weight w. a) Find the power regression model. (No ROUNDING) b) Use your model from part a to predict the pulse rate for a 450-kg horse. Mammal Body Weight Pulse Rate (kg) (beats/min) Rat Guinea pig Rabbit 05 Small dog 5 10 Large dog 0 85 Sheep Human 70 7 The remaining questions on this worksheet focus on the application of power functions to direct and inverse variation. 6. Multiple Choice: Which of the following ehibit inverse variation? A B C D The distance traveled as a function of speed. The total cost as a function of the number of items purchased. The area of a circular swimming pool as a function of its radius. The number of posts in a 0ft fence as a function of distance between posts. For questions 7 and 8, write a power function to model the situation. 7. The surface area S of a sphere varies directly as the square of the radius r. 8. The period of time T for the full swing of a pendulum varies directly as the square root of the pendulum s length L. 9. The time it takes for a group of volunteers to build a house varies inversely with the number of volunteers v. a) Write a power function to model this situation. b) If 0 volunteers can build a house in 6.5 working hours, how many volunteers would be needed to build a house in 50 working hours. (Hint: Find the value of k first.)

4 10. The power P (in watts) produced by a windmill is proportional to the cube of the wind speed v (in mph). a) Write a power function to model this situation. b) If a wind of 10 mph generates 15 watts of power, how much power is generated by a wind speed of 40mph? 11. Write a power function for the following situation, and then use the equation to solve for the missing information. The intensity I of light varies inversely as the square of the distance D from the source. If the intensity of illumination on a screen 5 feet from a light is foot-candles, find the intensity on a screen 15 feet from the light. 1. The volume V of a gas varies inversely as the pressure P and directly as the temperature T. A certain gas has a volume of 10 liters (L), a temperature of 00 kelvins (K), and a pressure of 1.5 atmospheres (atm). If the gas is compressed to a volume of 7.5 L and is heated to 50 K, what will the new pressure be? 1. The power P that must be delivered by a car engine varies directly as the distance d that the car moves and inversely as the time t required to move that distance. To move the car 500 m in 50 s, the engine must deliver 147 kilowatts (kw) of power. How many kilowatts must the engine deliver to move the car 700 m in 0 s?

5 Pre Calculus Worksheet.1 Before we can use what we know about polynomials, we need to check that we do have a polynomial function. Thinking about what a polynomial looks like, we need identify functions as polynomials or not. 1. Which of the following functions are polynomial functions? For those that are, state the degree and leading coefficient. For those that are not polynomials state why not. a) f ( ) 9 5 b) g 4 ( ) 17 c) h ( ) 5 d) k ( ) 9. Which of the polynomial functions above are also power functions (if any). Eplain.. Using function notation, if f (a) = b, then the function contains the point. The net few problems focus on Linear Functions 4. The average rate of change of a function between two points (a, f (a)) and (b, f (b)) is given by: 5. Write the equation of the linear function, f, if you know the given information. a) f () = 7 and f ( 1) = 5 b) f ( 5) = 1 and f () = 4 6. The table shows the average weekly earnings of construction workers for several years. Let = the number of years since Year Weekly Earnings a) Find the linear regression model for the data. b) What does the slope of the regression represent in terms of this situation? c) Use the linear regression model to predict the construction worker average salary in 005. DO NOT ROUND your regression model to determine your answer.

6 Now to focus on Quadratic Functions We have covered Quadratic functions etensively in Algebra, F.S.T. and in our study of parent functions in chapter 1. In order to put a standard form quadratic function into verte form, we use a process called Completing the Square. This method works well when we have an a value of 1 and/or a b value that is even. 7. Complete the square to rewrite each quadratic function in verte form, describe the transformation, and then graph the function. A sample is provided for your reference. Sample: y 1 7 y ( 4 ) 7 y +4 1 ( 4 ) 7 y ( ) 5 a) f( ) 4 6 b) f( ) 6 7 b 8. Use =- a to find the verte of the function in question 7a and 7b. What do you notice? 9. Find the equation of a quadratic function of the form f a h k with the given information: a) passes through (, 1) and has verte ( 1, 5) b) y (, 11) (1, ) 10. Among all the rectangles whose perimeters are 100 ft, find the dimensions of the one with the maimum area. 11. A large painting is feet longer than it is wide. If the wooden frame is 1 inches wide and the area of the painting is 08 ft, find the dimensions of the painting.

7 1. Jack was named manager of the month at the Sylvania Wire Company due to his hiring study. The study showed that each of the 0 salespersons he supervised averaged \$50,000 in sales each month, and that for each additional sales person he would hire, the average sales would decrease by \$1,000 per salesperson. a) Write a function for Jack s study showing the total sales as a function of the number of additional salespersons Jack hires. b) Use your function to determine how many salespersons Jack should hire to maimize the income from sales. What is this maimum sales Jack s company can anticipate according to the model? Projectile Motion All projectile motion can be represented by a quadratic function. Most quadratics are written in standard form based on a formula discovered by Sir Isaac Newton in the 17 th century. Newton discovered that the height h on an object at time t after it has been thrown upward with an initial velocity v o from an initial height h o satisfies the formula: 1 ht gt vt h where g is the acceleration due to gravity ( 9.8 meters per second or feet per second ) () o o 1. What is the leading coefficient for every projectile motion function whose height is measured in feet? What is the leading coefficient for every projectile motion function whose height is measured in meters? 14. A baseball throwing machine is use to train little league players to catch pop-ups. The machine throws baseballs straight upward with an initial velocity of 48 ft/sec from a height of.5 feet. a) Find an equation that models the height of the ball as a function of the ball t seconds after its thrown. b) What is the maimum height the ball will reach? How many seconds will it take to reach this height? c) If the player misses the catch, how long will it take the ball to hit the ground? 15. Consider the function f ( ) = -. a) Find the average rate of change between the points on the function when = 1 and =. b) On a graph, a line that connects any two points is called a secant line. Find the equation of the secant line that connects the points on the function when = 1 and =. c) In calculus, we find the slope between a point on the graph when = a and another point that is a small distance away. We use = a + h for the second point. Find the average rate of change between the points = a and = a + h.

8 Pre Calculus Worksheet. In lesson. we are putting together lots of skills from Algebra. If you are having difficulty with any of these individual skills, you need to check out a tetbook and do the OPTIONAL problems for your particular area of difficulty listed at the bottom of this worksheet BEFORE you attempt to put all the skills together. For each of the functions below a) State the degree of the polynomial, b) list the zeros of the polynomial function, c) state the multiplicity of each zero and whether the graph crosses the -ais or bounces off the -ais at the corresponding -intercept, and d) list the end behavior using limit notation and e) using all this information, sketch the function without a calculator. 1. k ( 1)( 4). h ( ) h (10 5 )(9 ) 4. f 6(4 ) ( 8) f( ) g ( ) 5

9 7. What is the y-intercept of the graph of f( ) 1 5? (NO, its not 5 and NO using your calculator!!) 8. Eplain why the function 9 g ( ) 50has at least one real zero. 9. Using only algebra, find a cubic function in standard form with zeros, 1 4 and 6. Check with a graphing calculator. 10. Many students begin to question why we spend all this time graphing by hand when we have a graphing calculator? 4 To answer this question graph the function f( ) using the given window and sketch the graph, then answer the questions that follow. Window A: , 1 f ( ) 1 Window B: 6 4, 000 f( ) 000 a) Which window matches what you know about the end behavior of the function? Why? b) What is missing from Window B that shows up in Window A? c) Why can t we just rely on what the calculator says? Net let s do some numerical analysis of polynomial functions. Given each function and corresponding table of values below, find the difference in the y values of the function. Then, find the difference in the results. Continue this process until you get a constant. How many levels of differences did you take? What can you tell about function according to the number of levels of differences? 11. f( ) f() f ( ) f()

10 1. Use the given table of values to determine the degree of the polynomial function whose curve will fit the data. Then, find the equation of that polynomial function using the regression capabilities of your graphing calculator f() Sometimes the numerical analysis does not work and the differences are never constant. In this situation, you can graph a scatterplot, as well as trying the possible regression equations (linear, quadratic, cubic or quartic) to determine which function best fits the given data. Using the data below, determine the best model and eplain why you chose this model. A graphing calculator should be used! f() Now for an application 15. A state highway patrol safety division collected data on stopping distances according to certain speeds of travel. This data is summarized in the table below. Speed (mph) Stopping Distance (ft) a) Find the quadratic regression model. b) Use the regression model to find the stopping distance for a vehicle traveling at 5 mph. Do NOT round your model, but round your answer to the thousandths. c) After an accident, a policeman measured the stopping distance (skid mark) of one car to be 00 feet. The driver claimed the car was traveling at the 55 mph speed limit when the brakes were applied. Use the regression model (and the graphing capabilities of your calculator) to predict the speed of a car when the stopping distance was 00 feet. Was the driver telling the truth? OPTIONAL EXTRA PRACTICE see your teacher to borrow a tetbook. Using limit notation to describe end behavior of polynomials page 0: #5-8 Find zeros of polynomial by factoring page: 0: #-8 Use end behavior and multiplicity of zeros to sketch polynomial by hand page: 0: #9-4, 7, 74

11 Pre Calculus Worksheet.4 A summary statement is another way to write the answer to a division problem. For eample, look back at eample from notes.4. We found that 9. Suppose we multiplied BOTH sides of this equation by. The fractions would cancel to leave 4 In other words, p( ) This is the summary statement. 1. Use long division to divide. Write a summary statement in polynomial form. a) b) Eplain why the following problems can be done using synthetic division; then use synthetic division to divide. Write a summary statement in polynomial form. a) 5 1 b) When we are only concerned with the remainder from division, we can use the remainder theorem to evaluate a polynomial when given a value or a factor. Find the remainder when f ( ) 5 5 is divided by k for the given value of k.. a) k b) k 1 4. What do you notice about question b above? Use this information to factor f ().

12 5. Find g( ) when 5 gw ( ) w 8w 5w 4using two different algebraic strategies from this lesson. Sometimes we can give you a graph to help you factor a polynomial. 6. For each given function, use the graph to guess and synthetic division to verify possible linear factor(s). Then, write the polynomial in factored form. a) f( ) b) g ( ) Use the factor theorem to decide if ( ) is a factor of theorem. 4. Eplain how your work relates to the factor 8. Find the polynomial function in factored form: y = a( b)( c)( d) for the given data y Find the polynomial function in standard form with leading coefficient that has the given degree and zeros. a) Degree, with 1, 0, and 4 as zeros b) Degree, with, 1, and 4 as zeros.

13 10. What does the Rational Zeros Theorem tell you? 11. Multiple Choice: Let f ( ) 7. Which of the following is NOT a possible rational root of f? A. B. 1 C. 1 D. 1/ E. / 1. Multiple Choice: Let f be a polynomial function with f () 0. Which of the following statements is NOT true? A. + is a factor of f(). B. is a factor of f(). C. = is a zero of f(). D. is an intercept of f(). E. The remainder when f() is divided by is zero. 1. List all possible rational zeros for the given function. Then, determine which, if any, possible zeros are actually zeros. NO CALCULATOR!! f ( ) For the function below, list the possible rational zeros. Then, use your calculator to determine which of these possible zeros are zeros and verify them algebraically (don t just say the calculator said so!!). Finally, rewrite the function in factored form. 4 h ( ) The Sunspot Small Appliance Company determines that the supply function for their EverCurl hair dryer is S( p) p and the its demand function is D( p) p, where p is the price of the hair dryer. Determine the price for which the supply is equal to the demand and the number of hair dryers corresponding to this equilibrium price.

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

### Solving Quadratic Equations

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Calculus 1st Semester Final Review

Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### 3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?

Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using

### Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

### Polynomial, Power, and Rational Functions

CHAPTER Polynomial, Power, and Rational Functions.1 Linear and Quadratic Functions and Modeling. Power Functions with Modeling.3 Polynomial Functions of Higher Degree with Modeling.4 Real Zeros of Polynomial

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Mathematics as Problem Solving The students will demonstrate the ability to gather information from a graphical representation of an equation.

Title: Another Way of Factoring Brief Overview: Students will find factors for quadratic equations with a leading coefficient of one. The students will then graph these equations using a graphing calculator

### Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

### 2.4 Real Zeros of Polynomial Functions

SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower

### 12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2

DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal

### TSI College Level Math Practice Test

TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)

### 6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

### Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

### A Resource for Free-standing Mathematics Qualifications

To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y say) in terms of one other variable (x). dy Find an expression for and put it equal to 0. Solve

### Use Square Roots to Solve Quadratic Equations

10.4 Use Square Roots to Solve Quadratic Equations Before You solved a quadratic equation by graphing. Now You will solve a quadratic equation by finding square roots. Why? So you can solve a problem about

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### 2-5 Rational Functions

-5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

### a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### MATH 100 PRACTICE FINAL EXAM

MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number

### Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

### Algebra II A Final Exam

Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

### FREE FALL. Introduction. Reference Young and Freedman, University Physics, 12 th Edition: Chapter 2, section 2.5

Physics 161 FREE FALL Introduction This experiment is designed to study the motion of an object that is accelerated by the force of gravity. It also serves as an introduction to the data analysis capabilities

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### Graphing Trigonometric Skills

Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

### Mathematical Modeling and Optimization Problems Answers

MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with -inch margins at the top and bottom of the poster

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Section 3-3 Approximating Real Zeros of Polynomials

- Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros

### Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.

### Introduction to Quadratic Functions

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### How To Factor Quadratic Trinomials

Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles

### Section 1-4 Functions: Graphs and Properties

44 1 FUNCTIONS AND GRAPHS I(r). 2.7r where r represents R & D ependitures. (A) Complete the following table. Round values of I(r) to one decimal place. r (R & D) Net income I(r).66 1.2.7 1..8 1.8.99 2.1

### Section 2-3 Quadratic Functions

118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### Roots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)

Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only

ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### Answers to Basic Algebra Review

Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### Analyzing Functions Intervals of Increase & Decrease Lesson 76

(A) Lesson Objectives a. Understand what is meant by the terms increasing/decreasing as it relates to functions b. Use graphic and algebraic methods to determine intervals of increase/decrease c. Apply

### COMPETENCY TEST SAMPLE TEST. A scientific, non-graphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh

BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, non-graphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### Sample Problems. Practice Problems

Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Factoring Quadratic Trinomials

Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists

### Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### 7.2 Quadratic Equations

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### Big Ideas in Mathematics

Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### 2.5 Zeros of a Polynomial Functions

.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

### M122 College Algebra Review for Final Exam

M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant

### 18.01 Single Variable Calculus Fall 2006

MIT OpenCourseWare http://ocw.mit.edu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Unit : Derivatives A. What

### Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED

Algebra Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED. Graph eponential functions. (Sections 7., 7.) Worksheet 6. Solve eponential growth and eponential decay problems. (Sections 7., 7.) Worksheet 8.

### Quadratic Equations and Functions

Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

### 1 of 7 9/5/2009 6:12 PM

1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### MATH 185 CHAPTER 2 REVIEW

NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..

### Quadratic Functions Unit

Quadratic Functions Unit (Level IV Academic Math) NSSAL (Draft) C. David Pilmer 009 (Last Updated: Dec, 011) Use our online math videos. YouTube: nsccalpmath This resource is the intellectual property

### 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.

BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's

### Algebra 2: Themes for the Big Final Exam

Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### MTH 100 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created June 6, 2011

MTH 00 College Algebra Essex County College Division of Mathematics Sample Review Questions Created June 6, 0 Math 00, Introductory College Mathematics, covers the mathematical content listed below. In

### The Physics of Kicking a Soccer Ball

The Physics of Kicking a Soccer Ball Shael Brown Grade 8 Table of Contents Introduction...1 What actually happens when you kick a soccer ball?...2 Who kicks harder shorter or taller people?...4 How much

### Slope and Rate of Change

Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify