College Algebra: Midterm Review

Size: px
Start display at page:

Transcription

3 Exercise 4. Solve the following quadratic equations by using any valid method of your choosing. a) (t+) 2 +t 2 9t = 8 b) 18 5 x x = 1 c) (w 5) 2 = d) x 2 +2x+5 = 0 e) (t+1) 2 = t 2 f) (x 1)(x+4) = (x+2)(x+1) Exercise 5. Solve each of the following word problems. a) The hypotenuse of a right triangle is 40 feet long. One leg of the triangle is ft longer than the other leg. Find the lengths of the legs of the triangle. b) Nolan throws a baseball straight up in the air from a cliff that is 52 feet high. The initial velocity is 72 ft sec. The height, in feet, of the object after t sec is given by h = 16t 2 +72t+52. Find the time at which the height of the object is 124 feet above the ground. Exercise 6. Multiple Choice Solve each quadratic equation by any valid method of your choosing. However, for these multiple choice problems, choose the answer choice that represents the sum of the squares of the solutions to the given equation. a) Solve 2x 2 +x 6 = 0. The sum of the squares of the solutions is. A. 1 2 B C. 1 4 D. 7 4 E. None of A. through D. is correct b) Solve y 2 8y +2 = 0. The sum of the squares of the solutions is. A. 5 B. 5 9 C. 7 9 D. 1 9 E. None of A. through D. is correct Exercise 7. Let m R. Consider the equation x 2 2mx 5 = 0. a) Which of the following represents solutions to this equation? A. x = 2m ± m C. x = m ± 2 m E. x = m ± m 2 15 B. x = m ± m D. x = m ± m F. None of A. through E. are correct b) Discuss the nature of the solutions to this equation (i.e., are the solutions real or imaginary?) c) Suppose you are told that x = 1 is a solution to this equation. i. Find m. ii. Find the other solution to the equation. Exercise 8. Let a R with a 0. Consider the equation ax 2 2x+ = 0. a) For this quadratic equation (in the variable x), find the discriminant. b) If this equation has one exactly real solution, then what can we say about a? c) What are the values of a, if any, that result in the equation having imaginary solutions? Created/Revised by Math Department 2/17/18 Page

4 Other Types Of Equations Exercise 9. Quadratic In Form a) Solve the equation (2x 2 5) 2 16(2x 2 5)+9 = 0. b) Solve the equation p 2/ +8 = 2p 1/. Exercise 10. Solve each of the cubic equations by factoring. a) Solve the equation x +9x 2 = 16(x+9). b) Solve the equation 75x +x 2 100x 4 = 0. Exercise 11. Solve each radical equation. a) Solve the equation 9x + 28 = x+4. The sum of the squares of the solutions is. A. 4 B. 25 C. 16 D. 1 E. None of A. through D. is correct b) Solve the equation 9 x x + 4 = 1. Exercise 12. Solve each absolute value equation. a) Solve the equation 2 = x b) Solve the equation 4 x = 2x+1. The sum of the squares of the solutions is. A. 17 B. 14 C. D. 16 E. None of A. through D. is correct Exercise 1. Solve each rational equation. a) Solve the equation b) Solve the equation x x x 4 = 2x2 14x x 2 2x 8. 2 (x + 2) 2 x + 2 = 5. Inequalities Exercise 14. Solve the following linear inequalities by using any valid method of your choosing. Write your answer in interval notation. a) 4(x + 2) 6 + 4(2x + 1) b) 2 5 (2x 1) > 10 c) 1 2 (x 9) 4 (x 1) 4 (x ) 2 d) 0 < x+2 19 e) 5 x f) 6+x 2x > 6+x Exercise 15. Solve the following linear inequalities by using any valid method of your choosing. Write your answer in interval notation. a) x b) 2 8 x +1 < 19 c) t+4 d) 10 < 5x+4 +2 e) x < x f) 5x+1 > 16 or 7x > 46 Created/Revised by Math Department 2/17/18 Page 4

5 Relations and Functions You should be familiar with the following terms, concepts, and/or expressions: the standard form of the equation of a circle center and radius function graph x-intercept y-intercept domain range Vertical-Line Test functional evaluation function addition, subtraction, multiplication, and division function composition piece-wise defined functions even and odd functions function transformations (reflection, translation, and scaling) Note 1. You will need to recall the technique of completing the square. Circles Exercise 16. Which of the following equations defines a circle? I. 2(x 7) 2 + (y + 5) 2 = 4 II. (x 7) 2 (y + 5) 2 = 2 2 III. (x 7) 2 + (y 5) 2 = 2 2 IV. x 2 + 2x + y 2 + 4y + 4 = 0 A. Only I. B. Only II. C. Only III. D. Only IV E. Only I and III. F. Only II and III. G. Only II and IV. H. Only III and IV. Exercise 17. Identify the the center and the radius of the circle whose equation is (x 2) 2 +(y+) 2 = 25. Exercise 18. Give the standard form of the equation of the circle that has its center at ( 5, 8) and a diameter of 12. Exercise 19. Write the standard form of the equation of the circle that has its center at (4, ) with the point ( 1, 9) on the circle. Exercise 20. Write the standard form of the equation of the circle given by x 2 6x+y 2 +8y+12 = 6. Identify the center and radius of this circle. Created/Revised by Math Department 2/17/18 Page 5

6 Functions Exercise 21. Given the function defined by f(x) = x x 2, evaluate and simplify each of the following: a) f( ) b) f(2a) where a is a real number (i.e., a R) c) f( x) d) f( + h) f( ) where h is a real number (i.e., h R) f( + h) f( ) e) where h R and h 0 h f(x + h) f( x) f) where h R and h 0 h Exercise 22. State the domain of each of functions defined by each of the following: a) p(x) = b) g(x) = x + 2 c) h(x) = x + 2 x + 2 d) f(t) = e) q(x) = x+2 f) r(x) = x t + 2 x 2 4 Exercise 2. Your grandfather has a blue spruce tree on his property that drops a lot of cones during the onset of winter. He will pay you fifteen cents (i.e., \$0.15) for each cone you pick up and bag. As a further enticement to assist with his work, he also gives you \$5 to show up to work no matter how many cones you ultimately bag. a) Write an expression that represents the total amount of money (in dollars) the you could earn as a function of the number of cones you pick up and bag. b) Identify the independent and dependent variables of your function. c) If you pick up and bag 250 cones today, how much money does Grandpa need to pay you for your work? Exercise 24. Let f(x) = x 2 x+2. If f(x) = 22, find x. Linear Equations in Two Variables and Linear Functions You should be familiar with the following terms, concepts, and expressions: midpoint slope x-intercept y-intercept quadrant slope-intercept form point-slope form horizontal line vertical line perpendicular parallel Created/Revised by Math Department 2/17/18 Page 6

7 Exercise 25. Let P = (2,) and Q = (4,6) and R = (2, 6). a) Find the distance between P and Q. b) Find the midpoint between R and Q. c) Find the slope of the line containing P and Q. d) Find the slope of the line containing P and R. e) Find the equation of the line containing P and Q. f) Find the equation of the line containing P and R. Exercise 26. Determine which of the following ordered pairs are solutions to the equation 2x+y = 8. P = (0, 4) Q = ( 4, 0) R = ( 10, 4) T = (10, 4) A. Only P. B. Only Q. C. Only R. D. Only T. E. Only P and R. F. Only Q and T. G. Only Q and R. H. Only R and T. Exercise 27. Let L be the line defined by 2x+4y = 8. a) Find the x-intercepts and y-intercepts of L. b) Find the slope of L. c) Write L in slope-intercept form. Exercise 28. Let L be the line defined by x 2 y = 1. a) Find the x-intercepts and y-intercepts of L. b) Find the slope of L. c) Write L in slope-intercept form. Exercise 29. Let P = ( 2,1+a) and Q = (,6+2a). a) Find the slope of the line L containing P and Q. b) If the L is parallel to 8x 2y = 6, find a. Exercise 0. Suppose that L 1 and L 2 are lines such that: L 1 contains P = (2, 5) and Q = (4, 9) and L 2 contains S = ( 1, 4) and T = (, 2). a) Let m 1 be the slope of L 1. Find m 1. b) Let m 2 be the slope of L 2. Find m 2. c) Without graphing the lines, determine whether L 1 and L 2 are parallel, perpendicular, or neither. Exercise 1. Which of the following represents the point-slope formula for the line with slope m that contains the point P = (x 1,y 1 )? A. y = mx + b B. y 1 y = m (x x 1 ) C. y y 1 = m (x x 1 ) D. m = y 2 y 1 x 2 x 1 Created/Revised by Math Department 2/17/18 Page 7

8 Exercise 2. Matching: Match the equation to the form of the line that it represents. You may assume that k, m, b, A, and B are constants and that A and B are not both equal to zero. Form Standard Form Horizontal Line Equation x = k y = mx + b Vertical Line y y 1 = m (x x 1 ) Slope-Intercept Form Point-Slope Formula Ax + By = C y = k Exercise. The speed of sound is approximately 1125 ft (i.e., 1100 feet per second). Thus, for every sec one-second difference between seeing lightning and hearing the associated thunder clap, we can estimate that a storm is approximately 1100 feet away. Let d represent the distance (in feet) that a storm is from you (the observer). Let t represent the difference in time between seeing lightning and hearing thunder (in seconds). a) Construct the linear model that represents the approximate distance (d) the lightning strike was is from the observer based on the time delay in hearing the hearing thunderclap (t). In this case, we know that t 0. b) Use the linear model to determine how far away a lightning strike is for the following differences in time between seeing lightning strike and hearing the associated thunder clap. i) 4 seconds, ii) 12 seconds, iii) 16 seconds. c) If the lightning strike is known to be 4.5 miles (i.e., 2, 760 feet) away, then how many seconds will pass between seeing lightning and hearing thunder? Exercise 4. The graph of the line L with equation ax+by = 1 is given below in Figure 1. Which of the following must be true? A. a > 0 and b < 0 B. a > 0 and b > 0 C. a < 0 and b < 0 D. a < 0 and b > 0 E. a = 0 and b > 0 F. None of A. through E. is true L y x Transformations of Functions and Piecewise-Defined Functions Exercise 5. The graph of f(x) = x 2 experiences the following successive transformations: (1) a reflection about the y-axis, (2) a translation 2 unit down, () a reflection about the x-axis. Identify the function that represents the resulting curve. A. g(x) = x B. g(x) = x C. g(x) = x 2 2 D. g(x) = x 2 2 E. None of A. through D. is correct Figure 1 Created/Revised by Math Department 2/17/18 Page 8

9 Exercise 6. Let g(x) = 2 f(x )+5. Which of the following successive transformations is a description of g? A. The graph of f shifted units to the left, reflected about the y-axis, stretched vertically by a factor of 2, and then shifted up 5 units. B. The graph of f shifted units to the right, reflected about the y-axis, stretched vertically by a factor of 2, and then shifted up 5 units. C. The graph of f shifted units to the left, reflected about the x-axis, stretched vertically by a factor of 2, and then shifted up 5 units. E. None of A. through D. is correct D. The graph of f shifted units to the right, reflected about the x-axis, stretched vertically by a factor of 2, and then shifted up 5 units. Exercise 7. Use the concepts of transformations to sketch the graphs of a) h(x) = (x+2) 2 b) g(x) = x + c) m(x) = 1 x 2 Exercise 8. Suppose that h is an odd function and h( 2) =, h(6) = 11, and h(1) = 5. Find h(2). x 7, for x 4 x Exercise 9. Let f(x) = 2 +, for 4 < x < 1. Evaluate each of the following: 7x + 2, for 1 < x 5 2 x 1, for x > 5 a) f(1) b)f( 6) c)f(4) d)f(7) e)f(2) f)f(a) where 2 < a.25 The Algebra of Functions Exercise 40. For f(x) = x + 6 and g(x) = 2x 2 7, find: a) (f + g)(10) b)f() g() c)(fg)( 2) d)( f g )( 5) e)( g f )( 5) f)( f g )(x) Exercise 41. For f(x) = x 2 x+2 and g(x) = 6x 7, find: a) f(g(1)) b)(f g)(0) c)(g f)(0) d)(g f)(1) e)(f g)(x) f)(g f)(0) Exercise 42. For f(x) = x 2 and g(x) = 2x 2 x 2, state the domain of h(x) = f(x) g(x). Write your answer using interval notation. Exercise 4. For f(x) = 1 x 2 and g(x) = 2 x, state the domain of h(x) = (f g)(x). Write your answer using interval notation. Created/Revised by Math Department 2/17/18 Page 9

10 Quadratic Functions Exercise 44. Find the vertex of f(x) = 6(x 4) Exercise 45. For f(x) = x 2 +6x+5, answer each of the following. a) Determine the y-intercept. b) Determine the x-intercept(s), if any. c) Identify the vertex. d) Determine the axis of symmetry. e) Write the range of the function in interval notation. f) Sketch the graph of the function. Exercise 46. The axis of symmetry of the graph of f(x) = ax 2 +8x+9 is x = 2. Find a. Exercise 47. If f(x) = 8x 2 +4x 1 is expressed in the form f(x) = a(x h) 2 +k where a, h, and k are constants (with a 0), what is the value of k? Exercise 48. Suppose that f is a quadratic function with axis of symmetry given by x = 5. If f(1) = 9, f(5) = 1, f(7) =, and f(9) = 9, then answer the following questions. a) Find the vertex of the parabola. b) Is the parabola open up or open down? c) Are there any x-intercepts of f? d) What is f()? e) (Bonus) Find constants a, b, and c such that f(x) = ax 2 + bx + c. Exercise 49. Find the points of intersection between the line L given by y x = 5 and the parabola given by y = x 2 2x 5. (See Figure 2.) y y = x 2 2x 5 y x = 5 x Figure 2 Created/Revised by Math Department 2/17/18 Page 10

Math ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying

Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the

A Partial List of Topics: Math Spring 2009

A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose

5.4 - Quadratic Functions

Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 92 5.4 - Quadratic Functions Definition: A function is one that can be written in the form f (x) = where a, b, and c are real numbers and a 0. (What

Important Math 125 Definitions/Formulas/Properties

Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition Students have the options of either purchasing the loose-leaf

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)

1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of

UMUC MATH-107 Final Exam Information

UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from

MATH 1113 Exam 1 Review

MATH 1113 Exam 1 Review Topics Covered Section 1.1: Rectangular Coordinate System Section 1.3: Functions and Relations Section 1.4: Linear Equations in Two Variables and Linear Functions Section 1.5: Applications

Chapter 2 Polynomial and Rational Functions

SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

SOLUTIONS FOR PROBLEMS 1-30

. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

MAC College Algebra

MAC 05 - College Algebra Name Review for Test 2 - Chapter 2 Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact distance between the

# %

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

INSTRUCTIONS USEFUL FORMULAS

MATH 1100 College Algebra Spring 18 Exam 1 February 15, 2018 Name Student ID Instructor Class time INSTRUCTIONS 1. Do not open until you are told to do so. 2. Do not ask questions during the exam. 3. CAREFULLY

Using the Laws of Exponents to Simplify Rational Exponents

6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify

North Carolina State University

North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical

Test 2 Review Math 1111 College Algebra

Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.

3 Inequalities Absolute Values Inequalities and Intervals... 18

Contents 1 Real Numbers, Exponents, and Radicals 1.1 Rationalizing the Denominator................................... 1. Factoring Polynomials........................................ 1. Algebraic and Fractional

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.

Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope

Please print the following information in case your scan sheet is misplaced:

MATH 1100 Common Final Exam FALL 010 December 10, 010 Please print the following information in case your scan sheet is misplaced: Name: Instructor: Student ID: Section/Time: The exam consists of 40 multiple

Instructor Notes for Chapters 3 & 4

Algebra for Calculus Fall 0 Section 3. Complex Numbers Goal for students: Instructor Notes for Chapters 3 & 4 perform computations involving complex numbers You might want to review the quadratic formula

Section 0.2 & 0.3 Worksheet. Types of Functions

MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

Summer Work for students entering PreCalculus

Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate

2. If the discriminant of a quadratic equation is zero, then there (A) are 2 imaginary roots (B) is 1 rational root

Academic Algebra II 1 st Semester Exam Mr. Pleacher Name I. Multiple Choice 1. Which is the solution of x 1 3x + 7? (A) x -4 (B) x 4 (C) x -4 (D) x 4. If the discriminant of a quadratic equation is zero,

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).

MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and

Pre-Calculus Summer Math Packet 2018 Multiple Choice

Pre-Calculus Summer Math Packet 208 Multiple Choice Page A Complete all work on separate loose-leaf or graph paper. Solve problems without using a calculator. Write the answers to multiple choice questions

Advanced Algebra II 1 st Semester Exam Review

dname Advanced Algebra II 1 st Semester Exam Review Chapter 1A: Number Sets & Solving Equations Name the sets of numbers to which each number belongs. 1. 34 2. 525 3. 0.875 4. Solve each equation. Check

Summer Work for students entering PreCalculus

Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate

(MATH 1203, 1204, 1204R)

College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:

Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

Chapter 3A -- Rectangular Coordinate System

Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page61 Chapter 3A -- Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3,

MATH College Algebra Review for Test 2

MATH 34 - College Algebra Review for Test 2 Sections 3. and 3.2. For f (x) = x 2 + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

Mock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}

Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the

Final Exam Review for DMAT 0310

Final Exam Review for DMAT 010 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor the polynomial completely. What is one of the factors? 1) x

Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities

Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,

Math Precalculus I University of Hawai i at Mānoa Spring

Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents

Chapter 8B - Trigonometric Functions (the first part)

Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of

Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website.

BTW Math Packet Advanced Math Name Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. Go to the BTW website

College Algebra Notes

Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

Pre Calculus with Mrs. Bluell

Welcome to Pre Calculus with Mrs. Bluell Quick Review Today's Topics include Interval Notation Exponent Rules Quadrants Distance Formula Midpoint Formula Circle Formula Alligator Mouths to Interval Notation

MATH College Algebra Review for Test 2

MATH 4 - College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of

Precalculus 1, 161. Fall 2018 CRN Section 010. Time: Saturday, 9:00 a.m. 12:05 p.m. Room BR-11

Precalculus 1, 161 Fall 018 CRN 4066 Section 010 Time: Saturday, 9:00 a.m. 1:05 p.m. Room BR-11 SYLLABUS Catalog description Functions and relations and their graphs, transformations and symmetries; composition

Math 120, Sample Final Fall 2015

Math 10, Sample Final Fall 015 Disclaimer: This sample final is intended to help students prepare for the final exam The final exam will be similar in structure and type of problems, however the actual

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions

MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms

Foundations of Math II Unit 5: Solving Equations

Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following

A-Level Notes CORE 1

A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is

8th Grade Math Definitions

8th Grade Math Definitions Absolute Value: 1. A number s distance from zero. 2. For any x, is defined as follows: x = x, if x < 0; x, if x 0. Acute Angle: An angle whose measure is greater than 0 and less

1. OBJECTIVE: Linear Equations

CUNY YORK COLLEGE FINAL EXAM REVIEW MATH 120: Precalculus Use the following questions to review for your final examimation for Math 120. Your ability to answer these questions will reflect what you learned

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections 3.1, 3.3, and 3.5

Department of Mathematics, University of Wisconsin-Madison Math 11 Worksheet Sections 3.1, 3.3, and 3.5 1. For f(x) = 5x + (a) Determine the slope and the y-intercept. f(x) = 5x + is of the form y = mx

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers. Example: Let A = {4, 8, 12, 16, 20,...} and let B = {6, 12, 18, 24, 30,...

Fry Texas A&M University!! Math 150! Spring 2015 Unit 1! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {4, 8, 12, 16, 20,...} and let B = {6, 12, 18, 24, 30,...} Then A B= and

Chapter Five Notes N P U2C5

Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have

Algebra 2 CP Semester 1 PRACTICE Exam

Algebra 2 CP Semester 1 PRACTICE Exam NAME DATE HR You may use a calculator. Please show all work directly on this test. You may write on the test. GOOD LUCK! THIS IS JUST PRACTICE GIVE YOURSELF 45 MINUTES

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)

Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,

Math 3C Midterm 1 Study Guide

Math 3C Midterm 1 Study Guide October 23, 2014 Acknowledgement I want to say thanks to Mark Kempton for letting me update this study guide for my class. General Information: The test will be held Thursday,

MHCA Math Summer Packet 2015

Directions: MHCA Math Summer Packet 2015 For students entering PreCalculus Honors You are to complete all the problems assigned in this packet by Friday, September 4 th. If you don t turn in your summer

COWLEY COLLEGE & Area Vocational Technical School

COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR COLLEGE ALGEBRA WITH REVIEW MTH 4421 5 Credit Hours Student Level: This course is open to students on the college level in the freshman

Math Precalculus I University of Hawai i at Mānoa Spring

Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2014 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.

Midterm Review (Honors Algebra 2) 4. Solve the compound inequality. Then graph its solution on a number line. 5 7 or 3x x

Midterm Review (Honors Algebra ) Name Chapter & : Equations Inequalities, and Linear Functions. The graph of function f(x) is shown at right. Find f(3).. Evaluate f( ) when f ( x) x 5x. 3. Solve the inequality

4 The Cartesian Coordinate System- Pictures of Equations

4 The Cartesian Coordinate System- Pictures of Equations Concepts: The Cartesian Coordinate System Graphs of Equations in Two Variables x-intercepts and y-intercepts Distance in Two Dimensions and the

MAT 107 College Algebra Fall 2013 Name. Final Exam, Version X

MAT 107 College Algebra Fall 013 Name Final Exam, Version X EKU ID Instructor Part 1: No calculators are allowed on this section. Show all work on your paper. Circle your answer. Each question is worth

Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2

Name: Math 114 Activity 1(Due by EOC Apr. 17) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please

MATH 1130 Exam 1 Review Sheet

MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This

UNLV University of Nevada, Las Vegas

UNLV University of Nevada, Las Vegas The Department of Mathematical Sciences Information Regarding Math 14 Final Exam Revised.8.018 While all material covered in the syllabus is essential for success in

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal

Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make

Algebra 1. Unit 3: Quadratic Functions. Romeo High School

Algebra 1 Unit 3: Quadratic Functions Romeo High School Contributors: Jennifer Boggio Jennifer Burnham Jim Cali Danielle Hart Robert Leitzel Kelly McNamara Mary Tarnowski Josh Tebeau RHS Mathematics Department

S4 (4.3) Quadratic Functions.notebook February 06, 2018

Daily Practice 2.11.2017 Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 6x + 4

Math1420 Review Comprehesive Final Assessment Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Add or subtract as indicated. x + 5 1) x2

College Algebra Through Problem Solving (2018 Edition)

City University of New York (CUNY) CUNY Academic Works Open Educational Resources Queensborough Community College Winter 1-25-2018 College Algebra Through Problem Solving (2018 Edition) Danielle Cifone

Section 3.2 Quadratic Functions & Their Graphs

Week 2 Handout MAC 1140 Professor Niraj Wagh J Section 3.2 Quadratic Functions & Their Graphs Quadratic Function: Standard Form A quadratic function is a function that can be written in the form: f (x)

Functions and Equations

Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN

Unit 1 Quadratic Functions Lecture Notes Introductory Algebra Page 1 of 8 1 Quadratic Functions In this unit we will learn many of the algebraic techniques used to work with the quadratic function fx)

Geometry 263 Prerequisites

Name Geometry 6 Prerequisites Dear Incoming Geometry Student, Listed below are fifteen skills that we will use throughout Geometry 6. You have likely learned and practiced these skills in previous math

PRACTICE FINAL , FALL What will NOT be on the final

PRACTICE FINAL - 1010-004, FALL 2013 If you are completing this practice final for bonus points, please use separate sheets of paper to do your work and circle your answers. Turn in all work you did to

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 2 nd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

Graphs of Polynomial Functions

Graphs of Polynomial Functions By: OpenStaxCollege The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link]. Year 2006 2007 2008 2009 2010 2011 2012 2013

Are you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*

Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course

Name I.D. Number. Select the response that best completes the statement or answers the question.

Name I.D. Number Unit 4 Evaluation Evaluation 04 Second Year Algebra 1 (MTHH 039 059) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus,

COWLEY COLLEGE & Area Vocational Technical School

COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR Student Level: This course is open to students on the college level in their freshman year. Catalog Description: MTH4410 - INTERMEDIATE

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know.

REVIEW EXAMPLES 1) Solve 9x + 16 = 0 for x. 9x + 16 = 0 9x = 16 Original equation. Subtract 16 from both sides. 16 x 9 Divide both sides by 9. 16 x Take the square root of both sides. 9 4 x i 3 Evaluate.

The ACCUPLACER (Elementary Algebra) is a 12 question placement exam. Its purpose is to make sure you are put in the appropriate math course.

About the ACCUPLACER Test The ACCUPLACER (Elementary Algebra) is a 12 question placement exam. Its purpose is to make sure you are put in the appropriate math course. A student whose score is 67 or higher

Summer Packet A Math Refresher For Students Entering IB Mathematics SL

Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school

Example: f(x) = 2x² + 1 Solution: Math 2 VM Part 5 Quadratic Functions April 25, 2017

Math 2 Variable Manipulation Part 5 Quadratic Functions MATH 1 REVIEW THE CONCEPT OF FUNCTIONS The concept of a function is both a different way of thinking about equations and a different way of notating

Advanced Algebra Scope and Sequence First Semester. Second Semester

Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs

Note: The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Sep 15 2:51 PM

2.1 Linear and Quadratic Name: Functions and Modeling Objective: Students will be able to recognize and graph linear and quadratic functions, and use these functions to model situations and solve problems.

Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.

Semester Review Packet

MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

Preliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics MATHS 101: Calculus I

Preliminaries 2 1 2 Lectures Department of Mathematics http://www.abdullaeid.net/maths101 MATHS 101: Calculus I (University of Bahrain) Prelim 1 / 35 Pre Calculus MATHS 101: Calculus MATHS 101 is all about

COWLEY COLLEGE & Area Vocational Technical School

COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR Student Level: This course is open to students on the college level in their freshman or sophomore year. Catalog Description: INTERMEDIATE

Instructional Materials for the WCSD Math Common Finals

201-2014 Algebra 2 Semester 1 Instructional Materials for the WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Math Common Final blueprint for

Maintaining Mathematical Proficiency

Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area

CURRICULUM GUIDE. Honors Algebra II / Trigonometry

CURRICULUM GUIDE Honors Algebra II / Trigonometry The Honors course is fast-paced, incorporating the topics of Algebra II/ Trigonometry plus some topics of the pre-calculus course. More emphasis is placed

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II. 1 st Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource