What to Expect on the Placement Exam

Transcription

1 What to Epect on the Placement Eam Placement into: MTH 45 The ACCUPLACER placement eam is an adaptive test created by the College Board Educational Testing Service. This document was created to give prospective CLC student and other interested parties an overview of how the placement eam works and the types of questions one might epect to see on it. General Information: The eam is computerized. The eam is not timed. Questions are presented in a multiple-choice format. No partial credit is awarded. Scratch paper and a scientific calculator are provided to assist the eaminee with calculations. There is a limit to the number of times a student may take the placement eam. CLC Mathematics department wants each student to do as well as possible on the placement eam and makes the following recommendations: Prepare for the eam. Don t take it cold. Studying for the eam may decrease the number of math classes you are required to take. Free practice questions/answers are available online at the Accuplacer website, located at Other review materials are available at the Math Center. (847) Take the eam when you are rested and refreshed. Allow plenty of time for testing so that you can rela and fully concentrate on what you are doing. Triple-check answers before moving on to the net question. The computer only knows if your answer is right or wrong. It cannot tell the difference between a careless mistake, a minor error, or a major error. Stay calm if you don t know the answer to a question. Remember that the placement test is adaptive and is used to test many levels of mathematics. It increases or decreases the level of the questions based on your previous answers. A student desiring to place into Calculus (MTH 45) must first pass the twelve question Basic Algebra portion of the eam. The student will then be given the opportunity to answer questions from the College Mathematics portion of the eam. Any student intending to place into Calculus via the Math Placement Eam must be proficient with the types of eercises illustrated as eamples for entry into College Algebra (MTH )/Precalculus (MTH 45). In addition, the student must be proficient with the types of eercises illustrated in this document. NOTE: The eamples provided below are not intended to be a complete list of problem types. The eamples are simply illustrations of problem types. As of June, 07

2 . The student must be completely proficient with all aspects of linear functions. In particular, concepts involving slope, y-intercept, parallelism and perpendicularity must be mastered. a) What is the equation of the linear function with slope / and y-intercept of (0, 6) b) Write the equation of a line perpendicular to the line with intercepts of (, 0) and (0,-). c) Write the equation of a line parallel to the line with intercepts of (, 0) and (0,-). d) Write the equation of the line which passes through the points (, -) and (-/7, /). Write your answer in the form of y = m + b.. The student must know how to perform horizontal and vertical translations of elementary functions such as f ( ), f ( ), f ( ), f () e, f ( ) sin( ), f ( ) cos( ), and f ( ) ln. a) What is the relationship between (y-k) = f(-h) to the graph of y = f() when h = 7 and k = b) What is the relationship between (y-k) = f(-h) to the graph of y = f() when h = - and k =.5. The student must be proficient with the determination of implicit real-valued domains of functions defined by equations. a) For what values of will f() be a real number if b) For what values of will f() be a real number if c) For what values of will f() be a real number if d) For what values of will f() be a real number if f () f () f () 6 f () The student must be able to complete the square and manipulate and interpret results. Eample: Complete the square on the quadratic function defined by compare its graph with the graph of g ( ). f( ) 8 00 and 5. The student must be proficient with logarithmic and eponential functions and epressions. a) What is the inverse function of f ( ) log (5 ) b) What is the eponential form of log (5) c) If log7, then log 7 As of June, 07

3 6. The student must be able to graph functions of the form f ( ) Acos( BC) k or f ( ) Asin( BC) k. Eample: a) Graph f( ) cos.5 on the interval from - to. f( ) sin on the interval from - to. b) Graph 7. The student must know all of the fundamental trigonometric identities. In particular: a) opposite the right triangle identities like sin() = hypotenuse, etc b) the reciprocal identities like sec() = cos, etc c) the Pythagorean identities like sin cos, etc 8. The student must know right triangle trigonometry and be able to apply it to practical applications. a) A tower 00-feet tall is located at the top of a hill. At a point 500 feet down the hill, the angle between the surface of the hill and the line of sight to the top of the tower is 0. Find the inclination of the hill. b) Two men stand 00 feet apart with a flagpole situated on the line between them. If the angles of elevation from the men to the top of the flagpole are 0 and 5 respectively, how tall is the pole and how far is each man from it 9. The student must be able to write functions representing geometric shapes and various physical phenomena. a) Given that the perimeter of the figure is 0 feet, write an epression for. (It may be assumed that the dashed portion is a semi-circle.) y b) In the figure shown, the circle is tangent to the rectangle and the diameter of the circle is equal to the width of the rectangle. The length of the rectangle is triple the width. Epress the area of the entire figure as a function of. As of June, 07

4 0. The student will be epected to solve non-linear systems of equations a) Solve the system. y 7 y b) Solve the system. y 7 y. The student will be epected to solve trigonometric equations. a) Solve sin = -cos on the interval (-, ). b) Solve sin sin 0 on the interval (-, ). c) Solve sin( ) on the interval (-, ).. The student must be familiar with basic theorems regarding polynomials of degree n in. 4 a) Any rational zeros of f ( ) 5 7 must look like what b) If + i is a zero of 4 0, what are the values of all the other zeros c) If i is a zero of a nd degree polynomial in, then what could this polynomial be. The student must be proficient with the arithmetic of comple numbers. Recall that i. a) Simplify i 5i b) Simplify ( + i)( i) 4. The student must know the algebraic and graphical relationship between a function and its inverse function when such an inverse eists. a) If f ( ) log ( ), what is the value of f () b) If f ( ) 5, what is the value of f 5. The student must be familiar with the Binomial theorem and be able to apply it. Eample: If 6 5 were epended and the terms were put in decreasing order according to the power of, what is the 9 th term of the polynomial As of June, 07 4

5 What to Epect on the Placement Eam Placement into: MTH 45 Solutions The ACCUPLACER placement eam is an adaptive test created by the College Board Educational Testing Service. This document was created to give prospective CLC student and other interested parties an overview of how the placement eam works and the types of questions one might epect to see on it. General Information: The eam is computerized. The eam is not timed. Questions are presented in a multiple-choice format. No partial credit is awarded. Scratch paper and a scientific calculator are provided to assist the eaminee with calculations. There is a limit to the number of times a student may take the placement eam. CLC Mathematics department wants each student to do as well as possible on the placement eam and makes the following recommendations: Prepare for the eam. Don t take it cold. Studying for the eam may decrease the number of math classes you are required to take. Free practice questions/answers are available online at the Accuplacer website, located at Other review materials are available at the Math Center. (847) Take the eam when you are rested and refreshed. Allow plenty of time for testing so that you can rela and fully concentrate on what you are doing. Triple-check answers before moving on to the net question. The computer only knows if your answer is right or wrong. It cannot tell the difference between a careless mistake, a minor error, or a major error. Stay calm if you don t know the answer to a question. Remember that the placement test is adaptive and is used to test many levels of mathematics. It increases or decreases the level of the questions based on your previous answers. A student desiring to place into Calculus (MTH 45) must first pass the twelve question Basic Algebra portion of the eam. The student will then be given the opportunity to answer questions from the College Mathematics portion of the eam. Any student intending to place into Calculus via the Math Placement Eam must be proficient with the types of eercises illustrated as eamples for entry into College Algebra (MTH )/Precalculus (MTH 45). In addition, the student must be proficient with the types of eercises illustrated in this document. NOTE: The eamples provided below are not intended to be a complete list of problem types. The eamples are simply illustrations of problem types. As of June, 07 5

6 . The student must be completely proficient with all aspects of linear functions. In particular, concepts involving slope, y-intercept, parallelism and perpendicularity must be mastered. a) What is the equation of the linear function with slope / and y-intercept of (0, 6) y = m + b gives y 6 b) Write the equation of a line perpendicular to the line with intercepts of (, 0) and (0,-). The slope of the line connecting the given points is. A line perpendicular to this would have a slope of. On possible answer is y =. c) Write the equation of a line parallel to the line with intercepts of (, 0) and (0,-). The slope of the line connecting the given points is have the same slope. On possible answer is y =.. A line parallel to this would d) Write the equation of the line which passes through the points (, -) and (-/7, /). Write your answer in the form of y = m + b. m = ( ) y y m( ) gives 77 y ( ) which simplifies to y = The student must know how to perform horizontal and vertical translations of elementary functions such as f ( ), f ( ), f ( ), f () e, f ( ) sin( ), f ( ) cos( ), and f ( ) ln. a) What is the relationship between (y k) = f( h) to the graph of y = f() when h = 7 and k = h shifts the graph to the right 7 units and k shifts the graph up units. b) What is the relationship between (y k) = f( h) to the graph of y = f() when h = - and k =.5 h shifts the graph left units and k shifts the graph up.5 units. As of June, 07 6

7 . The student must be proficient with the determination of implicit real-valued domains of functions defined by equations. a) For what values of will f() be a real number if f () Within the square root, 0 0 As a denominator, ( 5)( 4) 0 7 5, f () will be a real number if > 5 or if < -4. b) For what values of will f() be a real number if f () 6 Within the square root, 6 0 As a denominator, 0 (4 )(4 + ) > 0 is always positive f () will be a real number if -4 < < 4. c) For what values of will f() be a real number if f () 6 As a denominator, 0, so 0. As a denominator, 6 0, so 8. Within the square root, > 0, so > 0. f () will be a real number if > 0 but 8. d) For what values of will f() be a real number if f () 7 5 As a denominator within a square root, 5 > 0, so () f will be a real number if > 5 or if < -5. As of June, 07 7

8 4. The student must be able to complete the square and manipulate and interpret results. Eample: Complete the square on the quadratic function defined by compare its graph with the graph of g ( ). f( ) 8 00 and = ( 7) So if y = 8 00, then y + = ( 7) This is a parabola with a verte at (7, -), it opens down, and is vertically stretched by a factor of. 5. The student must be proficient with logarithmic and eponential functions and epressions. a) What is the inverse function of f ( ) log (5 ) f ( ) 5 b) What is the eponential form of log (5) 5 c) If log7, then log 7 7, so log 7 log The student must be able to graph functions of the form f ( ) Acos( BC) k or f ( ) Asin( BC) k. Eample: a) Graph f( ) cos.5 - to. on the interval from As of June, 07 8

9 ( ) sin on the interval from b) Graph f - to. 7. The student must know all of the fundamental trigonometric identities. In particular: a) opposite the right triangle identities like sin() = hypotenuse, etc b) the reciprocal identities like sec() = cos, etc c) the Pythagorean identities like sin cos, etc 8. The student must know right triangle trigonometry and be able to apply it to practical applications. a) A tower 00-feet tall is located at the top of a hill. At a point 500 feet down the hill, the angle between the surface of the hill and the line of sight to the top of the tower is 0. Find the inclination of the hill. Law of Sines says sin0 sin y, so y = y 00 ft y = 90, The angle of inclination of the hill is = ft b) Two men stand 00 feet apart with a flagpole situated on the line between them. If the angles of elevation from the men to the top of the flagpole are 0 and 5 respectively, how tall is the pole and how far is each man from it y y tan 5 and tan tan 5 (00 ) tan 0 So one man is = 0.6 ft, away from the flag pole, the other is 00 = 89.4 ft away and the flag pole is y = 5.6 ft tall. y As of June, 07 9

10 9. The student must be able to write functions representing geometric shapes and various physical phenomena. a) Given that the perimeter of the figure is 0 feet, write an epression for. (It may be assumed that the dashed portion is a semi-circle.) y Perimeter = y + y + + ()/ = 0 So = 0 y b) In the figure shown, the circle is tangent to the rectangle and the diameter of the circle is equal to the width of the rectangle. The length of the rectangle is triple the width. Epress the area of the entire figure as a function of. A ( ) 4 0. The student will be epected to solve non-linear systems of equations a) Solve the system. y 7 y Substitute. ( 7) = - 7 = 9 = 0 Use the quadratic formula to find = 5. Coordinate pairs are , and, b) Solve the system. y 7 y Substitute y = into the other equation. ( ) = 0 Use the quadratic formula to find = 9 9 pairs are,4 9 and,4 9. Coordinate As of June, 07 0

11 . The student will be epected to solve trigonometric equations. a) Solve sin = -cos on the interval (-, ). Knowledge of the unit circle directly shows that =,. 4 4 b) Solve sin sin 0 on the interval (-, ). (sin )(sin ) = 0 so sin = 0, meaning sin = ½ 5 This means =,. 6 6 or sin = 0 meaning sin = This means =. 5 =,, 6 6 c) Solve sin( ) on the interval (-, ). To make sin( ), = k or = 5 k 6 6 So k and 8 5 k 8 = 9 8,, 8 7, 8 8, 5 8, 8. The student must be familiar with basic theorems regarding polynomials of degree n in. a) Any rational zeros of 4 f ( ) 5 7 must look like what The Rational Zeros Theorem says that any rational zeros must come from the list,, 5, 5 As of June, 07

12 b) If + i is a zero of 4 0, what are the values of all the other zeros If + i is a solution, so is its conjugate, i. This means that ( ( + i)) and ( ( i)) are both factors of the polynomial. Division shows that the other factor is ( ). The other two solutions are i and /. c) If i is a zero of a nd degree polynomial in, then what could this polynomial be If i is a solution, so is its conjugate + i. This means that ( ( i)) and ( ( + i)) are both factors of the polynomial. Multiplying these factors together gives f ( ) 6. The student must be proficient with the arithmetic of comple numbers. Recall that i. a) Simplify i 5i i 5i 60ii5i i 5i 5i 95i5i5i 4 b) Simplify ( + i)( i) ( + i)( i) = 4i + i i = 4 i 4. The student must know the algebraic and graphical relationship between a function and its inverse function when such an inverse eists. a) If f ( ) log ( ), what is the value of f () f ( ), so 9 b) If f ( ) 5, what is the value of f f ( ) 5, so f = 7 As of June, 07

13 5. The student must be familiar with the Binomial theorem and be able to apply it. Eample: If 6 5 were epended and the terms were put in decreasing order according to the power of, what is the 9 th term of the polynomial 7 8 5C8 6 =,8,466,06,880 7 As of June, 07