JMAP REGENTS BY PERFORMANCE INDICATOR: TOPIC

Transcription

1 JMAP REGENTS BY PERFORMANCE INDICATOR: TOPIC NY Algebra /Trigonometry Regents Exam Questions from Spring 009 to January 06 Sorted by PI: Topic

2 TABLE OF CONTENTS TOPIC PI: SUBTOPIC QUESTION NUMBER GRAPHS AND STATISTICS PROBABILITY ABSOLUTE VALUE QUADRATICS A.S.-: Analysis of Data A.S.: Average Known with Missing Data...0- A.S.: Dispersion...-0 A.S.6-7: Regression...-9 A.S.8: Correlation Coefficient A.S.5: Normal Distributions A.S.0: Permutations A.S.: Combinations A.S.9: Differentiating Permutations and Combinations A.S.: Sample Space A.S.: Geometric Probability A.S.5: Binomial Probability 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... 8 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 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 i

3 RADICALS RATIONALS FUNCTIONS SEQUENCES AND SERIES TRIGONOMETRY A.N.: Operations with Irrational Expressions 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 A.N.7: Imaginary Numbers A.N.8: Conjugates of Complex Numbers A.N.9: Multiplication and Division of Complex Numbers A.A.6: Multiplication and Division of Rationals A.A.6: Addition and Subtraction of Rationals A.A.: Solving Rationals and Rational Inequalities A.A.7: Complex Fractions... - A.A.5: Inverse Variation A.A.0-: Functional Notation A.A.5: Families of Functions... 5 A.A.6: Properties of Graphs of Functions and Relations A.A.5: Identifying the Equation of a Graph A.A.7, 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 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.60: Finding the Terminal Side of an Angle 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 A.A.67: Simplifying & Proving Trigonometric Identities A.A.76: Angle Sum and Difference Identities ii

4 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 iii

5 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 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. 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 A market research firm needs to collect data on viewer preferences for local news programming in Buffalo. Which method of data collection is most appropriate? census survey observation controlled experiment 5 A school cafeteria has five different lunch periods. The cafeteria staff wants to find out which items on the menu are most popular, so they give every student in the first lunch period a list of questions to answer in order to collect data to represent the school. Which type of study does this represent? observation controlled experiment population survey sample survey 6 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 7 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

6 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 Which survey is least likely to contain bias? surveying a sample of people leaving a movie theater to determine which flavor of ice cream is the most popular surveying the members of a football team to determine the most watched TV sport surveying a sample of people leaving a library to determine the average number of books a person reads in a year surveying a sample of people leaving a gym to determine the average number of hours a person exercises per week 9 A survey is to be conducted in a small upstate village to determine whether or not local residents should fund construction of a skateboard park by raising taxes. Which segment of the population would provide the most unbiased responses? a club of local skateboard enthusiasts senior citizens living on fixed incomes a group opposed to any increase in taxes every tenth person 8 years of age or older walking down Main St. 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. What is the value of k for this table? 9 8 A.S.: DISPERSION The table below shows the first-quarter averages for Mr. Harper s statistics class. A.S.: AVERAGE KNOWN WITH MISSING DATA 0 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 What is the population variance for this set of data?

7 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The scores of one class on the Unit mathematics test are shown in the table below. 7 The following is a list of the individual points scored by all twelve members of the Webster High School basketball team at a recent game: Find the interquartile range for this set of data. 8 The table below shows five numbers and their frequency of occurrence. Find the population standard deviation of these scores, to the nearest tenth. 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 The interquartile range for these data is to 6 to 9 The table below shows the final examination scores for Mr. Spear s class last year. Determine the number of employees whose travel time is within one standard deviation of the mean. 5 The heights, in inches, of 0 high school varsity basketball players are 78, 79, 79, 7, 75, 7, 7, 7, 8, and 7. Find the interquartile range of this data set. 6 Ten teams competed in a cheerleading competition at a local high school. Their scores were 9, 8, 9, 7, 5, 0,, 8, 7, and 8. How many scores are within one population standard deviation from the mean? For these data, what is the interquartile range? Find the population standard deviation based on these data, to the nearest hundredth. Determine the number of students whose scores are within one population standard deviation of the mean.

8 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 The table below displays the number of siblings of each of the 0 students in a class. 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. What is the population standard deviation, to the nearest hundredth, for this group?....5 The table below shows the number of new stores in a coffee shop chain that opened during the years 986 through 99. A.S.6-7: REGRESSION Samantha constructs the scatter plot below from a set of data. 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. Based on her scatter plot, which regression model would be most appropriate? exponential linear logarithmic power

9 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 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. 6 The data collected by a biologist showing the growth of a colony of bacteria at the end of each hour are displayed in the table below. Write an exponential regression equation to model these data. Round all values to the nearest thousandth. Assuming this trend continues, use this equation to estimate, to the nearest ten, the number of bacteria in the colony at the end of 7 hours. 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. 7 The table below shows the concentration of ozone in Earth s atmosphere at different altitudes. Write the exponential regression equation that models these data, rounding all values to the nearest thousandth. 5 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. 8 The table below shows the amount of a decaying radioactive substance that remained for selected years after 990. Write an exponential regression equation for the data, rounding all values to the nearest thousandth. Write an exponential regression equation for this set of data, rounding all values to the nearest thousandth. Using this equation, determine the amount of the substance that remained in 00, to the nearest integer. 5

10 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 The table below gives the relationship between x and y. As shown in the table below, a person s target heart rate during exercise changes as the person gets older. Use exponential regression to find an equation for y as a function of x, rounding all values to the nearest hundredth. Using this equation, predict the value of x if y is 6., rounding to the nearest tenth. [Only an algebraic solution can receive full credit.] A.S.8: CORRELATION COEFFICIENT 0 Which value of r represents data with a strong negative linear correlation between two variables? Which calculator output shows the strongest linear relationship between x and y? 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? The relationship between t, a student s test scores, and d, the student s success in college, is modeled by the equation d = 0.8t Based on this linear regression model, the correlation coefficient could be between and 0 between 0 and equal to equal to 0 Which value of r represents data with a strong positive linear correlation between two variables?

11 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Determine which set of data given below has the stronger linear relationship between x and y. Justify your choice. 6 A study compared the number of years of education a person received and that person's average yearly salary. It was determined that the relationship between these two quantities was linear and the correlation coefficient was 0.9. Which conclusion can be made based on the findings of this study? There was a weak relationship. There was a strong relationship. There was no relationship. There was an unpredictable relationship. 7 Which statement regarding correlation is not true? The closer the absolute value of the correlation coefficient is to one, the closer the data conform to a line. A correlation coefficient measures the strength of the linear relationship between two variables. A negative correlation coefficient indicates that there is a weak relationship between two variables. A relation for which most of the data fall close to a line is considered strong. A.S.5: NORMAL DISTRIBUTIONS 8 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 9 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? 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. 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. 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 certain high school, a survey revealed the mean amount of bottled water consumed by students each day was 5 bottles with a standard deviation of bottles. Assuming the survey represented a normal distribution, what is the range of the number of bottled waters that approximately 68.% of the students drink?

12 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Liz has applied to a college that requires students to score in the top 6.7% on the mathematics portion of an aptitude test. The scores on the test are approximately normally distributed with a mean score of 576 and a standard deviation of 0. What is the minimum score Liz must earn to meet this requirement? In a certain school, the heights of the population of girls are normally distributed, with a mean of 6 inches and a standard deviation of inches. If there are 50 girls in the school, determine how many of the girls are shorter than 60 inches. Round the answer to the nearest integer. 6 On a test that has a normal distribution of scores, a score of 57 falls one standard deviation below the mean, and a score of 8 is two standard deviations above the mean. Determine the mean score of this test. 7 The scores on a standardized exam have a mean of 8 and a standard deviation of.6. Assuming a normal distribution, a student's score of 9 would rank below the 75 th percentile between the 75 th and 85 th percentiles between the 85 th and 95 th percentiles above the 95 th percentile 8 The scores of 000 students on a standardized test were normally distributed with a mean of 50 and a standard deviation of 5. What is the expected number of students who had scores greater than 60? PROBABILITY A.S.0: PERMUTATIONS 9 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!!! 50 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. 5 Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 5 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 How many different six-letter arrangements can be made using the letters of the word TATTOO? Find the number of possible different 0-letter arrangements using the letters of the word STATISTICS. 8

13 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 55 Which expression represents the total number of different -letter arrangements that can be made using the letters in the word MATHEMATICS?!!!!+!+!! 8!!!!! 56 The number of possible different -letter arrangements of the letters in the word TRIGONOMETRY is represented by!!! 6! P 8 P 6! 57 How many different -letter arrangements are possible using the letters in the word ARRANGEMENT?,9,800,989,600 9,958,00 9,96, What is the total number of different nine-letter arrangements that can be formed using the letters in the word TENNESSEE?,780 5,0 5,60 6, How many distinct ways can the eleven letters in the word "TALLAHASSEE" be arranged? 8,600,66,00,6,00 5,70,00 60 Determine how many eleven-letter arrangements can be formed from the word "CATTARAUGUS." A.S.: COMBINATIONS 6 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 6 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 6 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. 6 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 65 How many different ways can teams of four members be formed from a class of 0 students? 5 80,85 6,80 9

14 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 66 A customer will select three different toppings for a supreme pizza. If there are nine different toppings to choose from, how many different supreme pizzas can be made? A.S.9: DIFFERENTIATING BETWEEN PERMUTATIONS AND COMBINATIONS 67 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 68 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 69 There are eight people in a tennis club. Which expression can be used to find the number of different ways they can place first, second, and third in a tournament? 8 P 8 C 8 P 5 8 C 5 70 Which problem involves evaluating 6 P? How many different four-digit ID numbers can be formed using,,,, 5, and 6 without repetition? How many different subcommittees of four can be chosen from a committee having six members? How many different outfits can be made using six shirts and four pairs of pants? How many different ways can one boy and one girl be selected from a group of four boys and six girls? 7 A math club has 0 boys and 0 girls. Which expression represents the total number of different 5-member teams, consisting of boys and girls, that can be formed? 0 P 0 P 0 C 0 C 0 P + 0 P 0 C + 0 C 7 A video-streaming service can choose from six half-hour shows and four one-hour shows. Which expression could be used to calculate the number of different ways the service can choose four half-hour shows and two one-hour shows? 6 P P 6 P + P 6 C C 6 C + C 0

15 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 Six people met at a dinner party, and each person shook hands once with everyone there. Which expression represents the total number of handshakes? 6! 6!! 6!! 6!!! A.S.: GEOMETRIC PROBABILITY 76 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. A.S.: SAMPLE SPACE 7 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. 75 A school math team consists of three juniors and five seniors. How many different groups can be formed that consist of one junior and two seniors? 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 77 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?

16 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 78 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. 79 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. 80 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. 8 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? 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 On a multiple-choice test, Abby randomly guesses on all seven questions. Each question has four choices. Find the probability, to the nearest thousandth, that Abby gets exactly three questions correct. 8 Because Sam s backyard gets very little sunlight, the probability that a geranium planted there will flower is 0.8. Sam planted five geraniums. Determine the probability, to the nearest thousandth, that at least four geraniums will flower. 85 Whenever Sara rents a movie, the probability that it is a horror movie is Of the next five movies she rents, determine the probability, to the nearest hundredth, that no more than two of these rentals are horror movies. 86 The probability of Ashley being the catcher in a softball game is. Calculate the exact probability 5 that she will be the catcher in exactly five of the next six games. 87 The probability that Kay and Joseph Dowling will have a redheaded child is out of. If the Dowlings plan to have three children, what is the exact probability that only one child will have red hair?

17 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 88 The probability of winning a game is. Determine the probability, expressed as a fraction, of winning exactly four games if seven games are played. 89 In the diagram below, the spinner is divided into eight equal regions. 9 What is the solution set of x = x + 0? { } { 6} {, 6} 9 Which graph represents the solution set of 6x 7 5? Which expression represents the probability of the spinner landing on B exactly three times in five spins? 5 C C C C ABSOLUTE VALUE A.A.: ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 9 Solve the inequality 6 x < 5 for x. Graph the solution on the line below. 9 Which graph represents the solution set of x 5 >? 90 What is the solution set of the equation a + 6 a = 0? 0 0,

18 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 95 Determine the solution of the inequality x 7. [The use of the grid below is optional.] QUADRATICS A.A.0-: ROOTS OF QUADRATICS 0 Find the sum and product of the roots of the equation 5x + x = 0. 0 What are the sum and product of the roots of the equation 6x x = 0? sum = ; product = sum = ; product = sum = ; product = sum = ; product = 0 Determine the sum and the product of the roots of x = x What is the graph of the solution set of x >5? 97 Solve x + 5 < algebraically for x. 98 Solve x >5 algebraically. 99 Solve algebraically for x: x 5 x < 7 00 Which graph is the solution to the inequality x < 7? 0 Determine the sum and the product of the roots of the equation x + x 6 = What is the product of the roots of the quadratic equation x 7x = 5? What is the product of the roots of x 5x =? Given the equation x + x + k = 0, state the sum and product of the roots.

19 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 08 Which statement about the equation x + 9x = 0 is true? The product of the roots is. The product of the roots is. The sum of the roots is. The sum of the roots is What is the sum of the roots of the equation x + 6x = 0? 0 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 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 Write a quadratic equation such that the sum of its roots is 6 and the product of its roots is 7. 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 What is the product of the roots of x x + k = 0 if one of the roots is 7? 77 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 ) 6 Factored completely, the expression x + 0x x is equivalent to x (x + 6)(x ) (x + x)(x x) x (x )(x + ) x (x + )(x ) 7 Factor completely: 0ax ax 5a A.A.7: FACTORING THE DIFFERENCE OF PERFECT SQUARES 8 Factor the expression t 8 75t completely. A.A.7: FACTORING BY GROUPING 9 When factored completely, x + x x equals (x + )(x )(x ) (x + )(x )(x + ) (x )(x + ) (x )(x ) 5

20 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 When factored completely, the expression x 5x 8x + 80 is equivalent to (x 6)(x 5) (x + 6)(x 5)(x + 5) (x + )(x )(x 5) (x + )(x )(x 5)(x 5) The expression x (x + ) (x + ) is equivalent to x x x + x x + (x + )(x )(x + ) When factored completely, the expression x x 9x + 8 is equivalent to (x 9)(x ) (x )(x )(x + ) (x ) (x )(x + ) (x ) (x ) Factor completely: x 6x 5x + 50 Factor completely: x + x + x + 6 A.A.5: QUADRATIC FORMULA 5 The solutions of the equation y y = 9 are ± i ± i 5 ± 5 ± 5 6 The roots of the equation x + 7x = 0 are and and 7 ± 7 7 ± 7 7 Solve the equation 6x x = 0 and express the answer in simplest radical form. 8 A cliff diver on a Caribbean island jumps from a height of 05 feet, with an initial upward velocity of 5 feet per second. An equation that models the height, h(t), above the water, in feet, of the diver in time elapsed, t, in seconds, is h(t) = 6t + 5t How many seconds, to the nearest hundredth, does it take the diver to fall 5 feet below his starting point? A homeowner wants to increase the size of a rectangular deck that now measures feet by feet. The building code allows for a deck to have a maximum area of 800 square feet. If the length and width are increased by the same number of feet, find the maximum number of whole feet each dimension can be increased and not exceed the building code. [Only an algebraic solution can receive full credit.] A.A.: USING THE DISCRIMINANT 0 Use the discriminant to determine all values of k that would result in the equation x kx + = 0 having equal roots. 6

21 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The roots of the equation 9x + x = 0 are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal The roots of the equation x 0x + 5 = 0 are imaginary real and irrational real, rational, and equal real, rational, and unequal The discriminant of a quadratic equation is. The roots are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal The roots of the equation x + = 9x are real, rational, and equal real, rational, and unequal real, irrational, and unequal imaginary 5 For which value of k will the roots of the equation x 5x + k = 0 be real and rational numbers? Which equation has real, rational, and unequal roots? x + 0x + 5 = 0 x 5x + = 0 x x + = 0 x x + 5 = 0 7 The roots of x + x = are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal 8 The roots of the equation (x ) = x are imaginary real, rational, equal real, rational, unequal real, irrational, unequal A.A.: COMPLETING THE SQUARE 9 Solve x x + = 0 by completing the square, expressing the result in simplest radical form. 0 If x + = 6x is solved by completing the square, an intermediate step would be (x + ) = 7 (x ) = 7 (x ) = (x 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? Max solves a quadratic equation by completing the square. He shows a correct step: (x + ) = 9 What are the solutions to his equation? ± i ± i ± i ± i 7

22 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Which step can be used when solving x 6x 5 = 0 by completing the square? x 6x + 9 = x 6x 9 = 5 9 x 6x + 6 = x 6x 6 = 5 6 If x = x 7 is solved by completing the square, one of the steps in the process is (x 6) = (x + 6) = (x 6) = 9 (x + 6) = 9 5 Which value of k will make x x + k a perfect square trinomial? A.A.: QUADRATIC INEQUALITIES 7 Which graph best represents the inequality y + 6 x x? 6 Find the exact roots of x + 0x 8 = 0 by completing the square. 8

23 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 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 > } 9 Find the solution of the inequality x x > 5, algebraically. 50 What is the solution of the inequality 9 x < 0? {x < x < } {x x > or x < } {x x > } {x x < } SYSTEMS A.A.: QUADRATIC-LINEAR SYSTEMS 5 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 5 Solve the following systems of equations algebraically: 5 = y x x = 7x + y + 5 Which ordered pair is a solution of the system of equations shown below? x + y = 5 (,) (5,0) ( 5,0) (,9) (x + ) + (y ) = 5 5 Which ordered pair is in the solution set of the system of equations shown below? y x + = 0 (,6) (,) (, ) ( 6, ) y x = 0 55 Determine algebraically the x-coordinate of all points where the graphs of xy = 0 and y = x + intersect. 56 What is the total number of points of intersection of the graphs of the equations x y = 8 and y = x +? 0 POWERS A.N.: OPERATIONS WITH POLYNOMIALS 57 Express x as a trinomial. 58 When x x is subtracted from 5 x x +, the difference is x + x 5 x x + 5 x x x x 59 Express the product of y y and y + 5 as a trinomial. 9

24 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic x 60 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 6 When x + x is subtracted from x + x x, the difference is x + x 5x + x + x + x x + x + x x x + 5x + 6 The expression x + x x equivalent to 0 x x x is 6 When 7 8 x x is subtracted from 5 8 x x +, the difference is x x + x x + x + x + x x 65 Find the difference when x 5 8 x x is subtracted from x + x 9. A.N., A.8-9: NEGATIVE AND FRACTIONAL EXPONENTS 66 If a = and b =, what is the value of the expression a b? If n is a negative integer, then which statement is always true? 6n < n n > 6n 6n < n n > (6n) 0

25 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 68 What is the value of x + x 0 + x when x = 6? When simplified, the expression equivalent to w 7 w w 7 w w 5 w 9 70 Which expression is equivalent to 9x y 6? xy xy xy xy 7 Which expression is equivalent to x x x 9x 9x is? 7 The expression (a) is equivalent to 8a 6 a a 6a 7 The expression a b is equivalent to a b a 6 b 5 b 5 a 6 a b a b 7 When x is divided by x, the quotient is x x (x ) x y 5 75 Simplify the expression and write the (x y 7 ) answer using only positive exponents. 76 When x + is divided by x +, the quotient equals x x x

26 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 77 Which expression is equivalent to x y x y 5 x 5 y x y 5 y x 5 x 5 y? 78 Which expression is equivalent to x y y x y x x y x y y 5? 79 Which expression is equivalent to (5 a b )? 0b a 5b a a 5b a 5b 5 80 Which expression is equivalent to x y x y x y 6 y x y 6 x x y? A.A.: EVALUATING EXPONENTIAL EXPRESSIONS 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. 8 Evaluate e xlny 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 %.

27 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 If $5000 is invested at a rate of % interest compounded quarterly, what is the value of the investment in 5 years? (Use the formula A = P + r nt, where A is the amount accrued, P n is the principal, r is the interest rate, n is the number of times per year the money is compounded, and t is the length of time, in years.) $590. $ $ $ The formula to determine 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. Which equation could be used to determine the value of an account with an $8,000 initial investment, at an interest rate of.5% for months? A = 8,000e.5.5 A = 8,000e A = 8,000e A = 8,000e 86 A population, p(x), of wild turkeys in a certain area is represented by the function p(x) = 7(.5) x, where x is the number of years since 00. How many more turkeys will be in the population for the year 05 than 00? Yusef deposits $50 into a savings account that pays.5% interest compounded quarterly. The amount, A, in his account can be determined by the formula A = P + r nt, where P is the initial n amount invested, r is the interest rate, n is the number of times per year the money is compounded, and t is the number of years for which the money is invested. What will his investment be worth in years if he makes no other deposits or withdrawals? $55.0 $7.7 $. $ The amount of money in an account can be determined by the formula A = Pe rt, where P is the initial investment, r is the annual interest rate, and t is the number of years the money was invested. What is the value of a $5000 investment after 8 years, if it was invested at % interest compounded continuously? $967.0 $ $0,9.08 $0,7.7 A.A.8: EVALUATING LOGARITHMIC EXPRESSIONS 89 The expression log 8 6 is equivalent to 8 8

28 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 90 The expression log 5 5 is equivalent to 9 On the axes below, for x, graph y = x +. A.A.5: GRAPHING EXPONENTIAL FUNCTIONS 9 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.

29 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 An investment is earning 5% interest compounded quarterly. The equation represents the total amount of money, A, where P is the original investment, r is the interest rate, t is the number of years, and n represents the number of times per year the money earns interest. Which graph could represent this investment over at least 50 years? A.A.5: GRAPHING EXPONENTIAL FUNCTIONS 9 If a function is defined by the equation f(x) = x, which graph represents the inverse of this function? 5

30 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 95 Which graph represents the function log x = y? 96 Which sketch shows the inverse of y = a x, where a >? A.A.9: PROPERTIES OF LOGARITHMS 97 The expression log x ( log y + log z) is equivalent to log x y z log x z y log x yz log xz y 6

31 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A B 98 If r =, then log r can be represented by C 6 log A + log B log C (loga + log B log C) log(a + B) C log A + log B log C 99 If log x log a = log a, then log x expressed in terms of log a is equivalent to log 5a log 6 + log a log6 + loga log6 + loga 00 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 0 If log = a and log = b, the expression log 9 0 is equivalent to b a + b a b a + 0 b a + 0 The expression log m is equivalent to (log + logm) log + logm log + logm log6 + logm 0 If x = y, then log y equals log(x) + log log(x) log + logx log + logx 0 If log x = log a + log b, then x equals a b ab a + b a + b 05 If T = 0x, then log T is equivalent to y ( + logx) logy log( + x) logy ( logx) + logy ( logx) + logy A.A.8: LOGARITHMIC EQUATIONS 06 What is the solution of the equation log (5x) =? Solve algebraically for x: log x + x + x x = 7

32 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 08 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.] 09 What is the value of x in the equation log 5 x =? ,0 0 If log x =.5 and log y 5 =, find the numerical value of x, in simplest form. y Solve algebraically for all values of x: log (x + ) (7x ) = Solve algebraically for x: log 7 (x ) = Solve algebraically for all values of x: log (x + ) (x + ) + log (x + ) (x + 5) = Solve algebraically for x: log 5x = 6 If log x + 6 =, find the value of x. 7 Solve algebraically, to the nearest hundredth, for all values of x: log (x 7x + ) log (x 0) = 8 Solve algebraically for the exact value of x: log 8 6 = x + A.A.6, 7: EXPONENTIAL EQUATIONS 9 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? 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? The number of bacteria present in a Petri dish can be modeled by the function N = 50e t, where N is the number of bacteria present in the Petri dish after t hours. Using this model, determine, to the nearest hundredth, the number of hours it will take for N to reach 0,700. t 60 5 The equation log a x = y where x > 0 and a > is equivalent to x y = a y a = x a y = x a x = y 8

33 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Susie invests $500 in an account that is compounded continuously at an annual interest rate of 5%, according to the formula A = Pe rt, where A is the amount accrued, P is the principal, r is the rate of interest, and t is the time, in years. Approximately how many years will it take for Susie s money to double? The solution set of x + x = 6 is {,} {,} {, } {, } What is the value of x in the equation 9 x + = 7 x +? 5 Solve algebraically for x: 6 x + = 6 x + 6 The value of x in the equation x + 5 = 8 x is Solve algebraically for all values of x: 8 x + x = 7 5x 8 Which value of k satisfies the equation 8 k + = k? Solve e x = algebraically for x, rounded to the nearest hundredth. 0 Solve algebraically for x: 5 x = 5 x Solve for x: 6 = x A.A.6: BINOMIAL EXPANSIONS What is the fourth term in the expansion of (x ) 5? 70x 0x 70x,080x Write the binomial expansion of (x ) 5 as a polynomial in simplest form. What is the coefficient of the fourth term in the expansion of (a b) 9? 5, ,76 5 Which expression represents the third term in the expansion of (x y)? y 6x y 6x y x y 9

34 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 What is the middle term in the expansion of 6 x y? 0x y 5 x y 0x y 5 x y 7 What is the fourth term in the binomial expansion (x ) 8? 8x 5 8x 8x 5 8x What is the solution set of the equation x 5 8x = 0? {0,±} {0,±,} {0,±,±i} {±,±i} Solve algebraically for all values of x: x + x + x = 6x Solve x + 5x = x + 0 algebraically. 5 Solve the equation x x 8x + = 0 algebraically for all values of x. 6 The graph of y = f(x) is shown below. 8 What is the third term in the expansion of (x ) 5? 70x 80x 50x 080x 9 The ninth term of the expansion of (x + y) 5 is 5 C 9 (x)6 (y) 9 5 C 9 (x)9 (y) 6 5 C 8 (x)7 (y) 8 5 C 8 (x)8 (y) 7 A.A.6, 50: SOLVING POLYNOMIAL EQUATIONS 0 Solve the equation 8x + x 8x 9 = 0 algebraically for all values of x. Which values of x are solutions of the equation x + x x = 0? 0,, 0,, 0,, 0,, Which set lists all the real solutions of f(x) = 0? {,} {,} {,0,} {,0,} 0

35 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 The graph of y = x x + x + 6 is shown below. 9 What are the zeros of the polynomial function graphed below? 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 {,,} {,, } {, 8} { 6} RADICALS A.N.: OPERATIONS WITH IRRATIONAL EXPRESSIONS 50 The product of ( + 5) and ( 5) is A.A.: SIMPLIFYING RADICALS 5 Express in simplest form: a 6 b The expression 6a 6 is equivalent to 8a 8a 8 a 5 a a a 5

36 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.N., A.: OPERATIONS WITH RADICALS 5 Express 5 x 7x in simplest radical form. 5 The sum of 6a b and 6a b, expressed in simplest radical form, is 6 68a 8 b a b a b a 6ab 0a b 8 55 The expression 7x to x x x 6x x 6x 6x is equivalent 56 What is the product of a b and 6a b? ab a a b a 8ab a 8a b a 59 The expression x is equivalent to 9x x x + 9x x + 6x 60 The expression 7a 6b 8 is equivalent to 6ab 6ab ab ab 6 The legs of a right triangle are represented by x + and x. The length of the hypotenuse of the right triangle is represented by x + x + x + x A.N.5, A.5: RATIONALIZING DENOMINATORS 5 6 Express with a rational denominator, in simplest radical form. 57 The expression ab b a 8b + 7ab 6b is equivalent to ab 6b 6ab b 5ab + 7ab 6b 5ab b + 7ab 6b 58 Express 08x 5 y 8 6xy 5 in simplest radical form. 6 Which expression is equivalent to ?

37 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 The expression 5 5 (5 ) (5 + ) 8 65 The expression The expression is equivalent to is equivalent to is equivalent to 67 The expression The fraction a b b ab b ab a 6 6 a b 69 The expression x + x + (x + ) x x (x + ) x x x x + is equivalent to is equivalent to is equivalent to

38 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 70 Expressed with a rational denominator and in x simplest form, x x is x + x x x x x + x x x + x x x A.A.: SOLVING RADICALS 7 The solution set of the equation x + = x is {} {0} {,6} {,} 7 The solution set of x + 6 = x + is {,} {,} {} { } 7 Solve algebraically for x: x 5 = 7 What is the solution set for the equation 5x + 9 = x +? {} { 5} {,5} { 5,} 75 Solve algebraically for x: x + x + x = 7x + 76 The solution set of the equation x = x is {, } {,} {} { } 77 Solve algebraically for x: x + + = 8 A.A.0-: EXPONENTS AS RADICALS 78 The expression (x ) (x ) (x ) (x ) (x ) 5 79 The expression x x 5 5 x 5 x 5 x is equivalent to is equivalent to 80 The expression 6x y 7 is equivalent to 7 y x x 8 y 8 7 y x x 8 y 8 8 The expression 8x y 5 is equivalent to x x 9xy 9xy 5 y 5 y 5 5

39 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 The expression 7a 6 b c is equivalent to bc a b 9 c 6 a 8 b 6 c 5 a b c a A.N.6: SQUARE ROOTS OF NEGATIVE NUMBERS 8 In simplest form, 00 is equivalent to i 0 5i 0i i 5 8 Expressed in simplest form, 8 is 7 i 7i 85 The expression 80x 6 is equivalent to 6x 5 6x 8 5 6x i 5 6x 8 i 5 A.N.7: IMAGINARY NUMBERS 86 The product of i 7 and i 5 is equivalent to i i 87 The expression i + i is equivalent to i i + i + i 88 Determine the value of n in simplest form: i + i 8 + i + n = 0 89 Express xi + 5yi 8 + 6xi + yi in simplest a + bi form. 90 Express xi 8 yi 6 in simplest form. 9 The expression x(i ) + xi is equivalent to x + 7xi 7x 5x 9x A.N.8: CONJUGATES OF COMPLEX NUMBERS 9 What is the conjugate of + i? + i i i + i 9 The conjugate of 7 5i is 7 5i 7 + 5i 7 5i 7 + 5i 9 What is the conjugate of + i? + i i + i i 5

40 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 95 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 96 The expression ( 7i) is equivalent to 0 + 0i 0 i i 58 i 97 The expression (x + i) (x i) is equivalent to 0 + xi xi 98 If x = i, y = i, and z = m + i, the expression xy z equals mi 6 6mi mi 6 6mi 99 Multiply x + yi by its conjugate, and express the product in simplest form. 00 When i is multiplied by its conjugate, the result is 5 5 RATIONALS A.A.6: MULTIPLICATION AND DIVISION OF RATIONALS 0 Perform the indicated operations and simplify completely: x x + 6x 8 x x x x x x + x 8 6 x 0 The expression x + 9x ( x) is equivalent x to x x x x A.A.6: ADDITION AND SUBTRACTION OF RATIONALS 0 Expressed in simplest form, equivalent to 6y + 6y 5 (y 6)(6 y) y 9 y 6 y y y is 0 If x is a real number, express xi(i i ) in simplest a + bi form. 6

41 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic x 05 What is x x fraction? x + x x x x + (x ) x (x ) expressed as a single Solve algebraically for the exact values of x: 5x = x + x Which graph represents the solution set of x + 6 x 7? A.A.: SOLVING RATIONALS AND RATIONAL INEQUALITIES 06 Solve for x: x x = + x 07 Solve algebraically for x: x + x = x 9 08 Solve the equation below algebraically, and express the result in simplest radical form: x = 0 x 09 What is the solution set of the equation 0 x 9 + = 5 x? {,} {} {} { } 0 Which equation could be used to solve 5 x x =? x 6x = 0 x 6x + = 0 x 6x 6 = 0 x 6x + 6 = 0 A.A.7: COMPLEX FRACTIONS Written in simplest form, the expression equivalent to x x x x x + 5 Express in simplest form: 6 Express in simplest form: d d + d x x + 7x + x x + x x x + is Solve algebraically for x: x + x x + = x + 7

42 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 The simplest form of x x + x x x x x 8 x is The expression x + y xy xy x + y x + y xy is equivalent to a + b c 8 The expression d b c c + d a + b d b ac + b cd b ac + cd 9 Express in simplest terms: 0 Express in simplest form: is equivalent to + x 5 x x 6 x (x + 6) x x + x 8 A.A.5: INVERSE VARIATION 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. If p varies inversely as q, and p = 0 when q =, what is the value of p when q = 5? The quantities p and q vary inversely. If p = 0 when q =, and p = x when q = x +, then x equals and and 8

43 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 The points (,),,, and (6,d) lie on the graph of a function. If y is inversely proportional to the square of x, what is the value of d? 7 6 If d varies inversely as t, and d = 0 when t =, what is the value of t when d = 5? If p and q vary inversely and p is 5 when q is 6, determine q when p is equal to 0. 8 Given y varies inversely as x, when y is multiplied by, then x is multiplied by 9 A scholarship committee rewards the school's top math students. The amount of money each winner receives is inversely proportional to the number of scholarship recipients. If there are three winners, they each receive $00. If there are eight winners, how much money will each winner receive? $067 $00 $0 $50 FUNCTIONS A.A.0-: FUNCTIONAL NOTATION 0 The equation y sinθ = may be rewritten as f(y) = sinx + f(y) = sinθ + f(x) = sinθ + f(θ) = sinθ + If fx = x, what is the value of f( 0)? x 6 If g(x) = ax x, express g(0) in simplest form. If fx = x x +, then f(a + ) equals a a + 6 a a + a + 7a + 6 a + 7a + If fx = x x +, then fx+ is equal to x x + 7 x x + x + 9x + x + 9x + 5 9

44 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.5: FAMILIES OF FUNCTIONS 5 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 6 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,). 7 Theresa is comparing the graphs of y = x and y = 5 x. Which statement is true? The y-intercept of y = x is (0,), and the y-intercept of y = 5 x is (0,5). Both graphs have a y-intercept of (0,), and y = x is steeper for x > 0. Both graphs have a y-intercept of (0,), and y = 5 x is steeper for x > 0. Neither graph has a y-intercept. A.A.5: IDENTIFYING THE EQUATION OF A GRAPH 8 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 0

45 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 Which equation is represented by the graph below? The table of values below can be modeled by which equation? f(x) = x + f(x) = x + f(y) = y + f(y) = y + y = 5 x y = 0.5 x y = 5 x y = 0.5 x 0 What is the equation of the graph shown below? A.A.7, 8, : DEFINING FUNCTIONS Given the relation {(8,),(,6),(7,5),(k,)}, which value of k will result in the relation not being a function? y = x y = x x = y x = y

46 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Which graph does not represent a function? 5 Which graph does not represent a function? Which relation is not a function? (x ) + y = x + x + y = x + y = xy =

47 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 Which graph represents a relation that is not a function? 7 Which graph represents a function?

48 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 Which statement is true about the graphs of f and g shown below? 9 Which relation does not represent a function? f is a relation and g is a function. f is a function and g is a relation. Both f and g are functions. Neither f nor g is a function. 50 Which function is not one-to-one? {(0,),(,),(,),(,)} {(0,0),(,),(,),(,)} {(0,),(,0),(,),(,)} {(0,),(,0),(,0),(,)}

49 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Which graph represents a one-to-one function? 5 Which diagram represents a relation that is both one-to-one and onto? 55 Which relation is both one-to-one and onto? 5 Which function is one-to-one? f(x) = x f(x) = x f(x) = x f(x) = sinx 5 Which function is one-to-one? k(x) = x + g(x) = x + f(x) = x + j(x) = x + 5

50 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 56 Which list of ordered pairs does not represent a one-to-one function? (, ),(,0),(,),(,) (,),(,),(,),(,6) (,),(,),(,),(,) (,5),(,),(,),(,0) A.A.9, 5: DOMAIN AND RANGE 57 What is the domain of the function f(x) = x +? (, ) (, ) [, ) [, ) 58 What is the range of f(x) = (x + ) + 7? y y y = 7 y 7 6 The domain of f(x) = numbers greater than less than except between and is the set of all real x 6 What is the domain of the function gx = x -? (,] (,) (, ) (, ) 6 What are the domain and the range of the function shown in the graph below? 59 What is the range of f(x) = x +? {x x } {y y } {x x real numbers} {y y real numbers} 60 If f(x) = 9 x, what are its domain and range? domain: {x x }; range: {y 0 y } domain: {x x ±}; range: {y 0 y } domain: {x x or x }; range: {y y 0} domain: {x x }; range: {y y 0} 6 For y =, what are the domain and range? x {x x > } and {y y > 0} {x x } and {y y > 0} {x x > } and {y y 0} {x x } and {y y 0} {x x > }; {y y > } {x x }; {y y } {x x > }; {y y > } {x x }; {y y } 6

51 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 65 The graph below represents the function y = f(x). 67 What is the range of the function shown below? x 0 x 0 y 0 y 0 State the domain and range of this function. 68 The graph below shows the average price of gasoline, in dollars, for the years 997 to What is the domain of the function shown below? What is the approximate range of this graph? 997 x x y.8.7 y.8 x 6 y 6 x 5 y 5 7

52 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 69 Which value is in the domain of the function graphed below, but is not in its range? 0 7 A.A.: COMPOSITIONS OF FUNCTIONS 70 If f(x) = x and g(x) = x + 5, what is the value of (g f)()? 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 7 If f(x) = x 6 and g(x) = x, determine the value of (g f)( ). 7 If f(x) = x x and g(x) = x, then (f g) is equal to Which expression is equivalent to (n m p)(x), given m(x) = sin x, n(x) = x, and p(x) = x? sin(x) sin x sin (x) sin x 75 If g(x) = x + 8 and h(x) = x, what is the value of g(h( 8))? If f(x) = x x + and g(x) = x + 5, what is f(g(x))? x + 7x + 6 x + 7x + 66 x x + 6 x x If f(x) = x + and g(x) = x, what is the value of f(g( ))? If f(x) = x x and g(x) = x +, determine f(g(x)) in simplest form. 8

53 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: INVERSE OF FUNCTIONS 79 Which two functions are inverse functions of each other? f(x) = sin x and g(x) = cos(x) f(x) = + 8x and g(x) = 8x f(x) = e x and g(x) = ln x A.A.6: TRANSFORMATIONS WITH FUNCTIONS AND RELATIONS 8 The graph below shows the function f(x). f(x) = x and g(x) = x + 80 If f(x) = x 6, find f (x). 8 What is the inverse of the function f(x) = log x? f (x) = x f (x) = x f (x) = log x f (x) = log x 8 If m = {(,),(,),(,),(,),(,9),(,9)}, which statement is true? m and its inverse are both functions. m is a function and its inverse is not a function. m is not a function and its inverse is a function. Neither m nor its inverse is a function. Which graph represents the function f(x + )? 9

54 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 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, ) 85 The function f(x) is graphed on the set of axes below. On the same set of axes, graph f(x + ) +. SEQUENCES AND SERIES A.A.9-: SEQUENCES 87 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 88 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 86 Which transformation of y = f(x) moves the graph 7 units to the left and units down? y = f(x + 7) y = f(x + 7) + y = f(x 7) y = f(x 7) + 89 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 90 In an arithmetic sequence, a = 9 and a 7 =. Determine a formula for a n, the n th term of this sequence. 9 A theater has 5 seats in the first row. Each row has four more seats than the row before it. Which expression represents the number of seats in the nth row? 5 + (n + ) 5 + (n) 5 + (n + )() 5 + (n )() 50

55 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 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,... } 9 What is the common difference in the sequence a +, a +, 6a + 7, 8a + 0,...? a + a a + 5 a What is the common difference of the arithmetic sequence below? 7x, x, x,x,5x,... x x 96 Given the sequence: x,(x + y),(x + y),... Which expression can be used to determine the common difference of this sequence? x (x + y) (x + y) (x + y) x (x + y) (x + y) (x + y) 97 What is the common ratio of the geometric sequence whose first term is 7 and fourth term is 6? What is the common ratio of the geometric sequence shown below?,, 8,6, What is the common ratio of the sequence 6 a 5 b, a b, 9 6 ab5,...? b a 6b a a b 6a b 00 The common ratio of the sequence,, 9 8 is 5

56 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 What is the fifteenth term of the sequence 5, 0,0, 0,80,...? 6,80 8,90 8,90 7,680 0 What is the fifteenth term of the geometric sequence 5, 0, 5,...? An arithmetic sequence has a first term of 0 and a sixth term of 0. What is the 0th term of this sequence? Find the first four terms of the recursive sequence defined below. a = a n = a (n ) n 05 Find the third term in the recursive sequence a k + = a k, where a =. 06 Use the recursive sequence defined below to express the next three terms as fractions reduced to lowest terms. a = a n = a n 07 What is the fourth term of the sequence defined by a = xy 5 a n = x y a? n x y x y x y 8x 5 y 08 The first four terms of the sequence defined by a = and a n + = a n are,,,,,,,, 8, 6,,, 09 What is the third term of the recursive sequence below? a = 6 5 a n = a n n A.N.0, A.: SIGMA NOTATION 0 The value of the expression (n + n ) is 6 n = 0 5

57 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Evaluate: 0 + (n ) 5 n = The value of the expression ( r + r) is Evaluate: ( n n) n = 5 r = The expression + (k x) is equal to 58 x 6 x 58 x 6 x 5 k = 5 Which expression is equivalent to (a n)? a + 7 a + 0 a 0a + 7 a 0a What is the value of ( a) x? a a + a a + a a + a a + x = 0 7 Simplify: x a. a = n = 8 What is the value of cos nπ? 0 n = 9 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 = 0 Express the sum using sigma notation. Which summation represents ? n n = 5 0 (n + ) n = (n ) n = (n ) n = 5

58 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A jogger ran mile on day, and mile on day, and miles on day, and miles on day, and this pattern continued for more days. Which expression represents the total distance the jogger ran? 7 () d d = 7 d = () d 7 d = 7 d = d d Which expression is equivalent to the sum of the sequence 6,,0,0? n 0 7 n = 6 n = 5 n 5n n = 5 n + n n = A.A.5: SERIES 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? What is the sum of the first 9 terms of the sequence,0,7,,,...? Determine the sum of the first twenty terms of the sequence whose first five terms are 5,,,,. 7 The sum of the first eight terms of the series is,07,85 9, 65,55 TRIGONOMETRY A.A.55: TRIGONOMETRIC RATIOS 8 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' 5

59 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 Which ratio represents csc A in the diagram below? In the diagram below, the length of which line segment is equal to the exact value of sin θ? In the diagram below of right triangle JTM, JT =, JM = 6, and m JMT = 90. TO TS OR OS In the right triangle shown below, what is the measure of angle S, to the nearest minute? What is the value of cot J? 8 ' 8 ' 6 56' 6 9' By law, a wheelchair service ramp may be inclined no more than.76. If the base of a ramp begins 5 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building s entrance? sin.76 = h 5 sin.76 = 5 h tan.76 = h 5 tan.76 = 5 h 55

60 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.M.-: RADIAN MEASURE 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 5 The terminal side of an angle measuring π 5 radians lies in Quadrant I II III IV 6 Find, to the nearest minute, the angle whose measure is.5 radians. 7 What is the number of degrees in an angle whose radian measure is π? What is the radian measure of an angle whose measure is 0? 7π 7π 6 7π 6 7π 9 Find, to the nearest tenth of a degree, the angle whose measure is.5 radians. 0 What is the number of degrees in an angle whose measure is radians? 60 π π Find, to the nearest tenth, the radian measure of 6º. Convert radians to degrees and express the answer to the nearest minute. What is the number of degrees in an angle whose radian measure is 8π 5? Approximately how many degrees does five radians equal? π 6 5π 5 Convert.5 radians to degrees, and express the answer to the nearest minute. 6 Determine, to the nearest minute, the degree measure of an angle of 5 π radians. 7 Determine, to the nearest minute, the number of degrees in an angle whose measure is.5 radians. 8 Express 0 in radian measure, to the nearest hundredth. 56

61 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.60: UNIT CIRCLE, FINDING THE TERMINAL SIDE OF AN ANGLE 50 In which graph is θ coterminal with an angle of 70? 9 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. 57

62 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 If m θ = 50, which diagram represents θ drawn in standard position? 5 An angle, P, drawn in standard position, terminates in Quadrant II if cos P < 0 and csc P < 0 sin P > 0 and cos P > 0 csc P > 0 and cot P < 0 tanp < 0 and sec P > 0 5 If sin θ < 0 and cot θ > 0, in which quadrant does the terminal side of angle θ lie? I II III IV A.A.56, 6, 66: DETERMINING TRIGONOMETRIC FUNCTIONS 55 In the interval 0 x < 60, tanx is undefined when x equals 0º and 90º 90º and 80º 80º and 70º 90º and 70º 56 Express the product of cos 0 and sin 5 in simplest radical form. 5 Which angle does not terminate in Quadrant IV when drawn on a unit circle in standard position? If θ is an angle in standard position and its terminal side passes through the point (,), find the exact value of csc θ. 58 Angle θ is in standard position and (,0) is a point on the terminal side of θ. What is the value of sec θ? 0 undefined 58

63 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 59 Circle O has a radius of units. An angle with a measure of π radians is in standard position. If 6 the terminal side of the angle intersects the circle at point B, what are the coordinates of B?,,,, 60 If the terminal side of angle θ passes through point (, ), what is the value of sec θ? The value of csc 8 rounded to four decimal places is A.A.6: USING INVERSE TRIGONOMETRIC FUNCTIONS 6 What is the principal value of cos ? 65 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 6 The value of tan6 to the nearest ten-thousandth is Which expression, when rounded to three decimal places, is equal to.55? sec 5π 6 tan(9 0 ) sin π 5 csc( 8 ) What is m θ?

64 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 66 If sin 5 = A, then 8 sin A = 5 8 sin A = 8 5 cos A = 5 8 cos A = If tan Arccos k =, then k is 68 If sin A = 7 and A terminates in Quadrant IV, 5 tana equals What is the value of tan Arc cos 5 7? A.A.57: REFERENCE ANGLES 70 Expressed as a function of a positive acute angle, cos( 05 ) is equal to cos 55 cos 55 sin 55 sin 55 7 Expressed as a function of a positive acute angle, sin0 is equal to sin 0 sin 50 sin 0 sin 50 A.A.6: ARC LENGTH 7 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 7 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 60

65 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 Circle O shown below has a radius of centimeters. To the nearest tenth of a centimeter, determine the length of the arc, x, subtended by an angle of 8 50'. A.A.58-59: COFUNCTION AND RECIPROCAL TRIGONOMETRIC FUNCTIONS 78 If A is acute and tana =, then cot A = cot A = cot(90 A) = cot(90 A) = 75 A wheel has a radius of 8 inches. Which distance, to the nearest inch, does the wheel travel when it rotates through an angle of π 5 radians? 5 76 In a circle, an arc length of 6.6 is intercepted by a central angle of radians. Determine the length of the radius. 77 In a circle with a diameter of cm, a central angle of π radians intercepts an arc. The length of the arc, in centimeters, is 8π 9π 6π π 79 The expression sin θ + cos θ sin θ is equivalent to cos θ sin θ sec θ csc θ 80 Express cos θ(sec θ cos θ), in terms of sin θ. 8 If sec(a + 5) =csc(a), find the smallest positive value of a, in degrees. cot x sin x 8 Express as a single trigonometric sec x function, in simplest form, for all values of x for which it is defined. 8 The expression cot x is equivalent to csc x sinx cos x tanx sec x 8 Which trigonometric expression does not simplify to? sin x( + cot x) sec x( sin x) cos x(tan x ) cot x(sec x ) 6

66 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 85 Show that sec x sec x is equivalent to sin x. 86 If angles A and B are complementary, then sec B equals csc(90 B) csc(b 90 ) cos(b 90 ) cos(90 B) 87 Express the exact value of csc 60, with a rational denominator. 88 The exact value of csc 0 is A.A.67: SIMPLIFYING TRIGONOMETRIC EXPRESIONS, PROVING TRIGONOMETRIC IDENTITIES 89 Which expression always equals? cos x sin x cos x + sin x cos x sinx cos x + sinx 90 Starting with sin A + cos A =, derive the formula tan A + = sec A. 9 Show that sec θ sin θ cot θ = is an identity. 9 Prove that the equation shown below is an identity for all values for which the functions are defined: csc θ sin θ cot θ = cos θ A.A.76: ANGLE SUM AND DIFFERENCE IDENTITIES 9 The expression cos x cos x + sin x sin x is equivalent to sinx sin 7x cos x cos 7x 9 If tana = and sin B = 5 and angles A and B are in Quadrant I, find the value of tan(a + B). 95 Express as a single fraction the exact value of sin Given angle A in Quadrant I with sin A = and angle B in Quadrant II with cos B =, what is the 5 value of cos(a B)? The value of sin(80 + x) is equivalent to sinx sin(90 x) sinx sin(90 x) 98 The expression sin(θ + 90) is equivalent to sin θ cos θ sinθ cos θ 6

67 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 99 If sin x = sin y = a and cos x = cos y = b, then cos(x y) is b a b + a b a b + a A.A.77: DOUBLE AND HALF ANGLE IDENTITIES 500 The expression cos θ cos θ is equivalent to sin θ sin θ cos θ + cos θ 50 If sin A = where 0 <A < 90, what is the value of sin A? What is a positive value of tan x, when sin x = 0.8? If sin A =, what is the value of cos A? If sin A =, what is the value of cos A? The expression cot A tana sec A + cota + cos A sina is equivalent to 506 If cos θ =, then what is cos θ?

68 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.68: TRIGONOMETRIC EQUATIONS 507 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 508 Find all values of θ in the interval 0 θ < 60 that satisfy the equation sin θ = sin θ. 509 Solve the equation tanc = tanc algebraically for all values of C in the interval 0 C < What is the solution set for cos θ = 0 in the interval 0 θ < 60? {0,50 } {60,0 } {0,0 } {60,00 } 5 What is the solution set of the equation sec x = when 0 x < 60? {5,5,5,5 } {5,5 } {5,5 } {5,5 } 5 Find, algebraically, the measure of the obtuse angle, to the nearest degree, that satisfies the equation 5csc θ = 8. 5 Solve algebraically for all exact values of x in the interval 0 x < π : sin x + 5 sin x = 5 Solve sec x = 0 algebraically for all values of x in 0 x < Which values of x in the interval 0 x < 60 satisfy the equation sin x + sin x = 0? {0,70 } {0,50,70 } {90,0,0 } {90,0,70,0 } 57 Solve the equation cos x = cos x algebraically for all values of x in the interval 0 x < 60. A.A.69: PROPERTIES OF TRIGONOMETRIC FUNCTIONS 58 What is the period of the function y = sin x π? π 6π 59 What is the period of the function f(θ) = cos θ? π π π π 50 Which equation represents a graph that has a period of π? y = sin x y = sinx y = sin x y = sinx 55 In the interval 0 θ < 60, solve the equation 5cos θ = sec θ algebraically for all values of θ, to the nearest tenth of a degree. 6

69 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 What is the period of the graph y = sin6x? π 6 π π 6π 5 How many full cycles of the function y = sinx appear in radians? 5 What is the period of the graph of the equation y = sinx? π 6π 5 What are the amplitude and the period of the graph represented by the equation y = cos θ? A.A.7: IDENTIFYING THE EQUATION OF A TRIGONOMETRIC GRAPH 55 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 Write an equation for the graph of the trigonometric function shown below. amplitude: ; period: π amplitude: ; period: 6π amplitude: ; period: π amplitude: ; period: 6π 65

70 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 57 Which equation is represented by the graph below? 59 The periodic graph below can be represented by the trigonometric equation y = a cos bx + c where a, b, and c are real numbers. y = cos x y = sinx y = cos π x State the values of a, b, and c, and write an equation for the graph. y = sin π x 58 Which equation represents the graph below? y = sinx y = sin x y = cos x y = cos x 66

71 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.65, 70-7: GRAPHING TRIGONOMETRIC FUNCTIONS 5 Which graph shows y = cos x? 50 Which graph represents the equation y = cos x? 67

72 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Which graph represents one complete cycle of the equation y = sinπx? 5 Which equation is represented by the graph below? y = cotx y = csc x y = sec x y = tanx 5 Which equation is sketched in the diagram below? y = csc x y = sec x y = cotx y = tanx 68

73 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 55 Which is a graph of y = cot x? 57 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 58 Which statement regarding the inverse function is true? A domain of y = sin x is [0,π]. The range of y = sin x is [,]. A domain of y = cos x is (, ). The range of y = cos x is [0,π]. 59 When the inverse of tanθ is sketched, its domain is θ π θ π 0 θ π < θ < A.A.7: USING TRIGONOMETRY TO FIND AREA A.A.6: DOMAIN AND RANGE 56 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 50 In ABC, m A = 0, b = 0, and c = 8. What is the area of ABC to the nearest square inch? 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. 5 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?

74 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 In parallelogram BFLO, OL =.8, LF = 7., and m O = 6. If diagonal BL is drawn, what is the area of BLF? The two sides and included angle of a parallelogram are 8,, and 60. Find its exact area in simplest form. 55 The area of triangle ABC is. If AB = 8 and m B = 6, the length of BC is approximately A ranch in the Australian Outback is shaped like triangle ACE, with m A =, m E = 0, and AC = 5 miles. Find the area of the ranch, to the nearest square mile. 57 Find, to the nearest tenth of a square foot, the area of a rhombus that has a side of 6 feet and an angle of Two sides of a triangular-shaped sandbox measure feet and feet. If the angle between these two sides measures 55, what is the area of the sandbox, to the nearest square foot? What is the area of a parallelogram that has sides measuring 8 cm and cm and includes an angle of 0? Determine the area, to the nearest integer, of SRO shown below. A.A.7: LAW OF SINES 55 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. 55 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. 59 The area of a parallelogram is 59, and the lengths of its sides are and 6. Determine, to the nearest tenth of a degree, the measure of the acute angle of the parallelogram. 70

75 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 55 As shown in the diagram below, fire-tracking station A is 00 miles due west of fire-tracking station B. A forest fire is spotted at F, on a bearing 7 northeast of station A and 5 northeast of station B. Determine, to the nearest tenth of a mile, the distance the fire is from both station A and station B. [N represents due north.] 556 Given: DC = 0, AG = 5, BE = 6, FE = 0, m ABG = 0, m GBD = 90, m C < 90, BE ED, and GF FB 555 In PQR, p equals r sin P sin Q r sin P sin R r sin R sin P q sin R sin Q Find m A to the nearest tenth. Find BC to the nearest tenth. A.A.75: LAW OF SINES-THE AMBIGUOUS CASE 557 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 How many distinct triangles can be formed if m A = 5, a = 0, and b =? 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 7

76 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic In MNP, m = 6 and n = 0. Two distinct triangles can be constructed if the measure of angle M is In ABC, a = 5, b =, and c =, as shown in the diagram below. What is the m C, to the nearest degree? 56 In KLM, KL = 0, LM =, and m K = 0. The measure of M? must be between 0 and 90 must equal 90 must be between 90 and 80 is ambiguous 56 In DEF, d = 5, e = 8, and m D =. How many distinct triangles can be drawn given these measurements? 0 56 How many distinct triangles can be constructed if m A = 0, side a =, and side b =? one acute triangle one obtuse triangle two triangles none 56 In triangle ABC, determine the number of distinct triangles that can be formed if m A = 85, side a = 8, and side c =. Justify your answer. A.A.7: LAW OF COSINES 565 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. 566 In ABC, a =, b = 5, and c = 7. What is m C? Two sides of a parallelogram measure 7 cm and cm. The included angle measures 8. Find the length of the longer diagonal of the parallelogram, to the nearest centimeter. 569 In FGH, f = 6, g = 9, and m H = 57. Which statement can be used to determine the numerical value of h? h = (9)(h) cos 57 h = (6)(9) cos 57 6 = 9 + h (9)(h) cos 57 9 = 6 + h (6)(h) cos Find the measure of the smallest angle, to the nearest degree, of a triangle whose sides measure 8, 7, and. 57 In a triangle, two sides that measure 8 centimeters and centimeters form an angle that measures 8. To the nearest tenth of a degree, determine the measure of the smallest angle in the triangle. 7

77 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.7: VECTORS 57 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. 57 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. 57 Two forces of 0 pounds and 8 pounds act on an object. The angle between the two forces is 65. Find the magnitude of the resultant force, to the nearest pound. Using this answer, find the measure of the angle formed between the resultant and the smaller force, to the nearest degree. 578 Which equation represents a circle with its center at (, ) and that passes through the point (6,)? (x ) + (y + ) = (x + ) + (y ) = (x ) + (y + ) = (x + ) + (y ) = 579 What is the equation of a circle with its center at (0, ) and passing through the point (, 5)? x + (y + ) = 9 (x + ) + y = 9 x + (y + ) = 8 (x + ) + y = Write an equation of the circle shown in the graph below. CONICS A.A.7-9: EQUATIONS OF CIRCLES 575 The equation x + y x + 6y + = 0 is equivalent to (x ) + (y + ) = (x ) + (y + ) = 7 (x + ) + (y + ) = 7 (x + ) + (y + ) = What are the coordinates of the center of a circle whose equation is x + y 6x + 6y + 5 = 0? ( 8, ) ( 8,) (8, ) (8,) 577 What is the equation of the circle passing through the point (6,5) and centered at (, )? (x 6) + (y 5) = 8 (x 6) + (y 5) = 90 (x ) + (y + ) = 8 (x ) + (y + ) = 90 7

78 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 58 A circle shown in the diagram below has a center of ( 5,) and passes through point (,7). 58 Write an equation of the circle shown in the diagram below. 58 Which equation is represented by the graph below? Write an equation that represents the circle. 58 Which equation represents the circle shown in the graph below that passes through the point (0, )? (x ) + (y + ) = 5 (x + ) + (y ) = 5 (x ) + (y + ) = (x + ) + (y ) = (x ) + (y + ) = 6 (x ) + (y + ) = 8 (x + ) + (y ) = 6 (x + ) + (y ) = 8 7

79 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A circle with center O and passing through the origin is graphed below. What is the equation of circle O? x + y = 5 x + y = 0 (x + ) + (y ) = 5 (x + ) + (y ) = 0 75

80 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section 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: PTS: REF: 07a STA: A.S. TOP: Analysis of Data ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data 5 ANS: PTS: REF: 006a STA: A.S. TOP: Analysis of Data 6 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 7 ANS: PTS: REF: 00a STA: A.S. TOP: Analysis of Data 8 ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data 9 ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data 0 ANS: PTS: REF: 06a STA: A.S. TOP: Average Known with Missing Data 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

81 ANS: PTS: REF: fall09a STA: A.S. TOP: Dispersion KEY: basic, group frequency distributions ANS: 7. PTS: REF: 0609a STA: A.S. TOP: Dispersion KEY: basic, group frequency distributions 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 5 ANS: Ordered, the heights are 7, 7, 7, 7, 7, 75, 78, 79, 79, 8. Q = 7 and Q = = 7. PTS: REF: 0a STA: A.S. TOP: Central Tendency and Dispersion KEY: compute 6 ANS: σ x scores are within a population standard deviation of the mean. Q Q = 7 = x 8. PTS: REF: 068a STA: A.S. TOP: Dispersion KEY: advanced 7 ANS: Q =.5 and Q = = 7. PTS: REF: 00a STA: A.S. TOP: Central Tendency and Dispersion KEY: compute 8 ANS: 7 = 5 PTS: REF: 055a STA: A.S. TOP: Central Tendency and Dispersion KEY: frequency

82 9 ANS: ± = 7 PTS: REF: 0658a STA: A.S. TOP: Dispersion KEY: advanced, group frequency distributions 0 ANS: PTS: REF: 08509a STA: A.S. TOP: Dispersion KEY: basic, group frequency distributions ANS: PTS: REF: 067a STA: A.S.6 TOP: Regression ANS: y =.00x.98,,009. y =.00(5) PTS: REF: fall098a STA: A.S.7 TOP: Power Regression ANS: y = 0.596(.586) x PTS: REF: 080a STA: A.S.7 TOP: Regression KEY: exponential ANS: y = 7.05(.509) x. y = 7.05(.509) 8 PTS: REF: 08a STA: A.S.7 TOP: Regression KEY: exponential 5 ANS: y = 80.77(0.95) x PTS: REF: 06a STA: A.S.7 TOP: Regression KEY: exponential 6 ANS: y = 5.98(.65) x. 5.98(.65) PTS: REF: 07a STA: A.S.7 TOP: Regression KEY: exponential 7 ANS: y = 0.88(.6) x PTS: REF: 069a STA: A.S.7 TOP: Regression KEY: exponential 8 ANS: y = 7.66(0.786) x 7.66(0.786) PTS: REF: 056a STA: A.S.7 TOP: Regression KEY: exponential

83 9 ANS: y =.9(.) x 6. =.9(.) x 6..9 = (.) x log 6..9 = x log(.) log 6..9 log(.) x.5 = x PTS: REF: 067a STA: A.S.7 TOP: Exponential Regression 0 ANS: PTS: REF: 060a STA: A.S.8 TOP: Correlation Coefficient ANS: () shows the strongest linear relationship, but if r < 0, b < 0. The Regents announced that a correct solution was not provided for this question and all students should be awarded credit. PTS: REF: 0a STA: A.S.8 TOP: Correlation Coefficient ANS:. PTS: REF: 065a STA: A.S.8 TOP: Correlation Coefficient ANS: Since the coefficient of t is greater than 0, r > 0. PTS: REF: 00a STA: A.S.8 TOP: Correlation Coefficient ANS: PTS: REF: 066a STA: A.S.8 TOP: Correlation Coefficient 5 ANS: r A r B 0.99 Set B has the stronger linear relationship since r is higher. PTS: REF: 0655a STA: A.S.8 TOP: Correlation Coefficient 6 ANS: PTS: REF: 0850a STA: A.S.8 TOP: Correlation Coefficient 7 ANS: PTS: REF: 066a STA: A.S.8 TOP: Correlation Coefficient

84 8 ANS: PTS: REF: fall095a STA: A.S.5 TOP: Normal Distributions KEY: interval 9 ANS: 68% 50 = PTS: REF: 080a STA: A.S.5 TOP: Normal Distributions KEY: predict 0 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:.% + 9.% = 5.% PTS: REF: 0a STA: A.S.5 TOP: Normal Distributions KEY: probability ANS: x ± σ 5 ± 75 PTS: REF: 007a STA: A.S.5 TOP: Normal Distributions KEY: interval ANS: Top 6.7% =.5 s.d. + σ =.5(0) = 7 PTS: REF: 00a STA: A.S.5 TOP: Normal Distributions KEY: predict 5

85 5 ANS: Less than 60 inches is below.5 standard deviations from the mean PTS: REF: 068a STA: A.S.5 TOP: Normal Distributions KEY: predict 6 ANS: sd = = 65 8 (8) = = 8 PTS: REF: 05a STA: A.S.5 TOP: Normal Distributions KEY: mean and standard deviation 7 ANS: 9 8 =.5sd.6 PTS: REF: 085a STA: A.S.5 TOP: Normal Distributions KEY: interval 8 ANS: = standards above the mean or.%.% 000 = 5 PTS: REF: 06a STA: A.S.5 TOP: Normal Distributions KEY: predict 9 ANS: PTS: REF: fall095a STA: A.S.0 TOP: Permutations 50 ANS: 9 No. TENNESSEE: 9!!! = 6,880 =,780. VERMONT: 96 7 P 7 = 5,00 PTS: REF: 0608a STA: A.S.0 TOP: Permutations 5 ANS: 9,96,800. P!! = 79,00,600 = 9,96,800 PTS: REF: 0805a STA: A.S.0 TOP: Permutations 5 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 6

86 5 ANS: P 6 6!! = 70 = 60 PTS: REF: 0a STA: A.S.0 TOP: Permutations 5 ANS: P 0 0!!! =,68,800 = 50,00 7 PTS: REF: 060a STA: A.S.0 TOP: Permutations 55 ANS: PTS: REF: 009a STA: A.S.0 TOP: Permutations 56 ANS:!!! = 8 PTS: REF: 065a STA: A.S.0 TOP: Permutations 57 ANS: P!!!! = 9,96,800 =,9,800 6 PTS: REF: 058a STA: A.S.0 TOP: Permutations 58 ANS: P 9 9!!! = 6,880 =, PTS: REF: 065a STA: A.S.0 TOP: Permutations 59 ANS: P!!!! = 9,96,800 = 8,600 8 PTS: REF: 085a STA: A.S.0 TOP: Permutations 60 ANS:!!!! =,66,00 PTS: REF: 06a STA: A.S.0 TOP: Permutations 6 ANS: C = 6,5 5 8 PTS: REF: 080a STA: A.S. TOP: Combinations 6 ANS: C = 0 0 PTS: REF: 06a STA: A.S. TOP: Combinations 7

87 6 ANS: 5 C 0 = 5,0 PTS: REF: 0a STA: A.S. TOP: Combinations 6 ANS: C =,00. C = C = 5,0. C =,68, PTS: REF: 067a STA: A.S. TOP: Combinations 65 ANS: C =,85 0 PTS: REF: 0509a STA: A.S. TOP: Combinations 66 ANS: C = 8 9 PTS: REF: 085a STA: A.S. TOP: Combinations 67 ANS: PTS: REF: 06007a STA: A.S.9 TOP: Differentiating Permutations and Combinations 68 ANS: PTS: REF: 07a STA: A.S.9 TOP: Differentiating Permutations and Combinations 69 ANS: PTS: REF: 00a STA: A.S.9 TOP: Differentiating Permutations and Combinations 70 ANS: PTS: REF: 067a STA: A.S.9 TOP: Differentiating Permutations and Combinations 7 ANS: PTS: REF: 07a STA: A.S.9 TOP: Differentiating Permutations and Combinations 7 ANS: PTS: REF: 065a STA: A.S.9 TOP: Differentiating Permutations and Combinations 7 ANS: PTS: REF: 0856a STA: A.S.9 TOP: Differentiating Permutations and Combinations 7 ANS:,00. PTS: REF: fall095a STA: A.S. TOP: Sample Space 75 ANS: C C 5 = 0 = 0 PTS: REF: 06a STA: A.S. TOP: Combinations 8

88 76 ANS: π + π π = π π = PTS: REF: 008a STA: A.S. TOP: Geometric Probability 77 ANS: C C C PTS: REF: 0606a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 78 ANS: 6.%. 0 C C C PTS: REF: 0808a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 79 ANS: C C C PTS: REF: 08a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 80 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 9

89 8 ANS: C = 5 5 PTS: REF: 0a STA: A.S.5 TOP: Binomial Probability KEY: spinner 8 ANS: PTS: REF: 06a STA: A.S.5 TOP: Binomial Probability KEY: modeling 8 ANS: C 7 = = PTS: REF: 065a STA: A.S.5 TOP: Binomial Probability KEY: exactly 8 ANS: C C PTS: REF: 07a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 85 ANS: 5 C C C PTS: REF: 068a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 86 ANS: 5 C = = 576 5,65 PTS: REF: 05a STA: A.S.5 TOP: Binomial Probability KEY: exactly 87 ANS: C = 9 6 = 7 6 PTS: REF: 0650a STA: A.S.5 TOP: Binomial Probability KEY: exactly 88 ANS: C 7 = = PTS: REF: 085a STA: A.S.5 TOP: Binomial Probability KEY: exactly 89 ANS: PTS: REF: 0605a STA: A.S.5 TOP: Binomial Probability KEY: modeling 0

90 90 ANS: a + 6 = a 0. a + 6 = a = a = a = 8 = 8 0 PTS: REF: 006a STA: A.A. TOP: Absolute Value Equations 9 ANS: x = x is extraneous. x = x 0 = x 6 = x x = 8 x = PTS: REF: 065a STA: A.A. TOP: Absolute Value Equations 9 ANS: 6x 7 5 6x 7 5 6x x 6x x PTS: REF: fall0905a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 9 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 9 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

91 95 ANS: x 7 x x or x 7 x 0 x 5 PTS: REF: 0a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 96 ANS: x > 5. x < 5 x > 6 x > x > x < PTS: REF: 0607a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 97 ANS: x + 5 < x + 5 > < x <.5 x < 8 x > x > 8 x <.5 PTS: REF: 0a STA: A.A. TOP: Absolute Value Inequalities 98 ANS: x > 5 or x < 5 x > 8 x > x < x < PTS: REF: 060a STA: A.A. TOP: Absolute Value Inequalities 99 ANS: x 5 <x + 7 x 5 < x + 7 and x 5 > x 7 < x < x < x < x > x > PTS: REF: 0858a STA: A.A. TOP: Absolute Value Inequalities 00 ANS: x + 6 < x + 6 < 8 x + 6 > 8 x + 6 <8 x < x < x > x > 7 PTS: REF: 06a STA: A.A. TOP: Absolute Value Inequalities KEY: graph

92 0 ANS: Sum b a = 5. Product c a = 5 PTS: REF: 0600a STA: A.A.0 TOP: Roots of Quadratics 0 ANS: sum: b a = 6 =. product: c a = 6 = PTS: REF: 009a STA: A.A.0 TOP: Roots of Quadratics 0 ANS: x x + 6 = 0. Sum b a =. Product c a = 6 = PTS: REF: 09a STA: A.A.0 TOP: Roots of Quadratics 0 ANS: Sum b a =. Product c a = PTS: REF: 068a STA: A.A.0 TOP: Roots of Quadratics 05 ANS: x 7x 5 = 0 c a = 5 PTS: REF: 06a STA: A.A.0 TOP: Roots of Quadratics 06 ANS: c a = PTS: REF: 057a STA: A.A.0 TOP: Roots of Quadratics 07 ANS: Sum b a =. Product c a = k PTS: REF: 065a STA: A.A.0 TOP: Roots of Quadratics 08 ANS: P = c a = = PTS: REF: 08506a STA: A.A.0 TOP: Roots of Quadratics 09 ANS: b a = 6 = PTS: REF: 06a STA: A.A.0 TOP: Roots of Quadratics

93 0 ANS: S = b a = ( ) =. P = c a = 8 = PTS: REF: fall09a STA: A.A. TOP: Roots of Quadratics KEY: basic ANS: b a = 6 =. c a = = PTS: REF: 0a STA: A.A. TOP: Roots of Quadratics KEY: basic 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 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 ANS: b a = ( ) =. If the sum is, the roots must be 7 and. PTS: REF: 08a STA: A.A. TOP: Roots of Quadratics KEY: advanced 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 6 ANS: x + 0x x = x (6x + 5x 6) = x (x + )(x ) PTS: REF: 06008a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable 7 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

94 8 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 9 ANS: x + x x x (x + ) (x + ) (x )(x + ) (x + )(x )(x + ) PTS: REF: 06a STA: A.A.7 TOP: Factoring by Grouping 0 ANS: x 5x 8x + 80 x (x 5) 6(x 5) (x 6)(x 5) (x + )(x )(x 5) PTS: REF: 07a STA: A.A.7 TOP: Factoring by Grouping ANS: x (x + ) (x + ) (x )(x + ) (x + )(x )(x + ) PTS: REF: 06a STA: A.A.7 TOP: Factoring by Grouping ANS: x x 9x + 8 x (x ) 9(x ) (x 9)(x ) (x + )(x )(x ) PTS: REF: 05a STA: A.A.7 TOP: Factoring by Grouping ANS: x (x 6) 5(x 6) (x 5)(x 6) (x + 5)(x 5)(x 6) PTS: REF: 065a STA: A.A.7 TOP: Factoring by Grouping 5

95 ANS: x (x + ) + (x + ) = (x + )(x + ) PTS: REF: 069a STA: A.A.7 TOP: Factoring by Grouping 5 ANS: ± ( ) ()( 9) () = ± 5 = ± 5 PTS: REF: 06009a STA: A.A.5 TOP: Quadratics with Irrational Solutions 6 ANS: 7 ± 7 ()( ) () = 7 ± 7 PTS: REF: 08009a STA: A.A.5 TOP: Solving Quadratics KEY: quadratic formula 7 ANS: ± ( ) (6)( ) (6) = ± 76 = ± 9 = ± 9 = ± 9 6 PTS: REF: 0a STA: A.A.5 TOP: Solving Quadratics KEY: quadratic formula 8 ANS: 60 = 6t + 5t = 6t + 5t + 5 t = 5 ± 5 ( 6)(5) ( 6) 5 ± PTS: REF: 06a STA: A.A.5 TOP: Solving Quadratics KEY: quadratic formula 9 ANS: (x + )(x + ) = 800 x + 6x + 08 = 800 x = 6 ± ( 6) ()( 9) () = feet increase. x + 6x 9 = 0 PTS: 6 REF: 059a STA: A.A.5 TOP: Solving Quadratics KEY: quadratic formula 6

96 0 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 ANS: b ac = (9)( ) = 9 + = 5 PTS: REF: 0806a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation ANS: b ac = ( 0) ()(5) = = 0 PTS: REF: 00a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation ANS: PTS: REF: 0a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation ANS: b ac = ( 9) ()() = 8 = 9 PTS: REF: 0a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 5 ANS: ( 5) ()(0) = 5 PTS: REF: 06a STA: A.A. TOP: Using the Discriminant KEY: determine equation given nature of roots 6 ANS: ( 5) ()() = 9 PTS: REF: 0506a STA: A.A. TOP: Using the Discriminant 7 ANS: x + x = 0 ()( ) = + 68 = 69 = PTS: REF: 065a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 7

97 8 ANS: x + x = 0 b ac = ()( ) = = 7 PTS: REF: 068a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 9 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: Solving Quadratics KEY: completing the square 0 ANS: x + = 6x x 6x = x 6x + 9 = + 9 (x ) = 7 PTS: REF: 06a STA: A.A. TOP: Solving Quadratics KEY: completing the square ANS: PTS: REF: 06a STA: A.A. TOP: Solving Quadratics KEY: completing the square ANS: (x + ) = 9 x + = ± 9 x = ± i PTS: REF: 008a STA: A.A. TOP: Solving Quadratics KEY: completing the square ANS: PTS: REF: 0608a STA: A.A. TOP: Solving Quadratics KEY: completing the square 8

98 ANS: x = x 7 x x = 7 x x + 6 = (x 6) = 9 PTS: REF: 06505a STA: A.A. TOP: Solving Quadratics KEY: completing the square 5 ANS: = 6 PTS: REF: 0857a STA: A.A. TOP: Solving Quadratics KEY: completing the square 6 ANS: x + 0x + 5 = (x + 5) = x + 5 = ± x = 5 ± PTS: REF: 066a STA: A.A. TOP: Completing the Square 7 ANS: y x x 6 y (x )(x + ) PTS: REF: 0607a STA: A.A. TOP: Quadratic Inequalities KEY: two variables 8 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 9

99 9 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 50 ANS: 9 x < 0 or x + < 0 and x < 0 x 9 > 0 (x + )(x ) > 0 x + > 0 and x > 0 x > and x > x > x < and x < x < PTS: REF: 06507a STA: A.A. TOP: Quadratic Inequalities KEY: one variable 5 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: algebraically 5 ANS: 9, and,. y = x + 5 y = x + 7x. x + 7x = x + 5 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: algebraically 0

100 5 ANS: x + y = y = 5 y = x + 5 y = 0 (x + ) + ( x + 5 ) = 5 x + 6x x x + = 5 x + x 0 = 0 x + x 0 = 0 (x + 5)(x ) = 0 x = 5, PTS: REF: 00a STA: A.A. TOP: Quadratic-Linear Systems KEY: circle 5 ANS: x = y. y (y) + = 0. x = ( ) = 6 y 9y = 8y = y = y = ± PTS: REF: 06a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 55 ANS: x(x + ) = 0 x + x 0 = 0 (x + 5)(x ) = 0 x = 5, PTS: REF: 0a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations

101 56 ANS: x (x + ) = 8 x (x + x + ) 8 = 0 x x = 0 (x 6)(x + ) = 0 x = 6, PTS: REF: 0609a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 57 ANS: 9 x x +. x = x x = 9 x x x + = 9 x x + PTS: REF: 080a STA: A.N. TOP: Operations with Polynomials KEY: multiplication 58 ANS: PTS: REF: 0a STA: A.N. TOP: Operations with Polynomials KEY: subtraction 59 ANS: 6y 7 0 y 5 y. 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 KEY: multiplication 60 ANS: The binomials are conjugates, so use FL. PTS: REF: 006a STA: A.N. TOP: Operations with Polynomials KEY: multiplication 6 ANS: The binomials are conjugates, so use FL. PTS: REF: 060a STA: A.N. TOP: Operations with Polynomials KEY: multiplication 6 ANS: PTS: REF: 0a STA: A.N. TOP: Operations with Polynomials KEY: subtraction 6 ANS: x x + x x () = x PTS: REF: 05a STA: A.N. TOP: Operations with Polynomials KEY: multiplication 6 ANS: PTS: REF: 0655a STA: A.N. TOP: Operations with Polynomials KEY: subtraction

102 65 ANS: x + 8 x 7 9 x 9 PTS: REF: 065a STA: A.N. TOP: Operations with Polynomials KEY: subtraction 66 ANS: ( ) = 9 8 = 8 9 PTS: REF: 0600a STA: A.N. TOP: Negative and Fractional Exponents 67 ANS: 6n < n. Flip sign when multiplying each side of the inequality by n, since a negative number. 6 n < n 6 > PTS: REF: 06a STA: A.N. TOP: Negative and Fractional Exponents 68 ANS: f(6) = (6) = () + + = 7 PTS: REF: 0850a STA: A.N. TOP: Negative and Fractional Exponents 69 ANS: w 5 w 9 = (w ) = w PTS: REF: 080a STA: A.A.8 TOP: Negative and Fractional Exponents 70 ANS: PTS: REF: 006a STA: A.A.8 TOP: Negative and Fractional Exponents 7 ANS: PTS: REF: 00a STA: A.A.8 TOP: Negative and Fractional Exponents 7 ANS: PTS: REF: 060a STA: A.A.8 TOP: Negative and Fractional Exponents 7 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Negative and Fractional Exponents

103 7 ANS: x x = x x = x x x = (x ) x x = x PTS: REF: 0808a STA: A.A.9 TOP: Negative Exponents 75 ANS: x y 9. x y 5 y 5 (x y 7 ) = (x y 7 ) x = y 5 (x 6 y ) = x 6 y 9 = x x x y 9 PTS: REF: 06a STA: A.A.9 TOP: Negative Exponents 76 ANS: x + x + = x + + x x + = x x + = x PTS: REF: 0a STA: A.A.9 TOP: Negative Exponents 77 ANS: PTS: REF: 060a STA: A.A.9 TOP: Negative Exponents 78 ANS: PTS: REF: 06a STA: A.A.9 TOP: Negative Exponents 79 ANS: 5 a b = 5b a PTS: REF: 05a STA: A.A.9 TOP: Negative Exponents 80 ANS: PTS: REF: 06506a STA: A.A.9 TOP: Negative Exponents 8 ANS:, PTS: REF: fall09a STA: A.A. TOP: Evaluating Exponential Expressions 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

104 8 ANS: = 5000(.0075) PTS: REF: 00a STA: A.A. TOP: Evaluating Functions 85 ANS: PTS: REF: 066a STA: A.A. TOP: Evaluating Exponential Expressions 86 ANS: p(5) p(0) = 7(.5) (5) 7(.5) (0) PTS: REF: 0657a STA: A.A. TOP: Functional Notation 87 ANS: A = = 50(.0085) PTS: REF: 085a STA: A.A. TOP: Evaluating Functions 88 ANS: A = 5000e (.0)(8) 07.7 PTS: REF: 0607a STA: A.A. TOP: Evaluating Exponential Expressions 89 ANS: 8 = 6 PTS: REF: fall0909a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 90 ANS: PTS: REF: 0a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 9 ANS: y = 0 PTS: REF: 060a STA: A.A.5 TOP: Graphing Exponential Functions 5

105 9 ANS: PTS: REF: 0a STA: A.A.5 TOP: Graphing Exponential Functions 6

106 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section 9 ANS: PTS: REF: 0505a STA: A.A.5 TOP: Graphing Exponential Functions 9 ANS: f (x) = log x PTS: REF: fall096a STA: A.A.5 TOP: Graphing Logarithmic Functions 95 ANS: PTS: REF: 06a STA: A.A.5 TOP: Graphing Logarithmic Functions 96 ANS: PTS: REF: 0a STA: A.A.5 TOP: Graphing Logarithmic Functions 97 ANS: log x ( log y + log z) = log x log y log z = log x y z PTS: REF: 0600a STA: A.A.9 TOP: Properties of Logarithms 98 ANS: PTS: REF: 060a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 99 ANS: log x = log a + log a log x = log 6a log x = log 6 + log a log x = log 6 + log a log x = log 6 + log a PTS: REF: 0a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 00 ANS: PTS: REF: 0607a STA: A.A.9 TOP: Properties of Logarithms KEY: antilogarithms

107 0 ANS: log 9 log0 log log(0 ) log (log0 + log) b ( + a) b a PTS: REF: 06a STA: A.A.9 TOP: Properties of Logarithms KEY: expressing logs algebraically 0 ANS: log m = log + log m = log + log m PTS: REF: 06a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 0 ANS: log x = log + log x = log + log x PTS: REF: 066a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 0 ANS: log x = log a + log b log x = log a b x = a b PTS: REF: 0657a STA: A.A.9 TOP: Properties of Logarithms KEY: antilogarithms 05 ANS: log T = log 0x y = log0 + log x log y = + log x log y PTS: REF: 065a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs

108 06 ANS: log (5x) = log (5x) = 5x = 5x = 8 x = 8 5 PTS: REF: fall09a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 07 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 08 ANS: ln(t T 0 ) = kt+.78. 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

109 09 ANS: x = 5 = 65 PTS: REF: 0606a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 0 ANS: 800. x =.5 =. y = 5 y = 5 = 5. x y = 5 = 800 PTS: REF: 07a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced ANS: (x + ) = 7x x + 8x + 6 = 7x x 9x + 0 = 0 (x )(x 5) = 0 x =,5 PTS: REF: 06a STA: A.A.8 TOP: Logarithmic Equations KEY: basic ANS: x = 7 x = 8 x = 8 x = PTS: REF: 069a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced

110 ANS: log (x + ) (x + )(x + 5) = 6 is extraneous (x + ) = (x + )(x + 5) x + 6x + 9 = x + x + 5 x + 7x + 6 = 0 (x + 6)(x + ) = 0 x = PTS: 6 REF: 09a STA: A.A.8 TOP: Logarithmic Equations KEY: applying properties of logarithms ANS: (5x ) = 5x = 6 5x = 65 x = PTS: REF: 06a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 5 ANS: PTS: REF: 050a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 6 ANS: (x + ) = 6 x + = x = PTS: REF: 065a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 7 ANS: x 7x + log x 0 = x 7x + x 0 = 8 x 7x + = 6x 80 x x + 9 = 0 x = ± ( ) ()(9) () 7.8, 5.6 PTS: 6 REF: 0859a STA: A.A.8 TOP: Logarithmic Equations KEY: applying properties of logarithms 5

111 8 ANS: 8 x + = 6 (x + ) = x + = x = x = PTS: REF: 060a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 9 ANS: = 5000e.075t = e.075t ln = ln e.075t ln.075t ln e = t PTS: REF: 067a STA: A.A.6 TOP: Exponential Growth 0 ANS: 0 = 0() = () t 60 log = log() log = tlog 60 t 60 t log log = t 00 = t PTS: REF: 005a STA: A.A.6 TOP: Exponential Growth 6

112 ANS: 0700 = 50e t 6 = e t ln6 = ln e t ln6 = t ln e ln6 = t. t PTS: REF: 0a STA: A.A.6 TOP: Exponential Growth ANS: 000 = 500e.05t = e.05t ln = ln e.05t ln.05.9 t =.05t ln e.05 PTS: REF: 06a STA: A.A.6 TOP: Exponential Growth 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 7

113 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 5 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 6 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 8

114 7 ANS: 8 x + x = 7 5x x + x = x + 8x = 5x x + 8x 5x = 0 x(x + 8x 5) = 0 x(x )(x + 5) = 0 5x x = 0,, 5 PTS: 6 REF: 069a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 8 ANS: 8 k + = k. ( ) k + = ( ) k 9k + = k 9k + = k 5k = k = 5 PTS: REF: 009a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 9 ANS: ln e x = ln x = ln x = ln 0.6 PTS: REF: 050a STA: A.A.7 TOP: Exponential Equations KEY: without common base 9

115 0 ANS: 5 x = 5 x = x x = x PTS: REF: 0658a STA: A.A.7 TOP: Exponential Equations KEY: common base shown ANS: = x = x = x = x PTS: REF: 0859a STA: A.A.7 TOP: Exponential Equations KEY: common base shown ANS: 5 C (x) ( ) = 0 9x 8 = 70x PTS: REF: fall099a 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: C a 6 9 ( b) = 576a 6 b PTS: REF: 066a STA: A.A.6 TOP: Binomial Expansions 5 ANS: C (x ) ( y) = 6x y PTS: REF: 05a STA: A.A.6 TOP: Binomial Expansions 6 ANS: C x 6 ( y) = 0 x 8 8y = 0x y PTS: REF: 065a STA: A.A.6 TOP: Binomial Expansions 7 ANS: C x 8 8 ( ) = 56x 5 ( 8) = 8x 5 PTS: REF: 008a STA: A.A.6 TOP: Binomial Expansions 0

116 8 ANS: 5 C (x)5 ( ) = 70x PTS: REF: 059a STA: A.A.6 TOP: Binomial Expansions 9 ANS: PTS: REF: 0855a STA: A.A.6 TOP: Binomial Expansions 0 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 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: 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

117 ANS: x + x + x + 6x = 0 x(x + x + x + 6) = 0 x(x (x + ) + (x + )) = 0 x(x + )(x + ) = 0 x = 0,±i, PTS: 6 REF: 069a STA: A.A.6 TOP: Solving Polynomial Equations ANS: x + 5x x 0 = 0 x (x + 5) (x + 5) = 0 (x )(x + 5) = 0 (x + )(x )(x + 5) = 0 x = ±, 5 PTS: REF: 067a STA: A.A.6 TOP: Solving Polynomial Equations 5 ANS: x (x ) (x ) = 0 (x )(x ) = 0 (x + )(x )(x ) = 0 x = ±, PTS: REF: 0857a STA: A.A.6 TOP: Solving Polynomial Equations 6 ANS: PTS: REF: 06005a STA: A.A.50 TOP: Zeros of Polynomials 7 ANS: The roots are,,. PTS: REF: 080a STA: A.A.50 TOP: Zeros of Polynomials 8 ANS: PTS: REF: 06a STA: A.A.50 TOP: Solving Polynomial Equations 9 ANS: PTS: REF: 0850a STA: A.A.50 TOP: Zeros of Polynomials

118 50 ANS: ( + 5)( 5) = 9 5 = PTS: REF: 0800a STA: A.N. TOP: Operations with Irrational Expressions KEY: without variables index = 5 ANS: a b PTS: REF: 0a STA: A.A. TOP: Simplifying Radicals KEY: index > 5 ANS: a 5 a = a 5 a PTS: REF: 060a STA: A.A. TOP: Simplifying Radicals KEY: index > 5 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 5 ANS: 6a b + (7 6)a b a 6ab + a 6ab a 6ab PTS: REF: 09a STA: A.N. TOP: Operations with Radicals 55 ANS: 7x 6x = x 6 = x = 6x PTS: REF: 0a STA: A.N. TOP: Operations with Radicals 56 ANS: 6a 5 b 6 = a a b 6 = ab a PTS: REF: 056a STA: A.N. TOP: Operations with Radicals 57 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 =

119 58 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 = 59 ANS: PTS: REF: 0607a STA: A.A. TOP: Operations with Radicals KEY: with variables index = 60 ANS: 7a 6b 8 = a b = 6ab PTS: REF: 0650a STA: A.A. TOP: Operations with Radicals KEY: with variables index > 6 ANS: c = x + + x = x + x + + x x + = x + PTS: REF: 066a STA: A.A. TOP: Operations with Radicals KEY: with variables index = 6 ANS: 5( + ) = 5( + ) 9 = 5( + ) 7 PTS: REF: fall098a STA: A.N.5 TOP: Rationalizing Denominators 6 ANS: = = 5 = + 5 PTS: REF: 060a STA: A.N.5 TOP: Rationalizing Denominators 6 ANS: = (5 + ) 5 = 5 + PTS: REF: 066a STA: A.N.5 TOP: Rationalizing Denominators 65 ANS: = = PTS: REF: 00a STA: A.N.5 TOP: Rationalizing Denominators

120 66 ANS: = 5( + ) 6 = 5( + ) 5 = + PTS: REF: 06509a STA: A.N.5 TOP: Rationalizing Denominators 67 ANS: 8 = = 6 = PTS: REF: 0858a STA: A.N.5 TOP: Rationalizing Denominators 68 ANS: = a b a b b b = b ab = b ab PTS: REF: 0809a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 69 ANS: x + x + x + x + = (x + ) x + x + = x + PTS: REF: 0a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 70 ANS: x x x x + x x + x = x + x x x x = x(x + x ) x(x ) = x + x x PTS: REF: 065a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 7 ANS: PTS: REF: 0608a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 7 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 6 5

121 7 ANS: 7. x 5 = x 5 = x 5 = 9 x = x = 7 PTS: REF: 09a STA: A.A. TOP: Solving Radicals KEY: basic 7 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 75 ANS: x + x = x x + x = 6x x = 5x 5x = x 5x + 0 = (x )(x ) x =, x 0 () + < 0 is extraneous PTS: 6 REF: 09a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 6

122 76 ANS: x = x x = x x + 0 = x 6x = (x )(x ) x =, PTS: REF: 0606a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 77 ANS: x + = x + = 6 x = 5 x = 5 PTS: REF: 068a STA: A.A. TOP: Solving Radicals KEY: basic 78 ANS: PTS: REF: 060a STA: A.A.0 TOP: Fractional Exponents as Radicals 79 ANS: 5 x = x 5 = 5 x PTS: REF: 08a STA: A.A.0 TOP: Fractional Exponents as Radicals 80 ANS: 6x y 7 = 6 7 x y = x 7 y PTS: REF: 0607a STA: A.A. TOP: Radicals as Fractional Exponents 8 ANS: 8x y 5 = 8 5 x y = x 5 y PTS: REF: 0850a STA: A.A. TOP: Radicals as Fractional Exponents 8 ANS: 7a 6 b c = a bc = bc a PTS: REF: 0606a STA: A.A. TOP: Radicals as Fractional Exponents 7

123 8 ANS: 00 = 00 PTS: REF: 06006a STA: A.N.6 TOP: Square Roots of Negative Numbers 8 ANS: 9 6 = i i = i PTS: REF: 060a STA: A.N.6 TOP: Square Roots of Negative Numbers 85 ANS: 80x 6 = 6x 8 i 5 PTS: REF: 085a STA: A.N.6 TOP: Square Roots of Negative Numbers 86 ANS: PTS: REF: 0609a STA: A.N.7 TOP: Imaginary Numbers 87 ANS: i + i = ( ) + ( i) = i PTS: REF: 0800a STA: A.N.7 TOP: Imaginary Numbers 88 ANS: i + i 8 + i + n = 0 i + ( ) i + n = 0 + n = 0 n = PTS: REF: 068a STA: A.N.7 TOP: Imaginary Numbers 89 ANS: xi + 5yi 8 + 6xi + yi = xi + 5y 6xi + y = 7y xi PTS: REF: 0a STA: A.N.7 TOP: Imaginary Numbers 90 ANS: xi 8 yi 6 = x() y( ) = x + y PTS: REF: 065a STA: A.N.7 TOP: Imaginary Numbers 9 ANS: x(7i 6 ) + x(i ) = 7x + x = 5x PTS: REF: 060a STA: A.N.7 TOP: Imaginary Numbers 9 ANS: PTS: REF: 080a STA: A.N.8 TOP: Conjugates of Complex Numbers 9 ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers 9 ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers 8

124 95 ANS: PTS: REF: 069a STA: A.N.8 TOP: Conjugates of Complex Numbers 96 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 97 ANS: (x + i) (x i) = x + xi + i (x xi + i ) = xi PTS: REF: 07a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 98 ANS: (i)(i) (m + i) (i)(i )(m + i) (i)( )(m + i) ( i)(m + i) mi i mi + PTS: REF: 069a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 99 ANS: (x + yi)(x yi) = x y i = x + y PTS: REF: 06a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 00 ANS: ( i)( + i) = 9 i = 9 + = PTS: REF: 05a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 0 ANS: xi(i i ) = xi 8xi = xi 8xi = x + 8xi PTS: REF: 05a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 9

125 0 ANS: (x + 6) x. x (x ) + 6(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 0 ANS: x + 9x x ( x) = (x + )(x ) (x + )(x ) x = x PTS: REF: 0a STA: A.A.6 TOP: Multiplication and Division of Rationals KEY: division 0 ANS: y y y = y y 6 9 y 6 = y 9 (y ) = y 6 (y ) = PTS: REF: 05a STA: A.A.6 TOP: Addition and Subtraction of Rationals 05 ANS: x x + x = x (x ) + (x ) = x + (x ) PTS: REF: 0608a STA: A.A.6 TOP: Addition and Subtraction of Rationals 06 ANS: no solution. x x = + x x x = (x ) x = PTS: REF: fall090a STA: A.A. TOP: Solving Rationals KEY: rational solutions 0

126 07 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 08 ANS: x = 0 x = 0x x x 0x + = 0 0 ± 00 ()(). x = = () 0 ± 8 = 0 ± = 5 ± PTS: REF: 066a STA: A.A. TOP: Solving Rationals KEY: irrational and complex solutions 09 ANS: 0 (x + )(x ) + (x + )(x ) (x + )(x ) = 5(x + ) is an extraneous root. (x )(x + ) 0 + x 9 = 5x + 5 x 5x + 6 = 0 (x )(x ) = 0 x = PTS: REF: 067a STA: A.A. TOP: Solving Rationals KEY: rational solutions

127 0 ANS: 5x (x ) x(x ) x(x ) = x(x ) x(x ) 5x x + 6 = x x 0 = x 6x 6 PTS: REF: 05a STA: A.A. TOP: Solving Rationals KEY: irrational and complex solutions ANS: x + x x + = x + x + x + = x = x x = PTS: REF: 0657a STA: A.A. TOP: Solving Rationals KEY: rational solutions ANS: 0x = x + x 9x = x 9x = x = 9 x = ± PTS: REF: 085a STA: A.A. TOP: Solving Rationals KEY: rational solutions

128 ANS: x + 6 7(x ) x x 0 6x x 6() + 0 6(6) x + 0 = 0 6x = 0 x = 5 x = 0. Check points such that x <, < x < 5, and x > 5. If x =, x = = 6() + 0 =, which is less than 0. If x =, = =, which is greater than 0. If x = 6, = 6 =, which is less than 0. PTS: REF: 0a STA: A.A. TOP: Rational Inequalities ANS: x x x + = x x x + 8x = (x + )(x ) x 8x (x + ) = x PTS: REF: fall090a STA: A.A.7 TOP: Complex Fractions 5 ANS: d d + d = d 8 d d 8 = d + d d d 5d = d 8 5 d PTS: REF: 0605a STA: A.A.7 TOP: Complex Fractions 6 ANS: (x ) (x + )(x + ) x + (x ) = (x + )(x ) x + (x ) = (x + ) (x + ) PTS: REF: 066a STA: A.A.7 TOP: Complex Fractions 7 ANS: x x x 8 x = x x x x 8 = x(x ) (x )(x + ) = x x + x PTS: REF: 0605a STA: A.A.7 TOP: Complex Fractions 8 ANS: a + b ac + b c c d b = = ac + b c cd b c cd b = ac + b cd b c c PTS: REF: 005a STA: A.A.7 TOP: Complex Fractions

129 9 ANS: + x 5 x x x x = x + x x 5x = x(x + ) (x 8)(x + ) = x x 8 PTS: REF: 066a STA: A.A.7 TOP: Complex Fractions 0 ANS: (6 x)(6 + x) (x + 6)(x ) = 6 x (x + 6)(x + 6) x PTS: REF: 059a STA: A.A.7 TOP: Complex Fractions ANS: x + y xy xy = x + y xy xy = x + y PTS: REF: 060a STA: A.A.7 TOP: Complex Fractions ANS: 6 = 9w 8 = w PTS: REF: 00a STA: A.A.5 TOP: Inverse Variation ANS: 0 = 5 p 5 = 5 p 5 = p PTS: REF: 06a STA: A.A.5 TOP: Inverse Variation ANS: 0( ) = x( x + ) x x 0 = 0 0 = x + x x x 0 = 0 (x + )(x 5) = 0 x =,5 PTS: REF: 0a STA: A.A.5 TOP: Inverse Variation

130 5 ANS: = =. 6 d = 6d = d = PTS: REF: 060a STA: A.A.5 TOP: Inverse Variation 6 ANS: 0 = 5t 8 = t PTS: REF: 0a STA: A.A.5 TOP: Inverse Variation 7 ANS: 5 6 = 0q 5 = q PTS: REF: 058a STA: A.A.5 TOP: Inverse Variation 8 ANS: PTS: REF: 0650a STA: A.A.5 TOP: Inverse Variation 9 ANS: 00 = 8x 50 = x PTS: REF: 08507a STA: A.A.5 TOP: Inverse Variation 0 ANS: y sin θ = y = sin θ + f(θ) = sin θ + PTS: REF: fall097a STA: A.A.0 TOP: Functional Notation ANS: 0 f0 = ( 0) 6 = 0 8 = 5 PTS: REF: 060a STA: A.A. TOP: Functional Notation ANS: g(0) = a(0) 0 = 00a ( 9) = 900a PTS: REF: 06a STA: A.A. TOP: Functional Notation 5

131 ANS: fa + = (a + ) (a + ) + = (a + a + ) a = a + 8a + a = a + 7a + PTS: REF: 057a STA: A.A. TOP: Functional Notation ANS: f(x + ) = (x + ) (x + ) + = x + x + 8 x 9 + = x + 9x + PTS: REF: 069a STA: A.A. TOP: Functional Notation 5 ANS: PTS: REF: 09a STA: A.A.5 TOP: Families of Functions 6 ANS: PTS: REF: 09a STA: A.A.5 TOP: Properties of Graphs of Functions and Relations 7 ANS: As originally written, alternatives () and () had no domain restriction, so that both were correct. PTS: REF: 0605a STA: A.A.5 TOP: Properties of Graphs of Functions and Relations 8 ANS: PTS: REF: 0600a STA: A.A.5 TOP: Identifying the Equation of a Graph 9 ANS: PTS: REF: 0608a STA: A.A.5 TOP: Families of Functions 0 ANS: PTS: REF: 00a STA: A.A.5 TOP: Families of Functions ANS: PTS: REF: 050a STA: A.A.5 TOP: Identifying the Equation of a Graph ANS: PTS: REF: 005a STA: A.A.7 TOP: Defining Functions KEY: ordered pairs ANS: PTS: REF: fall0908a STA: A.A.8 TOP: Defining Functions KEY: graphs ANS: PTS: REF: 060a STA: A.A.8 TOP: Defining Functions 5 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 7 ANS: PTS: REF: 0609a STA: A.A.8 TOP: Defining Functions KEY: graphs 8 ANS: PTS: REF: 0507a STA: A.A.8 TOP: Defining Functions KEY: graphs 9 ANS: PTS: REF: 060a STA: A.A.8 TOP: Defining Functions KEY: ordered pairs 6

132 50 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 5 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 5 ANS: PTS: REF: 05a STA: A.A. TOP: Defining Functions 5 ANS: PTS: REF: 068a STA: A.A. TOP: Defining Functions 5 ANS: PTS: REF: 060a STA: A.A. TOP: Defining Functions 55 ANS: PTS: REF: 007a STA: A.A. TOP: Defining Functions 56 ANS: PTS: REF: 0650a STA: A.A. TOP: Defining Functions 57 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Domain and Range KEY: real domain, radical 58 ANS: PTS: REF: 06a STA: A.A.9 TOP: Domain and Range KEY: real domain, quadratic 59 ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain, absolute value 60 ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain, radical 6 ANS: PTS: REF: 06a STA: A.A.9 TOP: Domain and Range KEY: real domain, rational 6 ANS: PTS: REF: 05a STA: A.A.9 TOP: Domain and Range KEY: real domain, rational 6 ANS: PTS: REF: 0857a STA: A.A.9 TOP: Domain and Range KEY: real domain, exponential 6 ANS: PTS: REF: 0800a STA: A.A.5 TOP: Domain and Range KEY: graph 65 ANS: D: 5 x 8. R: y PTS: REF: 0a STA: A.A.5 TOP: Domain and Range KEY: graph 66 ANS: PTS: REF: 060a STA: A.A.5 TOP: Domain and Range KEY: graph 67 ANS: PTS: REF: 0608a STA: A.A.5 TOP: Domain and Range KEY: graph 68 ANS: PTS: REF: 068a STA: A.A.5 TOP: Domain and Range KEY: graph 7

133 69 ANS: PTS: REF: 0658a STA: A.A.5 TOP: Domain and Range KEY: graph 70 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 7 ANS: 7. f( ) = ( ) 6 =. g(x) = = 7. PTS: REF: 065a STA: A.A. TOP: Compositions of Functions KEY: numbers 7 ANS: g = =. f() = () = PTS: REF: 00a STA: A.A. TOP: Compositions of Functions KEY: numbers 7 ANS: PTS: REF: 066a STA: A.A. TOP: Compositions of Functions KEY: variables 75 ANS: h( 8) = ( 8) = = 6. g( 6) = ( 6) + 8 = + 8 = 5 PTS: REF: 00a STA: A.A. TOP: Compositions of Functions KEY: numbers 76 ANS: f(g(x)) = (x + 5) (x + 5) + = (x + 0x + 5) x 5 + = x + 7x + 6 PTS: REF: 069a STA: A.A. TOP: Compositions of Functions KEY: variables 77 ANS: g( ) = ( ) = 8 f( 8) = ( 8) + = 8 + = 9 PTS: REF: 0650a STA: A.A. TOP: Compositions of Functions KEY: numbers 8

134 78 ANS: (x + ) (x + ) = x + x + x = x + x PTS: REF: 0850a STA: A.A. TOP: Compositions of Functions KEY: variables 79 ANS: PTS: REF: 0807a STA: A.A. TOP: Inverse of Functions KEY: equations 80 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 8 ANS: PTS: REF: 065a STA: A.A. TOP: Inverse of Functions KEY: equations 8 ANS: PTS: REF: 085a STA: A.A. TOP: Inverse of Functions KEY: ordered pairs 8 ANS: PTS: REF: fall096a STA: A.A.6 TOP: Graphing Quadratic Functions 8 ANS: PTS: REF: 080a STA: A.A.6 TOP: Transformations with Functions 85 ANS: PTS: REF: 065a STA: A.A.6 TOP: Graphing Quadratic Functions 86 ANS: PTS: REF: 0656a STA: A.A.6 TOP: Transformations with Functions 87 ANS: PTS: REF: 0606a STA: A.A.9 TOP: Sequences 9

135 88 ANS: common difference is. b n = x + n 0 = x + () 8 = x PTS: REF: 080a STA: A.A.9 TOP: Sequences 89 ANS: 0 =.5 PTS: REF: 07a STA: A.A.9 TOP: Sequences 90 ANS: 9 7 = = x + ( ) = 9 a n = 7 + (n ) x + = 9 x = 7 PTS: REF: 0a STA: A.A.9 TOP: Sequences 9 ANS: PTS: REF: 0650a STA: A.A.9 TOP: Sequences 0

136 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section 9 ANS: PTS: REF: 0600a STA: A.A.0 TOP: Sequences 9 ANS: PTS: REF: 00a STA: A.A.0 TOP: Sequences 9 ANS: (a + ) (a + ) = a + PTS: REF: 00a STA: A.A.0 TOP: Sequences 95 ANS: PTS: REF: 06a STA: A.A.0 TOP: Sequences 96 ANS: PTS: REF: 060a STA: A.A.0 TOP: Sequences 97 ANS: 7r = 6 r = 6 7 r = PTS: REF: 0805a STA: A.A. TOP: Sequences 98 ANS: = PTS: REF: 00a STA: A.A. TOP: Sequences 99 ANS: a b 6b = 6 a 5 b a PTS: REF: 066a STA: A.A. TOP: Sequences 00 ANS: = PTS: REF: 0508a STA: A.A. TOP: Sequences

137 0 ANS: a n = 5( ) n a 5 = 5( ) 5 = 8,90 PTS: REF: 005a STA: A.A. TOP: Sequences 0 ANS: a n = 5( ) n a 5 = 5( ) 5 = 5( ) = 5 7 = 8 5 PTS: REF: 0609a STA: A.A. TOP: Sequences 0 ANS: = 0 5 = 6 a = 6n + n a 0 = 6(0) + = PTS: REF: 0850a STA: A.A. TOP: Sequences 0 ANS:, 5, 8, PTS: REF: fall09a STA: A.A. TOP: Sequences 05 ANS: a =. a = () = 5. a = (5) = 9. PTS: REF: 06a STA: A.A. TOP: Sequences 06 ANS: a = = a = = 6 a = 6 = 7 56 PTS: REF: 057a STA: A.A. TOP: Sequences 07 ANS: a = xy 5 x = xy 5 8x y y = x y PTS: REF: 065a STA: A.A. TOP: Sequences 08 ANS: PTS: REF: 0850a STA: A.A. TOP: Sequences 09 ANS: a = ( 6) = 5 a = ( 5) = PTS: REF: 06a STA: A.A. TOP: Sequences

138 0 ANS: n 0 Σ n + n = + = + = 8 = PTS: REF: fall09a STA: A.N.0 TOP: Sigma Notation KEY: basic ANS: ( ) + ( ) + ( ) + ( ) + (5 ) = = 0 PTS: REF: 0a STA: A.N.0 TOP: Sigma Notation KEY: basic ANS: n 5 Σ r + r + = 6 + = = 0 8 PTS: REF: 068a STA: A.N.0 TOP: Sigma Notation KEY: basic ANS: 0. PTS: REF: 00a STA: A.N.0 TOP: Sigma Notation KEY: basic ANS: + ( x) + ( x) + ( x) + (5 x) + 6 x + 9 x + x + 5 x 6 x PTS: REF: 065a STA: A.N.0 TOP: Sigma Notation KEY: advanced 5 ANS: (a ) + (a ) + (a ) + (a ) (a a + ) + (a a + ) + (a 6a + 9) + (a 8a + 6) a 0a + 0 PTS: REF: 0a STA: A.N.0 TOP: Sigma Notation KEY: advanced

139 6 ANS: ( a) 0 + ( a) + ( a) = + a + 9 a + a = a a + PTS: REF: 0656a STA: A.N.0 TOP: Sigma Notation KEY: advanced 7 ANS: x + x + x 9 + x 6 = x 0 PTS: REF: 0855a STA: A.N.0 TOP: Sigma Notation KEY: advanced 8 ANS: cos π + cos π + cos π = = PTS: REF: 067a STA: A.N.0 TOP: Sigma Notation KEY: advanced 9 ANS: PTS: REF: 0605a STA: A.A. TOP: Sigma Notation 0 ANS: 5 n = 7n PTS: REF: 0809a STA: A.A. TOP: Sigma Notation ANS: PTS: REF: 0605a STA: A.A. TOP: Sigma Notation ANS: PTS: REF: 060a STA: A.A. TOP: Sigma Notation ANS: PTS: REF: 050a STA: A.A. TOP: Sigma Notation ANS: S n = n [a + (n )d] = [(8) + ( )] = 798 PTS: REF: 060a STA: A.A.5 TOP: Series KEY: arithmetic 5 ANS: S n = n 9 [a + (n )d] = [() + (9 )7] = 5 PTS: REF: 00a STA: A.A.5 TOP: Series KEY: arithmetic

140 6 ANS: a n = 9n a = 9() = 5. S n = 0(5 + 76) = 80 a 0 = 9(0) = 76 PTS: REF: 08a STA: A.A.5 TOP: Series KEY: arithmetic 7 ANS: S 8 = ( ( ) 8 ) ( ) = 96,605 5 = 9, PTS: REF: 060a STA: A.A.5 TOP: Series KEY: geometric 8 ANS: cos K = 5 6 K = cos 5 6 K ' PTS: REF: 060a STA: A.A.55 TOP: Trigonometric Ratios 9 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 ANS: PTS: REF: 05a STA: A.A.55 TOP: Trigonometric Ratios 5

141 ANS: sin S = 8 7 S = sin 8 7 S 8 ' PTS: REF: 06a STA: A.A.55 TOP: Trigonometric Ratios ANS: PTS: REF: 065a STA: A.A.55 TOP: Trigonometric Ratios ANS: π 5 = 0π = 5π 6 PTS: REF: 065a STA: A.M. TOP: Radian Measure 5 ANS: PTS: REF: 0650a STA: A.M. TOP: Radian Measure 6 ANS: 97º π PTS: REF: fall09a STA: A.M. TOP: Radian Measure KEY: degrees 7 ANS: π 80 π = 65 PTS: REF: 0600a STA: A.M. TOP: Radian Measure KEY: degrees 8 ANS: 0 π 80 = 7π PTS: REF: 0800a STA: A.M. TOP: Radian Measure KEY: radians 6

142 9 ANS:.5 80 π. PTS: REF: 09a STA: A.M. TOP: Radian Measure KEY: degrees 0 ANS: 80 π = 60 π PTS: REF: 00a STA: A.M. TOP: Radian Measure KEY: degrees ANS: 6 π 80.8 PTS: REF: 06a STA: A.M. TOP: Radian Measure KEY: radians ANS: 80 π PTS: REF: 05a STA: A.M. TOP: Radian Measure KEY: degrees ANS: 8π 5 80 π = 88 PTS: REF: 060a STA: A.M. TOP: Radian Measure KEY: degrees ANS: 5 80 π 86 PTS: REF: 07a STA: A.M. TOP: Radian Measure KEY: degrees 7

143 5 ANS:.5 80 π ' PTS: REF: 06a STA: A.M. TOP: Radian Measure KEY: degrees 6 ANS: 5 π 80 π = 8 9' PTS: REF: 05a STA: A.M. TOP: Radian Measure KEY: degrees 7 ANS:.5 80 π = ' PTS: REF: 0858a STA: A.M. TOP: Radian Measure KEY: degrees 8 ANS: 0 π 80.7 PTS: REF: 06a STA: A.M. TOP: Radian Measure KEY: radians 9 ANS: PTS: REF: 060a STA: A.A.60 TOP: Unit Circle 50 ANS: PTS: REF: 08005a STA: A.A.60 TOP: Unit Circle 5 ANS: PTS: REF: 0606a STA: A.A.60 TOP: Unit Circle 8

144 5 ANS: =60, which terminates in Quadrant I. PTS: REF: 060a STA: A.A.60 TOP: Unit Circle 5 ANS: If csc P > 0, sin P > 0. If cot P < 0 and sin P > 0, cos P < 0 PTS: REF: 060a STA: A.A.60 TOP: Finding the Terminal Side of an Angle 5 ANS: PTS: REF: 06a STA: A.A.60 TOP: Finding the Terminal Side of an Angle 55 ANS: PTS: REF: 0a STA: A.A.56 TOP: Determining Trigonometric Functions KEY: degrees, common angles 56 ANS: = 6 PTS: REF: 06a STA: A.A.56 TOP: Determining Trigonometric Functions KEY: degrees, common angles 57 ANS:. sin θ = y = x + y ( ) + =. csc θ =. PTS: REF: fall09a STA: A.A.6 TOP: Determining Trigonometric Functions 58 ANS: sec θ = x + y x = ( ) + 0 = = PTS: REF: 050a STA: A.A.6 TOP: Determining Trigonometric Functions 59 ANS: x = = y = = PTS: REF: 0655a STA: A.A.6 TOP: Determining Trigonometric Functions 60 ANS: cos θ = 5 sec θ = 5 PTS: REF: 06a STA: A.A.6 TOP: Determining Trigonometric Functions 9

145 6 ANS: PTS: REF: 065a STA: A.A.66 TOP: Determining Trigonometric Functions 6 ANS: PTS: REF: 00a STA: A.A.66 TOP: Determining Trigonometric Functions 6 ANS: PTS: REF: 067a STA: A.A.66 TOP: Determining Trigonometric Functions 6 ANS: PTS: REF: 08007a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: basic 65 ANS: PTS: REF: 00a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: unit circle 66 ANS: PTS: REF: 0a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced 67 ANS: tan0 =. Arc cos k k = 0 = cos 0 k = PTS: REF: 06a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced 0

146 68 ANS: If sin A = sin A, cos A =, and tana = 5 cos A = 5 5 = 7 PTS: REF: 0a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced 69 ANS: If sin θ = 5 7, then cos θ = 8 7. tanθ = = 8 5 PTS: REF: 08508a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced 70 ANS: cos( ) = cos(55 ) PTS: REF: 060a STA: A.A.57 TOP: Reference Angles 7 ANS: PTS: REF: 0855a STA: A.A.57 TOP: Reference Angles 7 ANS: s = θ r = = 8 PTS: REF: fall09a STA: A.A.6 TOP: Arc Length KEY: arc length 7 ANS: s = θ r = π 8 6 = π PTS: REF: 06a STA: A.A.6 TOP: Arc Length KEY: arc length 7 ANS: 8 50' π.6 radians s = θ r = PTS: REF: 05a STA: A.A.6 TOP: Arc Length KEY: arc length 75 ANS: s = θ r = π 5 8 PTS: REF: 056a STA: A.A.6 TOP: Arc Length KEY: arc length

147 76 ANS: r = 6.6 = 9.9 PTS: REF: 085a STA: A.A.6 TOP: Arc Length KEY: radius 77 ANS: s = θ r = π = 6π PTS: REF: 06a STA: A.A.6 TOP: Arc Length KEY: arc length 78 ANS: Cofunctions tangent and cotangent are complementary PTS: REF: 060a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 79 ANS: sin θ + cos θ sin θ = cos θ = sec θ PTS: REF: 06a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 80 ANS: cos θ cos θ cos θ = cos θ = sin θ PTS: REF: 060a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 8 ANS: a a = 90 a + 5 = 90 a = 75 a = 5 PTS: REF: 00a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 8 ANS: cos x sin x sin x cot x sin x sec x = cos x = cos x PTS: REF: 06a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships

148 8 ANS: cot x csc x = cos x sin x sin x = cos x PTS: REF: 060a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 8 ANS: sin x + cos x sin x = sin x + cos x = cos x (cos x) = cos x sin x cos x = sin x cos x cos x sin x cos x = sin x cos x sin x = csc x cot x = PTS: REF: 055a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 85 ANS: cos x cos x cos x cos x = cos x = sin x PTS: REF: 085a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 86 ANS: Cofunctions secant and cosecant are complementary PTS: REF: 065a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 87 ANS:. If sin 60 =, then csc 60 = = PTS: REF: 05a STA: A.A.59 TOP: Reciprocal Trigonometric Relationships 88 ANS: sin 0 = csc 0 = = PTS: REF: 08505a STA: A.A.59 TOP: Reciprocal Trigonometric Relationships 89 ANS: PTS: REF: 008a STA: A.A.67 TOP: Simplifying Trigonometric Expressions 90 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

149 9 ANS: sec θ sin θ cot θ = cos θ sin θ cos θ sin θ = PTS: REF: 08a STA: A.A.67 TOP: Proving Trigonometric Identities 9 ANS: sin θ sin θ cos θ sin θ = cos θ cos θ = cos θ PTS: REF: 06a STA: A.A.67 TOP: Proving Trigonometric Identities 9 ANS: PTS: REF: fall090a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying 9 ANS: 5 cos B + sin B = tanb = sin B cos B = = 5 tan(a + B) = cos 5 5 = 0 = B + = cos B + 5 = cos B = 6 cos B = PTS: REF: 0807a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating 95 ANS: sin(5 + 0) = sin5cos 0 + cos 5 sin0 = = + = 6 + = 6 + PTS: REF: 066a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating 96 ANS: 5 cos(a B) = 5 5 = = 65 PTS: REF: 0a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating

150 97 ANS: sin(80 + x) = (sin80)(cos x) + (cos 80)(sinx) = 0 + ( sin x) = sin x PTS: REF: 08a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: identities 98 ANS: sin(θ + 90) = sin θ cos 90 + cos θ sin 90 = sin θ (0) + cos θ () = cos θ PTS: REF: 0609a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: identities 99 ANS: cos(x y) = cos x cos y + sin x sin y = b b + a a = b + a PTS: REF: 06a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying 500 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 50 ANS: cos A = 5 9 cos A = + + cos A = 5, sin A is acute. sin A = sin A cos A = 5 = 5 9 PTS: REF: 007a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 50 ANS: If sin x = 0.8, then cos x = 0.6. tan x = = 0..6 = 0.5. PTS: REF: 060a STA: A.A.77 TOP: Half Angle Identities 50 ANS: cos A = sin A = = 9 = 7 9 PTS: REF: 0a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 5

151 50 ANS: cos A = sin A = 8 = 9 = PTS: REF: 050a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 505 ANS: + cos A sin A = + cos A sin A cos A = cos A sin A = cot A PTS: REF: 065a STA: A.A.77 TOP: Double Angle Identities KEY: simplifying 506 ANS: cos θ = = 9 6 = = 8 PTS: REF: 085a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 507 ANS: tanθ = 0.. tanθ = θ = tan θ = 60, 0 PTS: REF: fall090a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 6

152 508 ANS: 0, 60, 80, 00. sin θ = sin θ sin θ sin θ = 0 sin θ cos θ sin θ = 0 sin θ(cos θ ) = 0 sin θ = 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 509 ANS: 5, 5 tanc = tanc = tanc tan = C C = 5,5 PTS: REF: 080a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 50 ANS: cos θ = cos θ = θ = cos = 60, 00 PTS: REF: 060a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 7

153 5 ANS: sec x = sec x = cos x = x = 5,5 PTS: REF: 0a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 5 ANS: 5csc θ = 8 csc θ = 8 5 sin θ = 5 8 θ PTS: REF: 06a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 5 ANS: sin x + 5 sin x = 0 ( sin x )(sin x + ) = 0 sin x = x = π 6, 5π 6 PTS: REF: 06a STA: A.A.68 TOP: Trigonometric Equations KEY: quadratics 5 ANS: sec x = cos x = cos x = x = 5,5 PTS: REF: 06a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 8

154 55 ANS: 5cos θ sec θ + = 0 5 cos θ cos θ + = 0 5 cos θ + cos θ = 0 (5 cos θ )(cos θ + ) = 0 cos θ = 5, θ 66.,9.6,80 PTS: 6 REF: 0659a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 56 ANS: ( sin x )(sin x + ) = 0 sin x =, x = 0,50,70 PTS: REF: 085a STA: A.A.68 TOP: Trigonometric Equations KEY: quadratics 57 ANS: cos x = cos x cos x cos x = 0 ( cos x + )(cos x ) = 0 cos x =, x = 0,0,0 PTS: REF: 068a STA: A.A.68 TOP: Trigonometric Equations KEY: double angle identities 58 ANS: π b = π = 6π PTS: REF: 0607a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions KEY: period 9

155 59 ANS: π b = π PTS: REF: 06a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 50 ANS: π b = π KEY: period b = PTS: REF: 05a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 5 ANS: π 6 = π PTS: REF: 06a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 5 ANS: π = π π π = PTS: REF: 0659a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 5 ANS: π = π KEY: period KEY: period KEY: period PTS: REF: 0859a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions KEY: period 5 ANS: PTS: REF: 067a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions KEY: period 55 ANS: π b = 0 b = π 5 PTS: REF: 07a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 0

156 56 ANS: y = sin x. 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 57 ANS: PTS: REF: 00a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 58 ANS: PTS: REF: 0606a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 59 ANS: a =, b =, c = y = cos x +. PTS: REF: 058a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 50 ANS: PTS: REF: fall09a STA: A.A.65 TOP: Graphing Trigonometric Functions 5 ANS: PTS: REF: 069a STA: A.A.65 TOP: Graphing Trigonometric Functions 5 ANS: period = π b = π π = PTS: REF: 0806a STA: A.A.70 TOP: Graphing Trigonometric Functions KEY: recognize 5 ANS: PTS: REF: 0600a STA: A.A.7 TOP: Graphing Trigonometric Functions 5 ANS: PTS: REF: 0a STA: A.A.7 TOP: Graphing Trigonometric Functions

157 55 ANS: PTS: REF: 007a STA: A.A.7 TOP: Graphing Trigonometric Functions 56 ANS: PTS: REF: 060a STA: A.A.6 TOP: Domain and Range 57 ANS: PTS: REF: 06a STA: A.A.6 TOP: Domain and Range 58 ANS: PTS: REF: 067a STA: A.A.6 TOP: Domain and Range 59 ANS: PTS: REF: 06a STA: A.A.6 TOP: Domain and Range 50 ANS: K = (0)(8) sin 0 = 5 78 PTS: REF: fall0907a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: basic 5 ANS: K = absinc = 0 sin57 60 PTS: REF: 060a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: parallelograms 5 ANS: K = (0)(8) sin6 9 PTS: REF: 080a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: parallelograms 5 ANS: (7.)(.8) sin6. PTS: REF: 08a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: basic 5 ANS: K = absinc = 8 sin60 = 96 = 98 PTS: REF: 06a STA: A.A.7 TOP: Using Trigonometry to Find Area KEY: parallelograms