# INVESTIGATIONS AND FUNCTIONS Example 1

Size: px
Start display at page:

Transcription

1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 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. Students are also introduced to their graphing calculators. The also learn the appropriate use of the graphing tool, so that time is not wasted using it when a problem can be solved more efficientl b hand. Not onl are students working on challenging, interesting problems, the are also reviewing topics from earlier math courses such as graphing, trigonometric ratios, and solving equations. The also practice algebraic manipulations b inputing values into function machines and calculating each output. For further information see the Math Notes boes in Lessons 1.1.1, 1.1.2, and Eample 1 Talula s function machine at right shows its inner workings, written in function notation. Note that = 10 2 is an equivalent form. What will the output be if: a. 2 is dropped in? b. 2 is dropped in? c. 10 is dropped in? d is dropped in? Solution: The number dropped in, that is, substituted for, takes the place of in the equation in the machine. Follow the Order of Operations to simplif the epression to determine the value of f(). a. f (2) = 10 (2) 2 = 10 4 = 6 c. f ( 10) = 10 ( 10) 2 = = 0 b. f ( 2) = 10 ( 2) 2 = 10 4 = 6 d. f ( 3.45) = 10 ( 3.45) 2 = = Parent Guide with Etra Practice 1

2 Eample 2 Consider the functions f () = 3 and g() = ( + 5 )2. a. What is f (4)? b. What is g(4)? c. What is the domain of f ()? d. What is the domain of g()? e. What is the range of f ()? f. What is the range of g()? Solution: Substitute the values of in the functions for parts (a) and (b): f (4) = = 2 1 = 2 g(4) = ( 4 + 5) 2 = (9) 2 = 81 The domain of f () is the set of -values that are allowable, and this function has some restrictions. First, we cannot take the square root of a negative number, so cannot be less than zero. Additionall, the denominator of a fraction cannot be zero, so 3. Therefore, the domain of f () is 0, 3. For g(), we could substitute an number for, add five, and then square the result. This function has no restrictions so the domain of g() is all real numbers. The range of these functions is the set of all possible values that result when substituting the domain, or -values. We need to decide if there are an values that the functions could never reach, or are impossible to produce. Consider the range of g() first. Since the function g squares the amount in the final step, the output will alwas be positive. It could equal zero (when = 5), but it will never be negative. Therefore, the range of g() is 0. The range of f () is more complicated. Tr finding some possibilities first. Can this function ever equal zero? Yes, when = 0, then f () = 0. Can the function ever equal a ver large positive number? Yes, this happens when < 3, but ver close to 3. (For eample, let = , then f () is approimatel equal to 17,320.) Can f () become an etremel negative number? Yes, when > 3, but ver close to 3. (Here, tr = , and then f () is approimatel equal to 17,320.) There does not seem to be an restrictions on the range of f (), therefore we can sa that the range is all real numbers. 2 Core Connections Algebra 2

3 Chapter 1 Eample 3 For each problem below, first decide how ou will answer the question: b using a graphing tool, our algebra skills, or a combination of the two. Use the most efficient method. Show our work, including a justification of the method ou chose for each problem. a. What is the -intercept of the graph of = ? b. Does the graph of = cross the -ais? If so, how man times? c. Where do the graphs of = and = intersect? 5 d. What are the domain and range of = ? Solutions: Part (a): The -intercept is the point (0, b). Therefore, the -intercept can be found b substituting 0 for. In this case, the -intercept can be found b calculating = 2 3 (0)+19 and hence the -intercept is the point (0, 19). Part (b): The most efficient method is to use a graphing calculator to see if the graph does or does not cross the -ais. The graph at right shows the complete graph, meaning we see everthing that is important about the graph, and the rest of the graph is predictable based on what we see. This graph shows us that the graph intersects the -ais, twice. Part (c): It is best to use algebra and solve this sstem of two equations with two unknowns to find where the graphs intersect. This can be done b using the Equal Values method = = (multipl all terms b 5) 26 = 130 (add and 100 to both sides) = 5 (divide both sides b 26) = 5(5)+ 20 = 45 (substitue 5 for in the first equation) Therefore the graphs intersect at the point (5, 45). 5 5 Part (d): We need to find the acceptable values for the input, and the possible outputs. The equation offers no restrictions, like dividing b zero, or taking the square root of a negative number, so the domain is all real numbers. Since this equation is a parabola when graphed, there will be restrictions on the range. Just as the equation = 2 can never be negative, the ranges of quadratic functions (parabolas) will have a lowest point, or a highest point. If we graph this parabola, we can see that the lowest point, the verte, is at (6, 10). The graph onl has values of 10, therefore the range is Parent Guide with Etra Practice 3

4 Most of the earl homework assignments review skills developed in Algebra 1 and geometr. The problems below include these tpes of problems. Problems Solve the following equations for and/or. 1. 5( + 7) = = 12 = = b( a) = c Find the error and show the correct solution = 2( 3) 5 9 = = 14 = = = 3 2( + 1) = 3 2 = 3 or + 1 = 3 = 3 2 or = 2 Sketch a complete graph of each of the following equations. Be sure to label the graph carefull so that all ke points are identified. What are the domain and range of each function? 7. = = 3 6 If f () = 3 2 6, find: 9. f (1) 10. f ( 3) 11. f (2.75) 4 Core Connections Algebra 2

5 Chapter 1 Use the graph at right to complete the following problems. 12. Briefl, what does the graph represent? 13. Based on the graph, give all the information ou can about the Ponda Concord. 14. Based on the graph, give all the information ou can about the Neo Brism. 15. Does it make sense to etend these lines into the second and fourth quadrants? Eplain. Gas in tank (gallons) Neo Brism Ponda Concord Distance traveled (miles) Answers 1. = = 2, = 6 3. = 7, 3 4. = c+ab = c b b + a 5. When distributing, ( 2)( 3) = 6. = Must set the equation equal to zero before factoring. = 1, The graph shows how much gas is in the tank of a Ponda Concord or a Neo Brism as the car is driven. 13. Ponda Concord s gas tank holds 16 gallons of gas, and the car has a driving range of about 350 miles on one tank of gas. It gets 22 miles per gallon. 14. The Neo Brism s gas tank holds onl 10 gallons of gas, but it has a driving range of 400 miles on a tank of gas. It gets 40 miles per gallon. 15. No, the car doesn t travel negative miles, nor can the gas tank hold negative gallons. Parent Guide with Etra Practice 5

8 Eample continued from previous page. The output will be the area of the triangle. To find the area, we need to know the length of the base of the triangle and for that we use the Pthagorean Theorem. Now that we know the length of the base, we can find the area of this triangle. A = 1 2 bh 1 2 (29.93)(20) A square inches b 2 = b 2 = 1296 b 2 = 896 b = This gives us one input and its corresponding output. Tr this again using an input (height) of 10 inches. You should get a corresponding output (area) of approimatel square inches. Each student needs to do as man eamples as s/he needs in order to understand the general case. In the general case, we let the height (input) be, solve for b, and then find the area of the triangle, all in terms of. b = 36 2 b = 1296 b 2 = b = A = 1 2 bh ( ) () A = in. b in. Now we have an equation to aid us with investigating the problem further. Note that this is just an aid. We will answer man of the questions b knowing the contet of the problem. To begin with, this is a function because ever input has one and onl one output. Net, we can determine the domain. In this situation, the acceptable inputs are the various heights along the wall that the ardstick can lean. The height could never be smaller than zero, and it could never be larger than 36 inches (because the ardstick will onl reach that high when it is flat against the wall, and no triangle is reall visible). Therefore the domain is 0 < < 36. Given this domain, the range is the values that the equation can take on within this restriction on. B looking at the graph of the equation representing this situation, we are looking for the -values this graph takes on. The area of the triangle can be zero (or ver close to zero) and has a maimum value. B using the zoom and trace buttons on our graphing calculator, ou can find the maimum value to be 324. (Note: At this point, we epect students to find the maimum value this wa. It will be some time before the can find the maimum value of a function algebraicall.) Therefore the range is 0 < < If we allow the triangle to have sides with length 0 inches, then the function has two -intercepts at (0, 0) and (36, 0), and the 300 one -intercept (0, 0). This function is also continuous (no breaks), has no asmptotes, and is not linear. At this point, we 200 do not know what tpe of function it is Core Connections Algebra 2 100

9 Chapter 1 Problems Solve the following equations for and/or = = 10 = = = 0 If g() = , find: 5. g( 2) 6. g(0.4) 7. g(18) Sketch a complete graph of each of the following equations. Be sure to label the graph carefull so that all ke points are identified. What are the domain and range of each function? 8. = = Investigate the function = Investigate the function = 1 6. Parent Guide with Etra Practice 9

10 Answers = 22, = = 1 4, =± This is a line with a slope of intercept: (300, 0) -intercept: (0, 30) This is a parabola opening upward. -intercepts: ( 40.88, 0) and (10.88, 0) -intercept: (0, 445) verte: ( 15, 670) Line of smmetr: = The graph of this function is curved. There are no - or -intercepts, it is a function, the domain is 6, the range is 1. The starting point is (6, 1). 11. This graph is a curve and it has two unconnected parts. It has no -intercepts, and the -intercept is ( 0, 1 6 ). The domain is all real values of ecept 6, and the range is all real values of ecept 0. The line = 0 is a horizontal asmptote, and = 6 is a vertical asmptote. This is a function. 10 Core Connections Algebra 2

11 Chapter 1 SAT PREP These problems are ver similar to actual SAT test questions. Use a calculator whenever ou need one. On multiple choice questions, choose the best answer from the ones provided. When a picture has a diagram, assume the diagram is drawn accuratel ecept when a problem sas it is not. These questions address more topics than ou have done in class so far. 1. If + 9 is an even integer, then which of the following could be the value of? a. 4 b. 2 c. 0 d. 1 e If (m + 5)(11 7) = 24, then m =? a. 1 b. 4 c. 8 d. 11 e The fractions 3 d, 4 d, and 5 d be the value of d? are in simplest reduced form. Which of the following could a. 20 b. 21 c. 22 d. 23 e A group of three numbers is called a j-triple for some number j, if ( 4 j, j, 5 4 j ). Which of the following is a j-triple? a. (0, 4, 5) b. ( 5 3 4,6,61 4 ) c. (9, 12, 15) d. (750, 1000, 1250) e. (575, 600, 625) 5. A ball is thrown straight up. The height of the ball can be modeled with the equation h = 38t 16t 2 where h is the height in feet and t is the number of second since the ball was thrown. How high is the ball two seconds after it is thrown? a. 12 b. 16 c. 22 d. 32 e In the figure at right, AC is a line segment with a length of 4 units. What is the value of k? 5k 3k A B C 7. Let the operation be defined as a b is the sum of all integers between a and b. For eample, 4 10 = = 35. What is the value of ( ) ( )? Parent Guide with Etra Practice 11

12 8. An isosceles triangle has a base of length 15. The length of each the other two equal sides is an integer. What is the shortest possible length of these other two sides? 9. Assume that 1 4 quart of cranberr concentrate is mied with 1 3 quarts of apple juice to 4 make cranapple juice for four people. How man quarts of cranberr concentrate are needed to make a cranapple drink at the same strength for 15 people? 10. A stack of five cards is labeled with a different integer ranging from 0 to 4. If two cards are selected at random without replacement, what is the probabilit that the sum will be 2? Answers 1. D 2. A 3. D 4. D 5. A Core Connections Algebra 2

### 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

### MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

### Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

### When I was 3.1 POLYNOMIAL FUNCTIONS

146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

### 1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

### 135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

### Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

### Graphing Linear Equations

6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

### C3: Functions. Learning objectives

CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

### 1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

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

### 5.2 Inverse Functions

78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

### SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

### Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### I think that starting

. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

### SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### 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

### Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

### Mathematical goals. Starting points. Materials required. Time needed

Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between

### Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### More Equations and Inequalities

Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

### Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial 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

### D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

### Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

### Start Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES

COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided

### THE PARABOLA 13.2. section

698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

### LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

### 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..

### 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

### 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

### 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

### LINEAR FUNCTIONS OF 2 VARIABLES

CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

### Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

### y intercept Gradient Facts Lines that have the same gradient are PARALLEL

CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

### DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

### 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

### Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

### 5.3 Graphing Cubic Functions

Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

### Linear Inequality in Two Variables

90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.

### Pre Calculus Math 40S: Explained!

Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph

### 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

### Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

### 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

### 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) -

### 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

### 9.5 CALCULUS AND POLAR COORDINATES

smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet

### Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.

CST/CAHSEE: Warm-Up Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

### COMPONENTS OF VECTORS

COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two

### 5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED

CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given

### Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter

### Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY

Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### Linear Equations in Two Variables

Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m.

INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesda, Januar 4, 01 9:15 a.m. to 1:15 p.m., onl Student Name: School Name: Print our name and

### 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) (

### Core Maths C3. Revision Notes

Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

### Implicit Differentiation

Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

### Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

### 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

### 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

### EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### 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

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract

### 2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

### sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj

Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.

Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### FACTORING QUADRATICS 8.1.1 through 8.1.4

Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten

### Section 5-9 Inverse Trigonometric Functions

46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

### 2.1 Three Dimensional Curves and Surfaces

. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

### SECTION 7-4 Algebraic Vectors

7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors

### Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

### Affine Transformations

A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates

### 2.3 TRANSFORMATIONS OF GRAPHS

78 Chapter Functions 7. Overtime Pa A carpenter earns \$0 per hour when he works 0 hours or fewer per week, and time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he

### Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

### SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT

. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail

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:

### PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers

### 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

### 1 Maximizing pro ts when marginal costs are increasing

BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market

### Systems of Linear Equations: Solving by Substitution

8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

### Math 152, Intermediate Algebra Practice Problems #1

Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work