JEFFERSON MATH PROJECT REGENTS BY PERFORMANCE INDICATOR: TOPIC

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1 JEFFERSON MATH PROJECT REGENTS BY PERFORMANCE INDICATOR: TOPIC NY Algebra /Trigonometry Regents Exam Questions from Fall 009 to June 0 Sorted by PI: Topic Dear Sir I have to acknolege the reciept of your favor of May. in which you mention that you have finished the 6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a further pursuit of that branch of science would be useful to you. there are some propositions in the latter books of Euclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted with them. trigonometry, so far as this, is most valuable to every man, there is scarcely a day in which he will not resort to it for some of the purposes of common life. the science of calculation also is indispensible as far as the extraction of the square & cube roots; Algebra as far as the quadratic equation & the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be indulged in by one who is to have a profession to follow for his subsistence. in this light I view the conic sections, curves of the higher orders, perhaps even spherical trigonometry, Algebraical operations beyond the d dimension, and fluxions. Letter from Thomas Jefferson to William G. Munford, Monticello, June 8, 799.

2 TABLE OF CONTENTS TOPIC PI: SUBTOPIC QUESTION NUMBER GRAPHS AND STATISTICS A.S.-: Analysis of Data A.S.: Average Known with Missing Data A.S.: Dispersion A.S.6-7: Regression A.S.8: Correlation Coefficient A.S.5: Normal Distributions PROBABILITY A.S.0: Permutations A.S.: Combinations A.S.9: Differentiating Permutations and Combinations... - A.S.: Sample Space... A.S.: Geometric Probability... 5 A.S.5: Binomial Probability ABSOLUTE VALUE QUADRATICS A.A.: Absolute Value Equations and Equalities A.A.0-: Roots of Quadratics A.A.7: Factoring Polynomials A.A.7: Factoring the Difference of Perfect Squares A.A.7: Factoring by Grouping A.A.5: Quadratic Formula A.A.: Using the Discriminant A.A.: Completing the Square A.A.: Quadratic Inequalities SYSTEMS A.A.: Quadratic-Linear Systems POWERS RADICALS A.N.: Operations with Polynomials A.N., A.8-9: Negative and Fractional Exponents A.A.: Evaluating Exponential Expressions A.A.8: Evaluating Logarithmic Expressions A.A.5: Graphing Exponential Functions A.A.5: Graphing Logarithmic Functions A.A.9: Properties of Logarithms A.A.8: Logarithmic Equations A.A.6, 7: Exponential Equations A.A.6: Binomial Expansions A.A.6, 50: Solving Polynomial Equations A.A.: Simplifying Radicals A.N., A.: Operations with Radicals A.N.5, A.5: Rationalizing Denominators A.A.: Solving Radicals A.A.0-: Exponents as Radicals A.N.6: Square Roots of Negative Numbers... 6 A.N.7: Imaginary Numbers A.N.8: Conjugates of Complex Numbers A.N.9: Multiplication and Division of Complex Numbers... -i-

3 RATIONALS A.N.9: Multiplication and Division of Rationals A.A.: Solving Rationals A.A.7: Complex Fractions A.A.5: Inverse Variation FUNCTIONS A.A.0-: Functional Notation A.A.5: Families of Functions A.A.6: Properties of Graphs of Functions and Relations A.A.5: Identifying the Equation of a Graph A.A.8, : Defining Functions A.A.9, 5: Domain and Range A.A.: Compositions of Functions A.A.: Inverse of Functions A.A.6: Transformations with Functions and Relations SEQUENCES AND SERIES TRIGONOMETRY A.A.9-: Sequences A.N.0, A.: Sigma Notation A.A.5: Series A.A.55: Trigonometric Ratios A.M.-: Radian Measure A.A.60: Unit Circle... - A.A.6, 66: Determining Trigonometric Functions A.A.6: Using Inverse Trigonometric Functions A.A.57: Reference Angles... A.A.6: Arc Length... - A.A.58-59: Cofunction/Reciprocal Trigonometric Functions -7 A.A.67: Proving Trigonometric Identities A.A.76: Angle Sum and Difference Identities A.A.77: Double and Half Angle Identities A.A.68: Trigonometric Equations A.A.69: Properties of Trigonometric Functions... - A.A.7: Identifying the Equation of a Trigonometric Graph - A.A.65, 70-7: Graphing Trigonometric Functions A.A.6: Domain and Range A.A.7: Using Trigonometry to Find Area A.A.7: Law of Sines A.A.75: Law of Sines - The Ambiguous Case A.A.7: Law of Cosines A.A.7: Vectors CONICS A.A.7, 9: Equations of Circles ii-

4 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic GRAPHS AND STATISTICS A.S.-: ANALYSIS OF DATA Which task is not a component of an observational study? The researcher decides who will make up the sample. The researcher analyzes the data received from the sample. The researcher gathers data from the sample, using surveys or taking measurements. The researcher divides the sample into two groups, with one group acting as a control group. A doctor wants to test the effectiveness of a new drug on her patients. She separates her sample of patients into two groups and administers the drug to only one of these groups. She then compares the results. Which type of study best describes this situation? census survey observation controlled experiment Howard collected fish eggs from a pond behind his house so he could determine whether sunlight had an effect on how many of the eggs hatched. After he collected the eggs, he divided them into two tanks. He put both tanks outside near the pond, and he covered one of the tanks with a box to block out all sunlight. State whether Howard's investigation was an example of a controlled experiment, an observation, or a survey. Justify your response. A survey completed at a large university asked,000 students to estimate the average number of hours they spend studying each week. Every tenth student entering the library was surveyed. The data showed that the mean number of hours that students spend studying was 5.7 per week. Which characteristic of the survey could create a bias in the results? the size of the sample the size of the population the method of analyzing the data the method of choosing the students who were surveyed 5 The yearbook staff has designed a survey to learn student opinions on how the yearbook could be improved for this year. If they want to distribute this survey to 00 students and obtain the most reliable data, they should survey every third student sent to the office every third student to enter the library every third student to enter the gym for the basketball game every third student arriving at school in the morning A.S.: AVERAGE KNOWN WITH MISSING DATA 6 The number of minutes students took to complete a quiz is summarized in the table below. If the mean number of minutes was 7, which equation could be used to calculate the value of x? 7 = 9 + x x 9 + 6x 7 = x 7 = 6 + x 6 + x 6 + 6x 7 = 6 + x

5 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 The table below displays the results of a survey regarding the number of pets each student in a class has. The average number of pets per student in this class is. 9 The scores of one class on the Unit mathematics test are shown in the table below. What is the value of k for this table? 9 8 A.S.: DISPERSION 8 The table below shows the first-quarter averages for Mr. Harper s statistics class. Find the population standard deviation of these scores, to the nearest tenth. 0 During a particular month, a local company surveyed all its employees to determine their travel times to work, in minutes. The data for all 5 employees are shown below Determine the number of employees whose travel time is within one standard deviation of the mean. What is the population variance for this set of data?

6 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.S.6-7: REGRESSION Samantha constructs the scatter plot below from a set of data. A population of single-celled organisms was grown in a Petri dish over a period of 6 hours. The number of organisms at a given time is recorded in the table below. Based on her scatter plot, which regression model would be most appropriate? exponential linear logarithmic power A cup of soup is left on a countertop to cool. The table below gives the temperatures, in degrees Fahrenheit, of the soup recorded over a 0-minute period. Determine the exponential regression equation model for these data, rounding all values to the nearest ten-thousandth. Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 8th hour. Write an exponential regression equation for the data, rounding all values to the nearest thousandth.

7 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The table below shows the number of new stores in a coffee shop chain that opened during the years 986 through Which calculator output shows the strongest linear relationship between x and y? Using x = to represent the year 986 and y to represent the number of new stores, write the exponential regression equation for these data. Round all values to the nearest thousandth. 5 The table below shows the results of an experiment involving the growth of bacteria. Write a power regression equation for this set of data, rounding all values to three decimal places. Using this equation, predict the bacteria s growth, to the nearest integer, after 5 minutes. A.S.8: CORRELATION COEFFICIENT 6 Which value of r represents data with a strong negative linear correlation between two variables?

8 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 As shown in the table below, a person s target heart rate during exercise changes as the person gets older. An amateur bowler calculated his bowling average for the season. If the data are normally distributed, about how many of his 50 games were within one standard deviation of the mean? 7 8 Assume that the ages of first-year college students are normally distributed with a mean of 9 years and standard deviation of year. To the nearest integer, find the percentage of first-year college students who are between the ages of 8 years and 0 years, inclusive. To the nearest integer, find the percentage of first-year college students who are 0 years old or older. Which value represents the linear correlation coefficient, rounded to the nearest thousandth, between a person s age, in years, and that person s target heart rate, in beats per minute? A.S.5: NORMAL DISTRIBUTIONS 9 The lengths of 00 pipes have a normal distribution with a mean of 0. inches and a standard deviation of 0. inch. If one of the pipes measures exactly 0. inches, its length lies below the 6 th percentile between the 50 th and 8 th percentiles between the 6 th and 50 th percentiles above the 8 th percentile 0 If the amount of time students work in any given week is normally distributed with a mean of 0 hours per week and a standard deviation of hours, what is the probability a student works between 8 and hours per week?.% 8.% 5.% 68.% In a study of 8 video game players, the researchers found that the ages of these players were normally distributed, with a mean age of 7 years and a standard deviation of years. Determine if there were 5 video game players in this study over the age of 0. Justify your answer. PROBABILITY A.S.0: PERMUTATIONS A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5? 8 50,0,50 5

9 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Which formula can be used to determine the total number of different eight-letter arrangements that can be formed using the letters in the word DEADLINE? 8! 8!! 8!!+! 8!!! 6 Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 7 The letters of any word can be rearranged. Carol believes that the number of different 9-letter arrangements of the word TENNESSEE is greater than the number of different 7-letter arrangements of the word VERMONT. Is she correct? Justify your answer. A.S.: COMBINATIONS 8 Ms. Bell's mathematics class consists of sophomores, 0 juniors, and 5 seniors. How many different ways can Ms. Bell create a four-member committee of juniors if each junior has an equal chance of being selected? 0,876 5,00 9,0 9 The principal would like to assemble a committee of 8 students from the 5-member student council. How many different committees can be chosen? 0 6,5,,00 59,59,00 0 If order does not matter, which selection of students would produce the most possible committees? 5 out of 5 5 out of 5 0 out of 5 5 out of 5 A blood bank needs twenty people to help with a blood drive. Twenty-five people have volunteered. Find how many different groups of twenty can be formed from the twenty-five volunteers. A.S.9: DIFFERENTIATING BETWEEN PERMUTATIONS AND COMBINATIONS Twenty different cameras will be assigned to several boxes. Three cameras will be randomly selected and assigned to box A. Which expression can be used to calculate the number of ways that three cameras can be assigned to box A? 0! 0!! C 0 P 0 Three marbles are to be drawn at random, without replacement, from a bag containing 5 red marbles, 0 blue marbles, and 5 white marbles. Which expression can be used to calculate the probability of drawing red marbles and white marble from the bag? 5 C 5 C 0 C 5 P 5 P 0 C 5 C 5 C 0 P 5 P 5 P 0 P 6

10 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.S.: SAMPLE SPACE A committee of 5 members is to be randomly selected from a group of 9 teachers and 0 students. Determine how many different committees can be formed if members must be teachers and members must be students. A.S.: GEOMETRIC PROBABILITY 5 A dartboard is shown in the diagram below. The two lines intersect at the center of the circle, and the central angle in sector measures π. If darts thrown at this board are equally likely to land anywhere on the board, what is the probability that a dart that hits the board will land in either sector or sector? 6 A.S.5: BINOMIAL PROBABILITY 6 A study finds that 80% of the local high school students text while doing homework. Ten students are selected at random from the local high school. Which expression would be part of the process used to determine the probability that, at most, 7 of the 0 students text while doing homework? 6 C C C C A spinner is divided into eight equal sections. Five sections are red and three are green. If the spinner is spun three times, what is the probability that it lands on red exactly twice? The members of a men s club have a choice of wearing black or red vests to their club meetings. A study done over a period of many years determined that the percentage of black vests worn is 60%. If there are 0 men at a club meeting on a given night, what is the probability, to the nearest thousandth, that at least 8 of the vests worn will be black? 9 A study shows that 5% of the fish caught in a local lake had high levels of mercury. Suppose that 0 fish were caught from this lake. Find, to the nearest tenth of a percent, the probability that at least 8 of the 0 fish caught did not contain high levels of mercury. 7

11 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 The probability that the Stormville Sluggers will win a baseball game is. Determine the probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games. The probability that a professional baseball player will get a hit is. Calculate the exact probability that he will get at least hits in 5 attempts. ABSOLUTE VALUE A.A.: ABSOLUTE VALUE EQUATIONS AND INEQUALITIES What is the solution set of the equation a + 6 a = 0? 0 0, Which graph represents the solution set of 6x 7 5? Which graph represents the solution set of x 5 >? 5 Graph the inequality 6 x < 5 for x. Graph the solution on the line below. QUADRATICS A.A.0-: ROOTS OF QUADRATICS 6 What are the sum and product of the roots of the equation 6x x = 0? sum = ; product = sum = ; product = sum = ; product = sum = ; product = 7 Find the sum and product of the roots of the equation 5x + x = 0. 8 For which equation does the sum of the roots equal and the product of the roots equal? x + x = 0 x x + = 0 x + 6x + = 0 x 6x + = 0 9 For which equation does the sum of the roots equal and the product of the roots equal? x 8x + = 0 x + 8x + = 0 x x 8 = 0 x + x = 0 8

12 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 50 Which equation has roots with the sum equal to 9 and the product equal to? x + 9x + = 0 x + 9x = 0 x 9x + = 0 x 9x = 0 5 Write a quadratic equation such that the sum of its roots is 6 and the product of its roots is 7. A.A.7: FACTORING POLYNOMIALS 5 Factored completely, the expression 6x x x is equivalent to x(x + )(x ) x(x )(x + ) x(x )(x + ) x(x + )(x ) 5 Factored completely, the expression x + 0x x is equivalent to x (x + 6)(x ) (x + x)(x x) x (x )(x + ) x (x + )(x ) 5 Factor completely: 0ax ax 5a A.A.7: FACTORING THE DIFFERENCE OF PERFECT SQUARES 55 Factor the expression t 8 75t completely. A.A.7: FACTORING BY GROUPING 56 When factored completely, x + x x equals (x + )(x )(x ) (x + )(x )(x + ) (x )(x + ) (x )(x ) A.A.5: QUADRATIC FORMULA 57 The roots of the equation x + 7x = 0 are and and 7 ± 7 7 ± 7 58 The solutions of the equation y y = 9 are ± i ± i 5 ± 5 ± 5 A.A.: USING THE DISCRIMINANT 59 The roots of the equation x 0x + 5 = 0 are imaginary real and irrational real, rational, and equal real, rational, and unequal 60 The roots of the equation 9x + x = 0 are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal 6 Use the discriminant to determine all values of k that would result in the equation x kx + = 0 having equal roots. 9

13 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: COMPLETING THE SQUARE 6 Brian correctly used a method of completing the square to solve the equation x + 7x = 0. Brian s first step was to rewrite the equation as x + 7x =. He then added a number to both sides of the equation. Which number did he add? A.A.: QUADRATIC INEQUALITIES 65 Which graph best represents the inequality y + 6 x x? 6 If x + = 6x is solved by completing the square, an intermediate step would be (x + ) = 7 (x ) = 7 (x ) = (x 6) = 6 Solve x x + = 0 by completing the square, expressing the result in simplest radical form. 0

14 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 66 The solution set of the inequality x x > 0 is {x < x < 5} {x 0 < x < } {x x < or x > 5} {x x < 5 or x > } 67 Find the solution of the inequality x x > 5, algebraically. SYSTEMS A.A.: QUADRATIC-LINEAR SYSTEMS 68 Which values of x are in the solution set of the following system of equations? y = x 6 0, 0, 6, 6, y = x x 6 69 Solve the following systems of equations algebraically: 5 = y x x = 7x + y + POWERS A.N.: OPERATIONS WITH POLYNOMIALS 70 When x x is subtracted from 5 x x +, the difference is x + x 5 x x + 5 x x x x x 7 What is the product of and x +? x 8 9 x 6 9 x 8 x 6 9 x 6 x What is the product of 5 x y and 5 x + y? 5 x 9 6 y 5 x 9 6 y 5 x y 5 x 7 Express x as a trinomial. 7 Express the product of y y and y + 5 as a trinomial. A.N., A.8-9: NEGATIVE AND FRACTIONAL EXPONENTS 75 If a = and b =, what is the value of the expression a b?

15 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 76 When simplified, the expression equivalent to w 7 w w 7 w w 5 w 9 77 The expression a b is equivalent to a b a 6 b 5 b 5 a 6 a b a b 78 Which expression is equivalent to x y x 5 y? x y 5 x 5 y x y 5 y x 5 x y 5 79 Simplify the expression and write the (x y 7 ) answer using only positive exponents. 80 When x is divided by x, the quotient is x x (x ) is 8 When x + is divided by x +, the quotient equals x x x A.A.: EVALUATING EXPONENTIAL EXPRESSIONS 8 Evaluate e xln y when x = and y =. 8 The formula for continuously compounded interest is A = Pe rt, where A is the amount of money in the account, P is the initial investment, r is the interest rate, and t is the time in years. Using the formula, determine, to the nearest dollar, the amount in the account after 8 years if $750 is invested at an annual rate of %. 8 Matt places $,00 in an investment account earning an annual rate of 6.5%, compounded continuously. Using the formula V = Pe rt, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, that Matt will have in the account after 0 years. A.A.8: EVALUATING LOGARITHMIC EXPRESSIONS 85 The expression log 8 6 is equivalent to 8 8

16 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 86 The expression log 5 5 is equivalent to 88 On the axes below, for x, graph y = x +. A.A.5: GRAPHING EXPONENTIAL FUNCTIONS 87 The graph of the equation y = x has an asymptote. On the grid below, sketch the graph of y = x and write the equation of this asymptote.

17 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.5: GRAPHING LOGARITHMIC FUNCTIONS 90 If a function is defined by the equation f(x) = x, which graph represents the inverse of this function? 89 Which graph represents the function log x = y?

18 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.9: PROPERTIES OF LOGARITHMS 9 The expression logx (logy + logz) is equivalent to log x y z log x z y log x yz log xz y A B 9 If r =, then logr can be represented by C 6 loga + logb logc (loga + logb logc) log(a + B) C loga + logb logc 9 If logx loga = loga, then logx expressed in terms of loga is equivalent to log5a log6 + loga log6 + loga log6 + loga 9 If log b x = log b p log b t + log r b, then the value of x is p t r p t r p t r p t r A.A.8: LOGARITHMIC EQUATIONS 95 What is the value of x in the equation log 5 x =? ,0 96 What is the solution of the equation log (5x) =? If log x =.5 and log y 5 =, find the numerical value of x, in simplest form. y 98 Solve algebraically for x: log x + x + x x = 5

19 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 99 The temperature, T, of a given cup of hot chocolate after it has been cooling for t minutes can best be modeled by the function below, where T 0 is the temperature of the room and k is a constant. ln(t T 0 ) = kt +.78 A cup of hot chocolate is placed in a room that has a temperature of 68. After minutes, the temperature of the hot chocolate is 50. Compute the value of k to the nearest thousandth. [Only an algebraic solution can receive full credit.] Using this value of k, find the temperature, T, of this cup of hot chocolate if it has been sitting in this room for a total of 0 minutes. Express your answer to the nearest degree. [Only an algebraic solution can receive full credit.] A.A.6, 7: EXPONENTIAL EQUATIONS 00 A population of rabbits doubles every 60 days according to the formula P = 0(), where P is the population of rabbits on day t. What is the value of t when the population is 0? Akeem invests $5,000 in an account that pays.75% annual interest compounded continuously. Using the formula A = Pe rt, where A = the amount in the account after t years, P = principal invested, and r = the annual interest rate, how many years, to the nearest tenth, will it take for Akeem s investment to triple? t 60 0 What is the value of x in the equation 9 x + = 7 x +? 0 The value of x in the equation x + 5 = 8 x is The solution set of x + x = 6 is {,} {,} {, } {, } 05 Solve algebraically for x: 6 x + = 6 x + 06 Solve algebraically for all values of x: 8 x + x = 7 5x A.A.6: BINOMIAL EXPANSIONS 07 What is the fourth term in the expansion of (x ) 5? 70x 0x 70x,080x 08 What is the coefficient of the fourth term in the expansion of (a b) 9? 5, ,76 6

20 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 09 Which expression represents the third term in the expansion of (x y)? y 6x y 6x y x y 5 The graph of y = f(x) is shown below. 0 What is the middle term in the expansion of 6 x y? 0x y 5 x y 0x y 5 x y Write the binomial expansion of (x ) 5 as a polynomial in simplest form. A.A.6, 50: SOLVING POLYNOMIAL EQUATIONS What is the solution set of the equation x 5 8x = 0? {0,±} {0,±,} {0,±,±i} {±,±i} Which set lists all the real solutions of f(x) = 0? {,} {,} {,0,} {,0,} Which values of x are solutions of the equation x + x x = 0? 0,, 0,, 0,, 0,, Solve the equation 8x + x 8x 9 = 0 algebraically for all values of x. 7

21 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 The graph of y = x x + x + 6 is shown below. A.N., A.: OPERATIONS WITH RADICALS 0 The product of ( + 5) and ( 5) is Express 5 x 7x in simplest radical form. What is the product of the roots of the equation x x + x + 6 = 0? How many negative solutions to the equation x x + x = 0 exist? 0 RADICALS A.A.: SIMPLIFYING RADICALS 8 The expression 6a 6 is equivalent to 8a 8a 8 a 5 a a a 5 The expression ab b a 8b + 7ab 6b is equivalent to ab 6b 6ab b 5ab + 7ab 6b 5ab b + 7ab 6b Express 08x 5 y 8 6xy 5 in simplest radical form. A.N.5, A.5: RATIONALIZING DENOMINATORS The expression is equivalent to 5 5 (5 ) (5 + ) 8 9 Express in simplest form: a 6 b 9 6 8

22 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Which expression is equivalent to ? 5 6 Express with a rational denominator, in simplest radical form. 7 The fraction a b ab b b ab a a b 8 The expression x + x + (x + ) x x (x + ) x x x x + is equivalent to is equivalent to A.A.: SOLVING RADICALS 9 The solution set of the equation x + = x is {} {0} {,6} {,} 0 The solution set of x + 6 = x + is {,} {,} {} { } What is the solution set for the equation 5x + 9 = x +? {} { 5} {,5} { 5,} Solve algebraically for x: x 5 = A.A.0-: EXPONENTS AS RADICALS 5 The expression x x 5 5 x 5 x 5 x is equivalent to 9

23 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The expression (x ) (x ) (x ) (x ) (x ) is equivalent to 5 The expression 6x y 7 is equivalent to x y 7 x 8 y 8 x y 7 x 8 y 8 A.N.6: SQUARE ROOTS OF NEGATIVE NUMBERS 6 In simplest form, 00 is equivalent to i 0 5i 0i i 5 A.N.7: IMAGINARY NUMBERS 7 The product of i 7 and i 5 is equivalent to i i 8 The expression i + i is equivalent to i i + i + i A.N.8: CONJUGATES OF COMPLEX NUMBERS 0 What is the conjugate of + i? + i i i + i The conjugate of 7 5i is 7 5i 7 + 5i 7 5i 7 + 5i What is the conjugate of + i? + i i + i i The conjugate of the complex expression 5x + i is 5x i 5x + i 5x i 5x + i A.N.9: MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS The expression ( 7i) is equivalent to 0 + 0i 0 i i 58 i 9 Determine the value of n in simplest form: i + i 8 + i + n = 0 0

24 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic RATIONALS A.A.6: MULTIPLICATION AND DIVISION OF RATIONALS 5 Express in simplest form: x x + 7x + x x + 6 Perform the indicated operations and simplify completely: x x + 6x 8 x x x x x x + x 8 6 x A.A.: SOLVING RATIONALS 7 Solve for x: x x = + x 8 Solve algebraically for x: A.A.7: COMPLEX FRACTIONS x + x = x 9 9 Written in simplest form, the expression equivalent to x x x x x + x x x + is A.A.5: INVERSE VARIATION 5 If p varies inversely as q, and p = 0 when q =, what is the value of p when q = 5? For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9. FUNCTIONS A.A.0-: FUNCTIONAL NOTATION 5 The equation y sinθ = may be rewritten as f(y) = sinx + f(y) = sinθ + f(x) = sinθ + f(θ) = sinθ + 5 If fx = x, what is the value of f( 0)? x 6 50 Express in simplest form: d d + d

25 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.5: FAMILIES OF FUNCTIONS 55 On January, a share of a certain stock cost $80. Each month thereafter, the cost of a share of this stock decreased by one-third. If x represents the time, in months, and y represents the cost of the stock, in dollars, which graph best represents the cost of a share over the following 5 months? A.A.5: PROPERTIES OF GRAPHS OF FUNCTIONS AND RELATIONS 56 Which statement about the graph of the equation y = e x is not true? It is asymptotic to the x-axis. The domain is the set of all real numbers. It lies in Quadrants I and II. It passes through the point (e,). A.A.5: IDENTIFYING THE EQUATION OF A GRAPH 57 Four points on the graph of the function f(x) are shown below. {(0,),(,),(,),(,8)} Which equation represents f(x)? f(x) = x f(x) = x f(x) = x + f(x) = log x 58 Which equation is represented by the graph below? y = 5 x y = 0.5 x y = 5 x y = 0.5 x

26 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.8, : DEFINING FUNCTIONS 60 Which graph does not represent a function? 59 Which graph does not represent a function?

27 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 Which graph represents a relation that is not a function? 6 Which graph represents a one-to-one function? 6 Which function is not one-to-one? {(0,),(,),(,),(,)} {(0,0),(,),(,),(,)} {(0,),(,0),(,),(,)} {(0,),(,0),(,0),(,)} 6 Which relation is not a function? (x ) + y = x + x + y = x + y = xy = 65 Which function is one-to-one? f(x) = x f(x) = x f(x) = x f(x) = sinx

28 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 66 Which function is one-to-one? k(x) = x + g(x) = x + f(x) = x + j(x) = x + 70 What is the domain of the function shown below? A.A.9, 5: DOMAIN AND RANGE 67 What is the domain of the function f(x) = x +? (, ) (, ) [, ) [, ) 68 What is the range of f(x) = (x + ) + 7? y y y = 7 y 7 69 What is the range of f(x) = x +? {x x } {y y } {x x realnumbers} {y y realnumbers} x 6 y 6 x 5 y 5 7 What are the domain and the range of the function shown in the graph below? {x x > }; {y y > } {x x }; {y y } {x x > }; {y y > } {x x }; {y y } 5

29 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 The graph below represents the function y = f(x). 76 Which expression is equivalent to (n m p)(x), given m(x) = sinx, n(x) = x, and p(x) = x? sin(x) sinx sin (x) sin x 77 If f(x) = x 6 and g(x) = x, determine the value of (g f)( ). A.A.: INVERSE OF FUNCTIONS State the domain and range of this function. A.A.: COMPOSITIONS OF FUNCTIONS 7 If f(x) = x and g(x) = x + 5, what is the value of (g f)()? Which two functions are inverse functions of each other? f(x) = sinx and g(x) = cos(x) f(x) = + 8x and g(x) = 8x f(x) = e x and g(x) = lnx f(x) = x and g(x) = x + 79 If f(x) = x 6, find f (x). 7 If f(x) = x 5 and g(x) = 6x, then g(f(x)) is equal to 6x 0x 6x 0 6x 5 x + 6x 5 75 If f(x) = x x and g(x) = x, then (f g) is equal to 7 7 6

30 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.6: TRANSFORMATIONS WITH FUNCTIONS AND RELATIONS 80 The graph below shows the function f(x). 8 The minimum point on the graph of the equation y = f(x) is (, ). What is the minimum point on the graph of the equation y = f(x) + 5? (,) (, 8) (, ) ( 6, ) SEQUENCES AND SERIES A.A.9-: SEQUENCES Which graph represents the function f(x + )? 8 What is a formula for the nth term of sequence B shown below? B = 0,,,6,... b n = 8 + n b n = 0 + n b n = 0() n b n = 0() n 8 A sequence has the following terms: a =, a = 0, a = 5, a = 6.5. Which formula represents the nth term in the sequence? a n = +.5n a n = +.5(n ) a n = (.5) n a n = (.5) n 8 What is the formula for the nth term of the sequence 5,8,6,...? a n = 6 n a n = 6 n a n = 5 n a n = 5 n 7

31 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 85 What is the common difference of the arithmetic sequence 5,8,,? Which arithmetic sequence has a common difference of? {0,n,8n,n,... } {n,n,6n,6n,... } {n +,n + 5,n + 9,n +,... } {n +,n + 6,n + 6,n + 56,... } 87 What is the common ratio of the geometric sequence whose first term is 7 and fourth term is 6? What is the fifteenth term of the sequence 5, 0,0, 0,80,...? 6,80 8,90 8,90 7, What is the fifteenth term of the geometric sequence 5, 0, 5,...? Find the third term in the recursive sequence a k + = a k, where a =. 9 Find the first four terms of the recursive sequence defined below. a = a n = a (n ) n A.N.0, A.: SIGMA NOTATION 9 The value of the expression ( r + r) is r = 9 The value of the expression (n + n ) is 6 9 Evaluate: ( n n) n = 95 Evaluate: 0 + (n ) 5 n = n = 0 96 Mrs. Hill asked her students to express the sum using sigma notation. Four different student answers were given. Which student answer is correct? 0 (k ) k = 0 (k ) k = 7 (k + ) k = 9 (k ) k = 8

32 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 97 Which summation represents ? n n = 5 0 (n + ) n = (n ) n = (n ) n = 98 Express the sum using sigma notation. A.A.5: SERIES 99 An auditorium has rows of seats. The first row has 8 seats, and each succeeding row has two more seats than the previous row. How many seats are in the auditorium? TRIGONOMETRY A.A.55: TRIGONOMETRIC RATIOS 0 In the diagram below of right triangle KTW, KW = 6, KT = 5, and m KTW = 90. What is the measure of K, to the nearest minute? ' ' 55' 56' 0 Which ratio represents csc A in the diagram below? 00 What is the sum of the first 9 terms of the sequence,0,7,,,...?

33 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 In the diagram below of right triangle JTM, JT =, JM = 6, and m JMT = 90. What is the value of cot J? A.M.-: RADIAN MEASURE 0 What is the radian measure of the smaller angle formed by the hands of a clock at 7 o clock? π π 5π 6 7π 6 06 What is the number of degrees in an angle whose measure is radians? 60 π π What is the number of degrees in an angle whose radian measure is π? Find, to the nearest minute, the angle whose measure is.5 radians. 09 Find, to the nearest tenth of a degree, the angle whose measure is.5 radians. 0 Find, to the nearest tenth, the radian measure of 6º. 05 What is the radian measure of an angle whose measure is 0? 7π 7π 6 7π 6 7π 0

34 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.60: UNIT CIRCLE In which graph is θ coterminal with an angle of 70? If m θ = 50, which diagram represents θ drawn in standard position?

35 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 0 and find the exact value of sin0. 7 Which expression, when rounded to three decimal places, is equal to.55? sec 5π 6 tan(9 0 ) sin π 5 csc( 8 ) A.A.6: USING INVERSE TRIGONOMETRIC FUNCTIONS A.A.6, 66: DETERMINING TRIGONOMETRIC FUNCTIONS If θ is an angle in standard position and its terminal side passes through the point (,), find the exact value of csc θ. 8 What is the principal value of cos ? 9 In the diagram below of a unit circle, the ordered pair, represents the point where the terminal side of θ intersects the unit circle. 5 The value of tan6 to the nearest ten-thousandth is The value of csc 8 rounded to four decimal places is What is m θ?

36 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 If sin 5 = A, then 8 sina = 5 8 sina = 8 5 cos A = 5 8 cos A = 8 5 A.A.57: REFERENCE ANGLES Expressed as a function of a positive acute angle, cos( 05 ) is equal to cos 55 cos 55 sin 55 sin 55 A.A.6: ARC LENGTH A circle has a radius of inches. In inches, what is the length of the arc intercepted by a central angle of radians? π 8π 8 A circle is drawn to represent a pizza with a inch diameter. The circle is cut into eight congruent pieces. What is the length of the outer edge of any one piece of this circle? π π π π A.A.58-59: COFUNCTION AND RECIPROCAL TRIGONOMETRIC FUNCTIONS If A is acute and tana =, then cot A = cot A = cot(90 A) = cot(90 A) = 5 The expression sin θ + cos θ sin θ is equivalent to cos θ sin θ sec θ csc θ 6 Express cos θ(sec θ cos θ), in terms of sin θ. 7 Express the exact value of csc 60, with a rational denominator. A.A.67: PROVING TRIGONOMETRIC IDENTITIES 8 Which expression always equals? cos x sin x cos x + sin x cos x sinx cos x + sinx 9 Starting with sin A + cos A =, derive the formula tan A + = sec A.

37 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.76: ANGLE SUM AND DIFFERENCE IDENTITIES 0 The expression cos x cos x + sinx sinx is equivalent to sinx sin7x cos x cos 7x Given angle A in Quadrant I with sina = and angle B in Quadrant II with cos B =, what is the 5 value of cos(a B)? If tana = and sinb = 5 and angles A and B are in Quadrant I, find the value of tan(a + B). Express as a single fraction the exact value of sin 75. A.A.77: DOUBLE AND HALF ANGLE IDENTITIES The expression cos θ cos θ is equivalent to sin θ sin θ cos θ + cos θ 5 If sina = where 0 < A < 90, what is the value of sina? What is a positive value of tan x, when sinx = 0.8? A.A.68: TRIGONOMETRIC EQUATIONS 7 What is the solution set for cos θ = 0 in the interval 0 θ < 60? {0,50 } {60,0 } {0,0 } {60,00 } 8 What are the values of θ in the interval 0 θ < 60 that satisfy the equation tanθ = 0? 60º, 0º 7º, 5º 7º, 08º, 5º, 88º 60º, 0º, 0º, 00º 9 Solve the equation tanc = tanc algebraically for all values of C in the interval 0 C < Find all values of θ in the interval 0 θ < 60 that satisfy the equation sinθ = sinθ.

38 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.69: PROPERTIES OF TRIGONOMETRIC FUNCTIONS What is the period of the function f(θ) = cos θ? π π π π What is the period of the function y = sin x π? π 6π A.A.7: IDENTIFYING THE EQUATION OF A TRIGONOMETRIC GRAPH Which equation is graphed in the diagram below? y = cos π 0 x + 8 y = cos π 5 x + 5 y = cos π 0 x + 8 y = cos π 5 x + 5 Write an equation for the graph of the trigonometric function shown below. 5

39 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.65, 70-7: GRAPHING TRIGONOMETRIC FUNCTIONS 6 Which graph shows y = cos x? 5 Which graph represents the equation y = cos x? 6

40 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 Which graph represents one complete cycle of the equation y = sin πx? 8 Which equation is represented by the graph below? y = cot x y = csc x y = sec x y = tanx 9 Which equation is sketched in the diagram below? y = csc x y = sec x y = cot x y = tanx 7

41 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 50 Which is a graph of y = cot x? 5 In which interval of f(x) = cos(x) is the inverse also a function? π < x < π π x π 0 x π π x π A.A.7: USING TRIGONOMETRY TO FIND AREA A.A.6: DOMAIN AND RANGE 5 The function f(x) = tanx is defined in such a way that f (x) is a function. What can be the domain of f(x)? {x 0 x π} {x 0 x π} x π < x < π x π < x < π 5 In ABC, m A = 0, b = 0, and c = 8. What is the area of ABC to the nearest square inch? The sides of a parallelogram measure 0 cm and 8 cm. One angle of the parallelogram measures 6 degrees. What is the area of the parallelogram, to the nearest square centimeter? In parallelogram BFLO, OL =.8, LF = 7., and m O = 6. If diagonal BL is drawn, what is the area of BLF? Two sides of a parallelogram are feet and 0 feet. The measure of the angle between these sides is 57. Find the area of the parallelogram, to the nearest square foot. 57 The two sides and included angle of a parallelogram are 8,, and 60. Find its exact area in simplest form. 8

42 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.7: LAW OF SINES 58 The diagram below shows the plans for a cell phone tower. A guy wire attached to the top of the tower makes an angle of 65 degrees with the ground. From a point on the ground 00 feet from the end of the guy wire, the angle of elevation to the top of the tower is degrees. Find the height of the tower, to the nearest foot. 6 In MNP, m = 6 and n = 0. Two distinct triangles can be constructed if the measure of angle M is A.A.7: LAW OF COSINES 6 In ABC, a = 5, b =, and c =, as shown in the diagram below. What is the m C, to the nearest degree? 59 In ABC, m A =, a =, and b = 0. Find the measures of the missing angles and side of ABC. Round each measure to the nearest tenth. A.A.75: LAW OF SINES-THE AMBIGUOUS CASE 60 How many distinct triangles can be formed if m A = 5, a = 0, and b =? 0 6 Given ABC with a = 9, b = 0, and m B = 70, what type of triangle can be drawn? an acute triangle, only an obtuse triangle, only both an acute triangle and an obtuse triangle neither an acute triangle nor an obtuse triangle 6 In ABC, m A = 7, a = 59., and c = 60.. What are the two possible values for m C, to the nearest tenth? 7.7 and and and and In ABC, a =, b = 5, and c = 7. What is m C? In a triangle, two sides that measure 6 cm and 0 cm form an angle that measures 80. Find, to the nearest degree, the measure of the smallest angle in the triangle. 9

43 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.7: VECTORS 67 Two forces of 5 newtons and 85 newtons acting on a body form an angle of 55. Find the magnitude of the resultant force, to the nearest hundredth of a newton. Find the measure, to the nearest degree, of the angle formed between the resultant and the larger force. 70 A circle shown in the diagram below has a center of ( 5,) and passes through point (,7). 68 The measures of the angles between the resultant and two applied forces are 60 and 5, and the magnitude of the resultant is 7 pounds. Find, to the nearest pound, the magnitude of each applied force. CONICS A.A.7, 9: EQUATIONS OF CIRCLES 69 Which equation represents the circle shown in the graph below that passes through the point (0, )? Write an equation that represents the circle. 7 Write an equation of the circle shown in the diagram below. (x ) + (y + ) = 6 (x ) + (y + ) = 8 (x + ) + (y ) = 6 (x + ) + (y ) = 8 0

44 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 Write an equation of the circle shown in the graph below. 7 The equation x + y x + 6y + = 0 is equivalent to (x ) + (y + ) = (x ) + (y + ) = 7 (x + ) + (y + ) = 7 (x + ) + (y + ) = 0

45 ID: A Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section ANS: PTS: REF: 07a STA: A.S. TOP: Analysis of Data ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data ANS: Controlled experiment because Howard is comparing the results obtained from an experimental sample against a control sample. PTS: REF: 0800a STA: A.S. TOP: Analysis of Data ANS: Students entering the library are more likely to spend more time studying, creating bias. PTS: REF: fall090a STA: A.S. TOP: Analysis of Data 5 ANS: PTS: REF: 00a STA: A.S. TOP: Analysis of Data 6 ANS: PTS: REF: 06a STA: A.S. TOP: Average Known with Missing Data 7 ANS: k + 5 = k + k + 6 k + = k + 6 = k + k = 8 k = PTS: REF: 06a STA: A.S. TOP: Average Known with Missing Data 8 ANS: PTS: REF: fall09a STA: A.S. TOP: Dispersion KEY: variance 9 ANS: 7. PTS: REF: 0609a STA: A.S. TOP: Dispersion KEY: basic, group frequency distributions

46 ID: A 0 ANS: σ x =.9. x = 0. There are 8 scores between 5. and 5.9. PTS: REF: 067a STA: A.S. TOP: Dispersion KEY: advanced ANS: PTS: REF: 067a STA: A.S.6 TOP: Regression ANS: y = 80.77(0.95) x PTS: REF: 06a STA: A.S.7 TOP: Exponential Regression ANS: y = 7.05(.509) x. y = 7.05(.509) 8 PTS: REF: 08a STA: A.S.7 TOP: Exponential Regression ANS: y = 0.596(.586) x PTS: REF: 080a STA: A.S.7 TOP: Exponential Regression 5 ANS: y =.00x.98,,009. y =.00(5) PTS: REF: fall098a STA: A.S.7 TOP: Power Regression 6 ANS: PTS: REF: 060a STA: A.S.8 TOP: Correlation Coefficient 7 ANS: () shows the strongest linear relationship, but if r < 0, b < 0. PTS: REF: 0a STA: A.S.8 TOP: Correlation Coefficient 8 ANS:. PTS: REF: 065a STA: A.S.8 TOP: Correlation Coefficient

47 ID: A 9 ANS: PTS: REF: fall095a STA: A.S.5 TOP: Normal Distributions KEY: interval 0 ANS:.% + 9.% = 5.% PTS: REF: 0a STA: A.S.5 TOP: Normal Distributions KEY: probability ANS: 68% 50 = PTS: REF: 080a STA: A.S.5 TOP: Normal Distributions KEY: predict ANS: 68% of the students are within one standard deviation of the mean. 6% of the students are more than one standard deviation above the mean. PTS: REF: 0a STA: A.S.5 TOP: Normal Distributions KEY: percent ANS: no. over 0 is more than standard deviation above the mean PTS: REF: 069a STA: A.S.5 TOP: Normal Distributions KEY: predict ANS: = 8. The first digit cannot be 0 or 5. The second digit cannot be 5 or the same as the first digit. The third digit cannot be 5 or the same as the first or second digit. PTS: REF: 05a STA: A.S.0 TOP: Permutations 5 ANS: PTS: REF: fall095a STA: A.S.0 TOP: Permutations 6 ANS: 9,96,800. P!! = 79,00,600 = 9,96,800 PTS: REF: 0805a STA: A.S.0 TOP: Permutations

48 ID: A 7 ANS: No. TENNESSEE: P 9 9!!! = 6,880 =,780. VERMONT: 96 7 P 7 = 5,00 PTS: REF: 0608a STA: A.S.0 TOP: Permutations 8 ANS: C = 0 0 PTS: REF: 06a STA: A.S. TOP: Combinations 9 ANS: C = 6,5 5 8 PTS: REF: 080a STA: A.S. TOP: Combinations 0 ANS: C =,00. C = C = 5,0. C =,68, PTS: REF: 067a STA: A.S. TOP: Combinations ANS: C = 5,0 5 0 PTS: REF: 0a STA: A.S. TOP: Combinations ANS: PTS: REF: 06007a STA: A.S.9 TOP: Differentiating Permutations and Combinations ANS: PTS: REF: 07a STA: A.S.9 TOP: Differentiating Permutations and Combinations ANS:,00. PTS: REF: fall095a STA: A.S. TOP: Sample Space

49 ID: A 5 ANS: π + π π = π π = PTS: REF: 008a STA: A.S. TOP: Geometric Probability 6 ANS: PTS: REF: 06a STA: A.S.5 TOP: Binomial Probability KEY: modeling 7 ANS: C 5 = PTS: REF: 0a STA: A.S.5 TOP: Binomial Probability KEY: spinner 8 ANS: C C C PTS: REF: 0606a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 9 ANS: 6.%. 0 C C C PTS: REF: 0808a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 0 ANS: C C C PTS: REF: 08a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 5

50 ID: A ANS: 5. C 5 5 C 5 C 5 0 = 0 = 0 = PTS: REF: 068a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most ANS: a + 6 = a 0. a + 6 = a = a = a = 8 = 8 0 PTS: REF: 006a STA: A.A. TOP: Absolute Value Equations ANS: 6x 7 5 6x 7 5 6x x 6x x PTS: REF: fall0905a STA: A.A. TOP: Absolute Value Inequalities KEY: graph ANS: x 5 > x 5 > x > 8 x > or x 5 < x 5 < x < x < PTS: REF: 0609a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 6

51 ID: A 5 ANS: 6 x < 5 6 x > 5 6 x > 5 or 6 x < 5 > x or < x. PTS: REF: 067a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 6 ANS: sum: b a = 6 =. product: c a = 6 = PTS: REF: 009a STA: A.A.0 TOP: Roots of Quadratics 7 ANS: Sum b a = 5. Product c a = 5 PTS: REF: 0600a STA: A.A.0 TOP: Roots of Quadratics 8 ANS: b a = 6 =. c a = = PTS: REF: 0a STA: A.A. TOP: Roots of Quadratics KEY: basic 9 ANS: S = b a = ( ) =. P = c a = 8 = PTS: REF: fall09a STA: A.A. TOP: Roots of Quadratics KEY: basic 50 ANS: sum of the roots, b a = ( 9) = 9. product of the roots, c a = PTS: REF: 0608a STA: A.A. TOP: Roots of Quadratics KEY: basic 5 ANS: x 6x 7 = 0, b a = 6. c = 7. If a = then b = 6 and c = 7 a PTS: REF: 060a STA: A.A. TOP: Roots of Quadratics KEY: basic 7

52 ID: A 5 ANS: 6x x x = x(x + x 6) = x(x + )(x ) PTS: REF: fall097a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable 5 ANS: x + 0x x = x (6x + 5x 6) = x (x + )(x ) PTS: REF: 06008a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable 5 ANS: 0ax ax 5a = a(0x x 5) = a(5x + )(x 5) PTS: REF: 0808a STA: A.A.7 TOP: Factoring Polynomials KEY: multiple variables 55 ANS: t 8 75t = t (t 5) = t (t + 5)(t 5) PTS: REF: 06a STA: A.A.7 TOP: Factoring the Difference of Perfect Squares 56 ANS: x + x x KEY: binomial x (x + ) (x + ) (x )(x + ) (x + )(x )(x + ) PTS: REF: 06a STA: A.A.7 TOP: Factoring by Grouping 57 ANS: 7 ± 7 ()( ) () = 7 ± 7 PTS: REF: 08009a STA: A.A.5 TOP: Quadratic Formula 58 ANS: ± ( ) ()( 9) () = ± 5 = ± 5 PTS: REF: 06009a STA: A.A.5 TOP: Quadratic Formula 59 ANS: b ac = ( 0) ()(5) = = 0 PTS: REF: 00a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 8

53 ID: A 60 ANS: b ac = (9)( ) = 9 + = 5 PTS: REF: 0806a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 6 ANS: b ac = 0 k ()() = 0 k 6 = 0 (k + )(k ) = 0 k = ± PTS: REF: 0608a STA: A.A. TOP: Using the Discriminant KEY: determine equation given nature of roots 6 ANS: PTS: REF: 06a STA: A.A. TOP: Completing the Square 6 ANS: x + = 6x x 6x = x 6x + 9 = + 9 (x ) = 7 PTS: REF: 06a STA: A.A. TOP: Completing the Square 6 ANS: ± 7. x x + = 0 x 6x + = 0 x 6x = x 6x + 9 = + 9 (x ) = 7 x = ± 7 x = ± 7 PTS: REF: fall096a STA: A.A. TOP: Completing the Square 9

54 ID: A 65 ANS: y x x 6 y (x )(x + ) PTS: REF: 0607a STA: A.A. TOP: Quadratic Inequalities KEY: two variables 66 ANS: x x 0 > 0 or (x 5)(x + ) > 0 x 5 > 0 and x + > 0 x > 5 and x > x > 5 x 5 < 0 and x + < 0 x < 5 and x < x < PTS: REF: 05a STA: A.A. TOP: Quadratic Inequalities KEY: one variable 67 ANS: x < or x > 5. x x 5 > 0. x 5 > 0 and x + > 0 or x 5 < 0 and x + < 0 (x 5)(x + ) > 0 x > 5 and x > x > 5 x < 5 and x < x < PTS: REF: 08a STA: A.A. TOP: Quadratic Inequalities KEY: one variable 68 ANS: x x 6 = x 6 x x = 0 x(x ) = 0 x = 0, PTS: REF: 0805a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 0

55 ID: A 69 ANS: 9, and,. y = x + 5. x + 7x = x + 5 y = x + 7x x + 6x 9 = 0 (x + 9)(x ) = 0 x = 9 and x = y = = and y = + 5 = PTS: 6 REF: 069a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 70 ANS: PTS: REF: 0a STA: A.N. TOP: Operations with Polynomials 7 ANS: The binomials are conjugates, so use FL. PTS: REF: 006a STA: A.N. TOP: Operations with Polynomials 7 ANS: The binomials are conjugates, so use FL. PTS: REF: 060a STA: A.N. TOP: Operations with Polynomials 7 ANS: 9 x x +. x = x x = 9 x x x + = 9 x x + PTS: REF: 080a STA: A.N. TOP: Operations with Polynomials 7 ANS: 6y 7 0 y 5 y. y y 5 = 6y + 0 y y 5 y = 6y 7 0 y 5 y PTS: REF: 068a STA: A.N. TOP: Operations with Polynomials 75 ANS: ( ) = 9 8 = 8 9 PTS: REF: 0600a STA: A.N. TOP: Negative and Fractional Exponents

56 ID: A 76 ANS: w 5 w 9 = (w ) = w PTS: REF: 080a STA: A.A.8 TOP: Negative and Fractional Exponents 77 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Negative and Fractional Exponents 78 ANS: PTS: REF: 060a STA: A.A.9 TOP: Negative Exponents 79 ANS: x y 9. x y 5 (x y 7 ) = y 5 (x y 7 ) = y 5 (x 6 y ) = x 6 y 9 = x x x x y 9 PTS: REF: 06a STA: A.A.9 TOP: Negative Exponents 80 ANS: x x = x x (x ) x = x x = x x = x PTS: REF: 0808a STA: A.A.9 TOP: Negative Exponents 8 ANS: x + x + = x + + x x + = x x + = x PTS: REF: 0a STA: A.A.9 TOP: Negative Exponents 8 ANS: e ln = e ln = e ln 8 = 8 PTS: REF: 06a STA: A.A. TOP: Evaluating Exponential Expressions 8 ANS: A = 750e (0.0)(8) 95 PTS: REF: 069a STA: A.A. TOP: Evaluating Exponential Expressions 8 ANS:, PTS: REF: fall09a STA: A.A. TOP: Evaluating Exponential Expressions

57 ID: A 85 ANS: 8 = 6 PTS: REF: fall0909a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 86 ANS: PTS: REF: 0a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 87 ANS: y = 0 PTS: REF: 060a STA: A.A.5 TOP: Graphing Exponential Functions 88 ANS: PTS: REF: 0a STA: A.A.5 TOP: Graphing Exponential Functions 89 ANS: PTS: REF: 06a STA: A.A.5 TOP: Graphing Logarithmic Functions 90 ANS: f (x) = log x PTS: REF: fall096a STA: A.A.5 TOP: Graphing Logarithmic Functions 9 ANS: logx (logy + logz) = logx logy logz = log x PTS: REF: 0600a STA: A.A.9 TOP: Properties of Logarithms y z

58 ID: A 9 ANS: PTS: REF: 060a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 9 ANS: logx = loga + loga logx = log6a logx = log6 + loga logx = log6 + loga logx = log6 + loga PTS: REF: 0a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 9 ANS: PTS: REF: 0607a STA: A.A.9 TOP: Properties of Logarithms KEY: antilogarithms 95 ANS: x = 5 = 65 PTS: REF: 0606a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 96 ANS: log (5x) = log (5x) = 5x = 5x = 8 x = 8 5 PTS: REF: fall09a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 97 ANS: 800. x =.5 =. y = 5. x y = y = 5 = 5 5 = 800 PTS: REF: 07a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced

59 ID: A 98 ANS: x =, log x + x + x x x + x x x + x x = = (x + ) = x + 6x + 9 x + x = x + 6x + 9x 0 = 6x + 8x + 0 = x + x + 0 = (x + )(x + ) x =, PTS: 6 REF: 0809a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 99 ANS: ln(t T 0 ) = kt ln(t 68) = 0.0(0) ln(50 68) = k() k +.78 k 0.0 ln(t 68) =.678 T T 08 PTS: 6 REF: 09a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 00 ANS: 0 = 0() = () t 60 log = log() log = t log 60 60log log = t 00 = t t 60 t 60 PTS: REF: 005a STA: A.A.6 TOP: Exponential Growth 5

60 ID: A 0 ANS: = 5000e.075t = e.075t ln = lne.075t ln.075t lne = t PTS: REF: 067a STA: A.A.6 TOP: Exponential Growth 0 ANS: 9 x + = 7 x +. ( ) x + = ( ) x + 6x + = x + 6 6x + = x + 6 x = x = PTS: REF: 08008a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 0 ANS: x + 5 = 8 x. x + 5 = x + 0 = 9x x + 0 = 9x 0 = 5x = x x PTS: REF: 0605a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 6

61 ID: A 0 ANS: x + x = 6. x + 8x = 6 ( ) x + x = 6 x + 8x = 6 x + 8x + 6 = 0 x + x + = 0 (x + )(x + ) = 0 x = x = PTS: REF: 0605a STA: A.A.7 TOP: Exponential Equations KEY: common base shown 05 ANS: 6 x + = 6 x + ( ) x + = ( ) x + x + 6 = x + 6 x = 0 PTS: REF: 08a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 06 ANS: 8 x + x = 7 x + x = x + 8x = 5x x + 8x 5x = 0 x(x + 8x 5) = 0 x(x )(x + 5) = 0 5x 5x x = 0,, 5 PTS: 6 REF: 069a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 07 ANS: 5 C (x) ( ) = 0 9x 8 = 70x PTS: REF: fall099a STA: A.A.6 TOP: Binomial Expansions 08 ANS: C a 6 9 ( b) = 576a 6 b PTS: REF: 066a STA: A.A.6 TOP: Binomial Expansions 7

62 ID: A 09 ANS: C (x ) ( y) = 6x y PTS: REF: 05a STA: A.A.6 TOP: Binomial Expansions 0 ANS: C x 6 ( y) = 0 x 8 8y = 0x y PTS: REF: 065a STA: A.A.6 TOP: Binomial Expansions ANS: x 5 80x + 80x 0x + 0x. 5 C 0 (x) 5 ( ) 0 = x 5. 5 C (x) ( ) = 80x. 5 C (x) ( ) = 80x. 5 C (x) ( ) = 0x. 5 C (x) ( ) = 0x. 5 C 5 (x) 0 ( ) 5 = PTS: REF: 06a STA: A.A.6 TOP: Binomial Expansions ANS: x 5 8x = 0 x(x 6) = 0 x(x + )(x ) = 0 x(x + )(x + )(x ) = 0 PTS: REF: 06a STA: A.A.6 TOP: Solving Polynomial Equations ANS: x + x x = 0 x(x + x ) = 0 x(x + )(x ) = 0 x = 0,, PTS: REF: 00a STA: A.A.6 TOP: Solving Polynomial Equations ANS: ±,. 8x + x 8x 9 = 0 x (x + ) 9(x + ) = 0 (x 9)(x + ) = 0 x 9 = 0 or x + = 0 (x + )(x ) = 0 x = x = ± PTS: REF: fall097a STA: A.A.6 TOP: Solving Polynomial Equations 8

63 ID: A 5 ANS: PTS: REF: 06005a STA: A.A.50 TOP: Solving Polynomial Equations 6 ANS: The roots are,,. PTS: REF: 080a STA: A.A.50 TOP: Solving Polynomial Equations 7 ANS: PTS: REF: 06a STA: A.A.50 TOP: Solving Polynomial Equations 8 ANS: a 5 a = a 5 a PTS: REF: 060a STA: A.A. TOP: Simplifying Radicals KEY: index > 9 ANS: a b PTS: REF: 0a STA: A.A. TOP: Simplifying Radicals KEY: index > 0 ANS: ( + 5)( 5) = 9 5 = PTS: REF: 0800a STA: A.N. TOP: Operations with Radicals ANS: 5 x 7x = 5 x x 9x x = 5x x 6x x = x x PTS: REF: 060a STA: A.N. TOP: Operations with Radicals ANS: ab b a 9b b + 7ab 6b = ab b 9ab b + 7ab 6b = 5ab b + 7ab 6b PTS: REF: fall098a STA: A.A. TOP: Operations with Radicals KEY: with variables index = ANS: 08x 5 y 8 6xy 5 = 8x y = x y y PTS: REF: 0a STA: A.A. TOP: Operations with Radicals KEY: with variables index = 9

64 ID: A ANS: = (5 + ) 5 = 5 + PTS: REF: 066a STA: A.N.5 TOP: Rationalizing Denominators 5 ANS: = = 5 = + 5 PTS: REF: 060a STA: A.N.5 TOP: Rationalizing Denominators 6 ANS: 5( + ) = 5( + ) 9 = 5( + ) 7 PTS: REF: fall098a STA: A.N.5 TOP: Rationalizing Denominators 7 ANS: a b = a b b b = b ab = b ab PTS: REF: 0809a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 8 ANS: x + x + x + x + = (x + ) x + x + = x + PTS: REF: 0a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 9 ANS: PTS: REF: 0608a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 0 ANS: x + 6 = (x + ). is an extraneous solution. x + 6 = x + x + 0 = x + x 0 = (x + )(x ) x = x = PTS: REF: 06a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 0

65 ID: A ANS: 5x + 9 = (x + ). ( 5) + shows an extraneous solution. 5x + 9 = x + 6x = x + x 0 0 = (x + 5)(x ) x = 5, PTS: REF: 06a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions ANS: 7. x 5 = x 5 = x 5 = 9 x = x = 7 PTS: REF: 09a STA: A.A. TOP: Solving Radicals KEY: basic ANS: 5 x = x 5 = 5 x PTS: REF: 08a STA: A.A.0 TOP: Fractional Exponents as Radicals ANS: PTS: REF: 060a STA: A.A.0 TOP: Fractional Exponents as Radicals 5 ANS: 6x y 7 = 6 7 x y = x 7 y PTS: REF: 0607a STA: A.A. TOP: Radicals as Fractional Exponents 6 ANS: 00 = 00 PTS: REF: 06006a STA: A.N.6 TOP: Square Roots of Negative Numbers 7 ANS: PTS: REF: 0609a STA: A.N.7 TOP: Imaginary Numbers 8 ANS: i + i = ( ) + ( i) = i PTS: REF: 0800a STA: A.N.7 TOP: Imaginary Numbers

66 ID: A 9 ANS: i + i 8 + i + n = 0 i + ( ) i + n = 0 + n = 0 n = PTS: REF: 068a STA: A.N.7 TOP: Imaginary Numbers 0 ANS: PTS: REF: 080a STA: A.N.8 TOP: Conjugates of Complex Numbers ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers ANS: PTS: REF: 069a STA: A.N.8 TOP: Conjugates of Complex Numbers ANS: ( 7i)( 7i) = 9 i i + 9i = 9 i 9 = 0 i PTS: REF: fall090a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 5 ANS: (x ) (x + )(x + ) x + (x ) = (x + )(x ) x + (x + ) = (x ) (x + ) PTS: REF: 066a STA: A.A.6 TOP: Multiplication and Division of Rationals KEY: division 6 ANS: (x + 6). x (x ) + 6(x ) x x x x x x x + x 8 6 x (x + 6)(x ) x(x ) (x ) x (x ) ( + x)( x) (x + )(x ) (x + 6) x PTS: 6 REF: 09a STA: A.A.6 TOP: Multiplication and Division of Rationals KEY: division

67 ID: A 7 ANS: no solution. x x = + x x x = (x ) x = PTS: REF: fall090a STA: A.A. TOP: Solving Rationals KEY: rational solutions 8 ANS: x + x = x 9 x + + x = x 9 x + (x + ) (x + )(x ) = x + x + 6 = x = (x + )(x ) x = PTS: REF: 0806a STA: A.A. TOP: Solving Rationals KEY: rational solutions 9 ANS: x x x + = x x x + 8x = (x + )(x ) x 8x (x + ) = x PTS: REF: fall090a STA: A.A.7 TOP: Complex Fractions 50 ANS: d d + d = d 8 d d + d d = d 8 d d 5d = d 8 5 PTS: REF: 0605a STA: A.A.7 TOP: Complex Fractions

68 ID: A 5 ANS: 0 = 5 p 5 = 5 p 5 = p PTS: REF: 06a STA: A.A.5 TOP: Inverse Variation 5 ANS: 6 = 9w 8 = w PTS: REF: 00a STA: A.A.5 TOP: Inverse Variation 5 ANS: y sinθ = y = sinθ + f(θ) = sinθ + PTS: REF: fall097a STA: A.A.0 TOP: Functional Notation 5 ANS: 0 f0 = ( 0) 6 = 0 8 = 5 PTS: REF: 060a STA: A.A. TOP: Functional Notation 55 ANS: PTS: REF: 09a STA: A.A.5 TOP: Families of Functions 56 ANS: PTS: REF: 09a STA: A.A.5 TOP: Properties of Graphs of Functions and Relations 57 ANS: PTS: REF: 0600a STA: A.A.5 TOP: Identifying the Equation of a Graph 58 ANS: PTS: REF: 0608a STA: A.A.5 TOP: Identifying the Equation of a Graph 59 ANS: PTS: REF: fall0908a STA: A.A.8 TOP: Defining Functions KEY: graphs 60 ANS: PTS: REF: 00a STA: A.A.8 TOP: Defining Functions KEY: graphs 6 ANS: PTS: REF: 06a STA: A.A.8 TOP: Defining Functions KEY: graphs 6 ANS: PTS: REF: 060a STA: A.A.8 TOP: Defining Functions

69 ID: A 6 ANS: () and () fail the horizontal line test and are not one-to-one. Not every element of the range corresponds to only one element of the domain. () fails the vertical line test and is not a function. Not every element of the domain corresponds to only one element of the range. PTS: REF: 0800a STA: A.A. TOP: Defining Functions 6 ANS: () fails the horizontal line test. Not every element of the range corresponds to only one element of the domain. PTS: REF: fall0906a STA: A.A. TOP: Defining Functions 65 ANS: PTS: REF: 05a STA: A.A. TOP: Defining Functions 66 ANS: PTS: REF: 068a STA: A.A. TOP: Defining Functions 67 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Domain and Range KEY: real domain 68 ANS: PTS: REF: 06a STA: A.A.9 TOP: Domain and Range KEY: real domain 69 ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain 70 ANS: PTS: REF: 060a STA: A.A.5 TOP: Domain and Range 7 ANS: PTS: REF: 0800a STA: A.A.5 TOP: Domain and Range 7 ANS: D: 5 x 8. R: y PTS: REF: 0a STA: A.A.5 TOP: Domain and Range 7 ANS: f() = () =. g( ) = ( ) + 5 = PTS: REF: fall090a STA: A.A. TOP: Compositions of Functions KEY: numbers 7 ANS: 6(x 5) = 6x 0 PTS: REF: 009a STA: A.A. TOP: Compositions of Functions KEY: variables 75 ANS: g = =. f() = () = PTS: REF: 00a STA: A.A. TOP: Compositions of Functions KEY: numbers 5

70 ID: A 76 ANS: PTS: REF: 066a STA: A.A. TOP: Compositions of Functions KEY: variables 77 ANS: 7. f( ) = ( ) 6 =. g(x) = = 7. PTS: REF: 065a STA: A.A. TOP: Compositions of Functions KEY: numbers 78 ANS: PTS: REF: 0807a STA: A.A. TOP: Inverse of Functions KEY: equations 79 ANS: y = x 6. f (x) is not a function. x = y 6 x + 6 = y ± x + 6 = y PTS: REF: 06a STA: A.A. TOP: Inverse of Functions KEY: equations 80 ANS: PTS: REF: fall096a STA: A.A.6 TOP: Transformations with Functions and Relations 8 ANS: PTS: REF: 080a STA: A.A.6 TOP: Transformations with Functions and Relations 8 ANS: common difference is. b n = x + n 0 = x + () 8 = x PTS: REF: 080a STA: A.A.9 TOP: Sequences 8 ANS: 0 =.5 PTS: REF: 07a STA: A.A.9 TOP: Sequences 8 ANS: PTS: REF: 0606a STA: A.A.9 TOP: Sequences 85 ANS: PTS: REF: 0600a STA: A.A.0 TOP: Sequences 86 ANS: PTS: REF: 00a STA: A.A.0 TOP: Sequences 6

71 ID: A 87 ANS: 7r = 6 r = 6 7 r = PTS: REF: 0805a STA: A.A. TOP: Sequences 88 ANS: a n = 5( ) n a 5 = 5( ) 5 = 8,90 PTS: REF: 005a STA: A.A. TOP: Sequences 89 ANS: a n = 5( ) n a 5 = 5( ) 5 = 5( ) = 5 7 = 8 5 PTS: REF: 0609a STA: A.A. TOP: Sequences 90 ANS: a =. a = () = 5. a = (5) = 9. PTS: REF: 06a STA: A.A. TOP: Recursive Sequences 9 ANS:, 5, 8, PTS: REF: fall09a STA: A.A. TOP: Recursive Sequences 9 ANS: n 5 Σ r + r + = 6 + = = 0 8 PTS: REF: 068a STA: A.N.0 TOP: Sigma Notation KEY: basic 9 ANS: n 0 Σ n + n = + = + = 8 = PTS: REF: fall09a STA: A.N.0 TOP: Sigma Notation KEY: basic 7

72 ID: A 9 ANS: 0. PTS: REF: 00a STA: A.N.0 TOP: Sigma Notation KEY: basic 95 ANS: ( ) + ( ) + ( ) + ( ) + (5 ) = = 0 PTS: REF: 0a STA: A.N.0 TOP: Sigma Notation KEY: basic 96 ANS: PTS: REF: 0605a STA: A.A. TOP: Sigma Notation 97 ANS: PTS: REF: 0605a STA: A.A. TOP: Sigma Notation 98 ANS: 5 n = 7n PTS: REF: 0809a STA: A.A. TOP: Sigma Notation 99 ANS: S n = n [a + (n )d] = [(8) + ( )] = 798 PTS: REF: 060a STA: A.A.5 TOP: Series KEY: arithmetic 00 ANS: S n = n 9 [a + (n )d] = [() + (9 )7] = 5 PTS: REF: 00a STA: A.A.5 TOP: Summations KEY: arithmetic 8

73 ID: A 0 ANS: cos K = 5 6 K = cos 5 6 K ' PTS: REF: 060a STA: A.A.55 TOP: Trigonometric Ratios 0 ANS: PTS: REF: 0800a STA: A.A.55 TOP: Trigonometric Ratios 0 ANS: 6 = 08 = 6 = 6. cot J = A O = 6 6 = PTS: REF: 00a STA: A.A.55 TOP: Trigonometric Ratios 0 ANS: π 5 = 0π = 5π 6 PTS: REF: 065a STA: A.M. TOP: Radian Measure 05 ANS: 0 π 80 = 7π PTS: REF: 0800a STA: A.M. TOP: Radian Measure KEY: radians 06 ANS: 80 π = 60 π PTS: REF: 00a STA: A.M. TOP: Radian Measure KEY: degrees 07 ANS: π 80 π = 65 PTS: REF: 0600a STA: A.M. TOP: Radian Measure KEY: degrees 9

74 ID: A 08 ANS: 97º π PTS: REF: fall09a STA: A.M. TOP: Radian Measure KEY: degrees 09 ANS:.5 80 π. PTS: REF: 09a STA: A.M. TOP: Radian Measure KEY: degrees 0 ANS: 6 π 80.8 PTS: REF: 06a STA: A.M. TOP: Radian Measure KEY: radians ANS: PTS: REF: 08005a STA: A.A.60 TOP: Unit Circle ANS: PTS: REF: 0606a STA: A.A.60 TOP: Unit Circle ANS: PTS: REF: 060a STA: A.A.60 TOP: Unit Circle 0

75 ID: A ANS:. sinθ = y = x + y ( ) + =. csc θ =. PTS: REF: fall09a STA: A.A.6 TOP: Determining Trigonometric Functions 5 ANS: PTS: REF: 065a STA: A.A.66 TOP: Determining Trigonometric Functions 6 ANS: PTS: REF: 067a STA: A.A.66 TOP: Determining Trigonometric Functions 7 ANS: PTS: REF: 00a STA: A.A.66 TOP: Determining Trigonometric Functions 8 ANS: PTS: REF: 08007a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: basic 9 ANS: PTS: REF: 00a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: unit circle 0 ANS: PTS: REF: 0a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced ANS: cos( ) = cos(55 ) PTS: REF: 060a STA: A.A.57 TOP: Reference Angles ANS: s = θ r = = 8 PTS: REF: fall09a STA: A.A.6 TOP: Arc Length KEY: arc length

76 ID: A ANS: s = θ r = π 8 6 = π PTS: REF: 06a STA: A.A.6 TOP: Arc Length KEY: arc length ANS: Cofunctions tangent and cotangent are complementary PTS: REF: 060a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 5 ANS: sin θ + cos θ sin θ = cos θ = sec θ PTS: REF: 06a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 6 ANS: cos θ cos θ cos θ = cos θ = sin θ PTS: REF: 060a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 7 ANS:. If sin60 =, then csc 60 = = PTS: REF: 05a STA: A.A.59 TOP: Reciprocal Trigonometric Relationships 8 ANS: PTS: REF: 008a STA: A.A.67 TOP: Proving Trigonometric Identities 9 ANS: sin A cos A + cos A cos A = cos A tan A + = sec A PTS: REF: 05a STA: A.A.67 TOP: Proving Trigonometric Identities 0 ANS: PTS: REF: fall090a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying ANS: 5 cos(a B) = 5 5 = = 65 PTS: REF: 0a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating

77 ID: A ANS: cos B + sin B = cos B + 5 = tanb = sinb cos B = 5 = 5 tan(a + B) = = = = cos B + 5 = cos B = 6 cos B = PTS: REF: 0807a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating ANS: sin(5 + 0) = sin5cos 0 + cos 5sin0 = + = 6 + = 6 + PTS: REF: 066a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating ANS: cos θ cos θ = cos θ (cos θ sin θ) = sin θ PTS: REF: 060a STA: A.A.77 TOP: Double Angle Identities KEY: simplifying 5 ANS: + cos A = sina = sina cos A cos A = 5 9 cos A = + 5, sin A is acute. = = PTS: REF: 007a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating

78 ID: A 6 ANS: If sinx = 0.8, then cos x = 0.6. tan x = = 0..6 = 0.5. PTS: REF: 060a STA: A.A.77 TOP: Half Angle Identities 7 ANS: cos θ = cos θ = θ = cos = 60, 00 PTS: REF: 060a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 8 ANS: tanθ = 0.. tanθ = θ = tan θ = 60, 0 PTS: REF: fall090a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 9 ANS: 5, 5 tanc = tanc = tanc tan = C C = 5,5 PTS: REF: 080a STA: A.A.68 TOP: Trigonometric Equations KEY: basic

79 ID: A 0 ANS: 0, 60, 80, 00. sinθ = sinθ sinθ sinθ = 0 sinθ cos θ sinθ = 0 sinθ(cos θ ) = 0 sinθ = 0 cos θ = 0 θ = 0,80 cos θ = θ = 60,00 PTS: REF: 0607a STA: A.A.68 TOP: Trigonometric Equations KEY: double angle identities 5

80 ID: A Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section ANS: π b = π PTS: REF: 06a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions ANS: π b = π = 6π PTS: REF: 0607a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions ANS: π b = 0 KEY: period KEY: period b = π 5 PTS: REF: 07a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph ANS: y = sinx. The period of the function is π, the amplitude is and it is reflected over the x-axis. PTS: REF: 065a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 5 ANS: PTS: REF: fall09a STA: A.A.65 TOP: Graphing Trigonometric Functions 6 ANS: PTS: REF: 069a STA: A.A.65 TOP: Graphing Trigonometric Functions 7 ANS: period = π b = π π = PTS: REF: 0806a STA: A.A.70 TOP: Graphing Trigonometric Functions KEY: recognize

81 ID: A 8 ANS: PTS: REF: 0600a STA: A.A.7 TOP: Graphing Trigonometric Functions 9 ANS: PTS: REF: 0a STA: A.A.7 TOP: Graphing Trigonometric Functions 50 ANS: PTS: REF: 007a STA: A.A.7 TOP: Graphing Trigonometric Functions 5 ANS: PTS: REF: 060a STA: A.A.6 TOP: Domain and Range 5 ANS: PTS: REF: 06a STA: A.A.6 TOP: Domain and Range 5 ANS: K = (0)(8) sin0 = 5 78 PTS: REF: fall0907a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: basic 5 ANS: K = (0)(8) sin6 9 PTS: REF: 080a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: parallelograms 55 ANS: (7.)(.8) sin6. PTS: REF: 08a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: basic