30S Pre-Calculus Final Exam Review Chapters 1-4

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1 30S Pre-Calculus Final Exam Review Chapters 1 - Name:

2 30S Pre-Calculus Final Exam Formula Sheet

3 30S Pre-Calculus Exam Review- Chapter 1 Sequences and Series Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (1 point) Which term below is a term of an arithmetic sequence with and? A. 11 C. 10 B. 13 D (1 point) Two terms of an arithmetic sequence are and. What is t 21? A. C. B. D. 3. (1 point) Determine the common difference, d, of this arithmetic sequence: 2, 2.5, 7, 11.5,... A. C. B. D. The sequence has no common difference.. (1 point) In the arithmetic sequence: 18, 10, 2, 6,...; which term has the value 222? A. C. B. D. 5. (1 point) Two terms of an arithmetic sequence are and. What is t 1? A. C. B. D. 6. (1 point) Determine the number of terms in an arithmetic sequence with and. The final term in the sequence is A. C. B. D. 7. (1 point) The number 25. is a term of an arithmetic sequence with and. What is its term number. A. t 27. C. t 29. B. t 25. D. t (1 point) Determine the sum of the given terms of this arithmetic series: A. 10 C. 12 B. 520 D (1 point) An arithmetic series has and ; determine t 1. A. C. B. D. 10. (1 point) The sum of the first 25 terms of an arithmetic series is The sum of the first 26 terms is The common difference is 5. Determine t 1. A. C. B. D.

4 11. (1 point) Determine the sum of the first 17 terms of this arithmetic series:... A. C. B. D. 12. (1 point) An arithmetic series has,, and ; determine the first 3 terms of the series. A. C. B. D. 13. (1 point) Which sequence could be geometric? A. 2 6, 2, 3, 2 9,... C. 2 6, 2, 3, 2 9,... B. 6, 9 2, 18 5, 3,... D. 6, 3, 2, 3 2, (1 point) is a term in which geometric sequence? A. 2,, 8, 16,... C. 2, 12, 72, 32,... B., 2, 1, 86,... D. 3, 6, 12, 2, (1 point) Determine r and t 6 of this geometric sequence: 32, 108, 36, 12,... A. 1 r = 3, t6 = 1 C. r = 3, t6 = 2 27 B. r = 1 6, t6 = D. 1 r = 3 3, t6 = (1 point) The sum of the first 12 terms of which geometric series is 560? A. C. B. D. 17. (1 point) The sum of the first 13 terms of a geometric series is The common ratio is 3. Determine the 1st term. A. C. B. D. 18. (1 point) Determine the common ratio of a geometric series that has these partial sums:,, A. C. B. D. 19. (1 point) The term values of a geometric sequence increase as more points are plotted. Choose the most appropriate value for the common ratio. A. r = 0.3 C. r = 5 B. r = 9 D. r = (1 point) The term values of a geometric sequence decrease and approach 0 as more points are plotted. Choose the most appropriate value for the common ratio. A. r = C. r = 0.8 B. r = 0. D. r = 8

5 21. (1 point) The common ratio of a geometric sequence is 1.5. Which graph could represent this geometric sequence? A. C. Geometric Sequence Geometric Sequence Term value Term number Term value Term number B. Geometric Sequence D. Geometric Sequence Term value Term number Term value Term number 22. (1 point) Determine whether this infinite geometric series has a finite sum: If it does, determine the sum. A. This series does not have a finite sum. C. B. D. 23. (1 point) An infinite geometric series has and. Determine. A. C. B. The sum is not finite. D. 2. (1 point) A geometric series has and. Determine t 1. A. C. B. D. Short Answer 25. (1 point) Determine r, t 5, and t 6 of this geometric sequence: 3125, 625, 125, 25, (1 point) Calculate the sum of this geometric series:

6 27. (1 point) Describe the graph of the partial sums of this geometric series: (1 point) Determine whether this infinite geometric series has a finite sum: Problem 29. (1 point) Cicadas are insects that spend many years growing underground before emerging as adults for a few weeks or months. Entire populations of cicadas of the genus Magicicada emerge at the same time. Each emerging group is collectively known as a brood. The cicada broods are numbered using Roman numerals. Brood XIII cicadas appear every 17 years and were last seen in the year Determine whether Brood XIII cicadas should appear in (1 point) Use the given data about each arithmetic series to determine the indicated value. a)...; determine S 16 b) ; determine S 17 c) ; determine S (1 point) Use the given data about each arithmetic series to determine the indicated value. a) ; determine t 1 b) ; determine n 32. (1 point) a) A geometric sequence has these terms: t = 8, t 5 = 2, t 6 = 1 2 State the common ratio, then write the first 3 terms of the sequence. b) Identify the sequence as convergent or divergent. Explain. 33. (1 point) An infinite geometric series with is represented by this equation: a) Determine the first terms of the series. b) Determine whether the series diverges or converges. c) If the series has a finite sum, determine the sum.

7 30S Pre-Calculus Exam Review- Chapter 1 Sequences and Series Answer Section MULTIPLE CHOICE 1. D 2. C 3. B. D 5. A 6. D 7. B 8. C 9. B 10. B 11. D 12. D 13. C 1. A 15. D 16. D 17. D 18. A 19. C 20. C 21. C 22. A 23. B 2. D SHORT ANSWER 25. r = 1, t5 = 5, t6 = The common ratio is, which is between 0 and 1, so the partial sums increase and approach a constant value. 28., so the sum is not finite. PROBLEM 29. The years in which each brood appears form an arithmetic sequence. The arithmetic sequence for Brood XIII cicadas has and. To determine whether Brood XIII cicadas should appear in the year 236, determine whether 236 is a term of its sequence.

8 30. a) Since n is a natural number, the year 236 is a term in the arithmetic sequence. Brood XIII cicadas should appear in 236. b) c) 31. a)

9 b) 32. a) The first 3 terms are: 33. a) b) The sequence is convergent because the terms approach a constant value of 0. t 2 is: 5 t 3 is: 5 9 t is:

10 b), so the series converges. c)

11 30S Pre-Calculus Exam Review - Chapter 2 Absolute Value and Radicals Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which point on the number line has an absolute value of? Z Y W X A. X B. W C. Z D. Y What is the distance between 18.1 and 9.7 on a number line? A B C D Evaluate: A. 8 9 B C. 9 1 D Evaluate: A. 7.3 B..1 C D Evaluate: A B C D Write this entire radical as a mixed radical: A. B. C. D. 7. Arrange these radicals in order from greatest to least. i) ii) iii) iv) A. iv, i, iii, ii B. ii, iii, i, iv C. iii, ii, i, iv D. ii, iv, i, iii 8. For which values of the variable, x, is this radical defined? A. B. C. D. the radical is never defined

12 9. Which radical expression simplifies to? A. B. C. D. 10. Which radical expression simplifies to? A. B. C. D. 11. Which radical expression simplifies to? A. B. C. D. 12. Which radical expression simplifies to A. B. C. D. 13. Simplify this radical, if possible: A. B. C. D. cannot be simplified 1. Simplify by adding or subtracting like terms: A. B. C. D. 15. Simplify by adding or subtracting like terms: A. B. C. D. 16. Simplify: A. B. C. D. 17. Simplify by adding or subtracting like terms: A. B. C. D. 18. Expand and simplify this expression: A. B. C. D. 19. Expand and simplify this expression: A. B. C. D. 20. Expand and simplify this expression: A. B. C. D.

13 21. Expand and simplify this expression: A. B. C. D. 22. Solve this equation: A. x = 16 B. x = 6 C. x = 1 8 D. x = Solve this equation: A. B. C. D. 2. Solve this equation: A. x = 5 2 B. x = 25 C. x = 5 D. x = 10 Short Answer 25. Determine the root of each equation. a) b) c) d) 26. The speed of sound in air is given by the equation, where s is the speed in metres per second and t is the temperature in degrees Celsius. a) To the nearest degree Celsius, determine the temperature when the speed of sound is 318 m/s. b) To the nearest metre per second, determine the speed of sound when the temperature is 20 C. Problem 27. Order the absolute values of the numbers in this set from least to greatest. Describe the strategy you used., 16, 13, 10.5, 18.9, 20.7, When 8 is added to an integer, x, the absolute value of the sum is 5. Determine a value for x. How many different values of x are possible? Show how you solved the problem. 29. Sixteen congruent squares are placed together to form a large square. The middle squares are removed. The final shape is a square within a square. The area of the large square is 80 square units.

14 a) What is the area of the inner square? b) What is the difference between the perimeters of the outer square and the inner square? Explain your work. 30. Expand and simplify this expression: Show your work. 31. Write this expression in simplest form: Describe your strategy. 32. a) Identify the values of the variables for which this expression is defined. b) Write the expression in simplest form. Show your work. 33. Determine whether the given value of x is a root of this equation. Justify your answer. ; 3. Does this equation have a real root? Justify your answer. 35. The volume of water in Lake Ontario is about 1710 km 3. a) To the nearest kilometre, determine the edge length of a cube with the same volume. b) To the nearest kilometre, determine the radius of a sphere with the same volume.

15 30S Pre-Calculus Exam Review - Chapter 2 Absolute Value and Radicals Answer Section MULTIPLE CHOICE 1. C 2. B 3. A. D 5. C 6. D 7. B 8. D 9. B 10. D 11. A 12. A 13. D 1. A 15. D 16. D 17. B 18. A 19. C 20. A 21. A 22. A 23. B 2. C SHORT ANSWER 25. a) b) c) The equation has no real root. d) 26. a) about 20 C b) about 32 m/s PROBLEM 27. Determine the absolute value of each number, then order the results from least to greatest. = 16 = = = = = = 11.1

16 So, the absolute values from least to greatest are:, 10.5, 11.1, 13, 16, 18.9, Write, then solve an equation: Since 5 = 5 and = 5 then, or So, two values of x are possible: 3 or a) The area of the large square is 80 square units. So, the area of each small square is: The inner square has the area of small squares: The area of the inner square is 20 square units., or 5 square units. b) The side length of a small square is the square root of its area: units The perimeter of the outer square is equal to 16 times the side length of the small square: units The perimeter of the inner square is equal to 8 times the side length of the small square: 30. Difference between perimeters: The difference between the perimeters of the outer square and the inner square is units. 31. Simplify the denominators. Rationalize the denominators. 32. a)

17 b) 33. Since the left side does not equal the right side, 3. Since, then 2 is not a root of the equation. Since, then 2 3 So, for both radicals to be defined, 2 3 Since x = 2 lies in the set of possible values for x, is a root of the equation. 35. a) Use the formula for the volume, V, of a cube:, where s represents the edge length of the cube. A cube with the same volume as Lake Ontario would have an edge length of about 12.0 km. b) Use the formula for the volume, V, of a sphere:, where r represents the radius of the sphere. A sphere with the same volume as Lake Ontario would have a radius of about 7. km.

18 Chapter 3 Solving Quadratic Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor: A. ( )( ) B. ( )( ) C. ( )( ) D. ( )( ) 2. Factor this polynomial: A. ( )( ) B. ( )( ) C. ( )( ) D. ( )( ) 3. Factor this polynomial: A. B. C. D.. Factor this polynomial expression: A. B. C. D. 5. Which equations have only one root? i) ii) iii) iv) A. i and ii B. i, ii, and iv C. ii and iii D. i and iv 6. Solve by factoring: A. x = 2 3 or x = 2 B. x = 2 or x = 2 C. x = 2 or x = 2 D. x = 2 3 or x = 2 7. Solve by completing the square. A. x = or x = B. x = 7 2 or x = 1 C. x = 2 or x = 7 2 D. x = 1 or x = 7 8. Which expression is a solution of the equation? A. B. C. D. 9. Solve this quadratic equation: A. B. C. D. 10. The quadratic equation has only one root. Use the quadratic formula to determine the value of d. A. B. C. D.

19 11. The coefficients of a quadratic equation are all integers. The discriminant is 9. Which statement best describes its roots? A. Two rational roots B. One rational root C. No real roots D. Two irrational roots 12. The coefficients of a quadratic equation are all integers. Which discriminant indicates that the equation has two irrational roots? A. B. 6 C. 0.6 D Without solving, determine the number of real roots of this equation: A. 2 B. 0 C. 1 Short Answer 1. Is a factor of the trinomial? 15. Factor this trinomial: 16. Factor this trinomial: 17. Factor this polynomial expression: 18. Factor this polynomial expression: 19. A baseball is hit upward. The approximate height of the baseball, h metres, after t seconds is modelled by this formula: When is the baseball 11 m high? 20. When 3 times a number is added to the square of the number, the result is 0. Determine the number. 21. Solve this equation: 22. The formula models the height, h metres, of an object t seconds after it is dropped from the top of a tower that is 2 m tall. a) When will the object hit the ground? Give the answer to the nearest tenth of a second. b) What is the height of the object 5 s after it is dropped from the top of the tower? 23. Consider the quadratic equation, where b is a constant. Determine the possible values of b so that this equation has real solutions. 2. When 7 times a number is added to the square of the number, the sum is 3. What is the number? 25. Consider the quadratic equation. Determine the possible values of c so that this equation has no solution.

20 26. Consider this quadratic equation: 1 x2 2x = 0 a) Rewrite the equation so that it does not contain fractions. b) Solve the equation. Give the answer to 3 decimal places. 27. a) Determine the value of the discriminant for this equation: b) Use the value of the discriminant to choose a solution strategy, then solve the equation. Problem 28. The perimeter, P, of a rectangular concrete slab is 6 m and its area, A, is 90 m 2. Use the formula. Determine the dimensions of the slab. Show your work. 29. Solve this equation, then verify the solution: Explain your steps. 30. A ball is thrown in the air. The approximate height of the ball, h metres, after t seconds can be modelled by the equation. Will the ball ever reach a height of 15 m? Explain your answer. 31. Solve this equation using each strategy below. Which strategy do you prefer? Explain why. a) factoring b) completing the square c) using the quadratic formula 32. For each equation, choose a solution strategy, justify your choice, then solve the equation. a) b) c) d) 33. a) Solve this quadratic equation by expanding, simplifying, then applying the quadratic formula: b) Solve the equation in part a using the quadratic formula without expanding. 3. Determine the values of k for which the equation has two real roots, then write a possible equation. 35. Determine the values of k for which the equation has exactly one real root, then write a possible equation.

21 Chapter 3 Solving Quadratic Equations Review Answer Section MULTIPLE CHOICE 1. C 2. C 3. D. D 5. D 6. D 7. D 8. C 9. D 10. D 11. C 12. D 13. B SHORT ANSWER 1. is not a factor of the trinomial The baseball is 11 m high after 1 s and after 2 s. 20. There are 2 numbers: 5 and a) The object will hit the ground after approximately 9. s. b) The height of the object is 317 m. 23. or 2. There are two numbers: or a) b) or 27. a) b) The discriminant is a perfect square, so use factoring. x = 3 or x = 3

22 PROBLEM 28. The length of the slab, in metres, is. The area of the slab is 90 m 2. Use the formula. Either w = 5 or w = 18 Determine the value of l when w = 5. The width of the slab is 5 m and its length is 18 m. Or Determine the value of l when w = The width of the slab is 18 m and its length is 5 m. So, there is one slab of dimensions 18 m by 5 m. Square each side of the equation. Either or Combine like terms. Factor. Solve using the zero product property. Check for extraneous roots. In, substitute: and

23 L.S. L.S. R.S. R.S. 30. For, the left side does not equal the right side, so is not a root of the radical equation. For, the left side is equal to the right side, so this solution is verified. The root is: Substitute: Divide each term by 5. Complete the square. The left side is a perfect square and the right side is positive, so there is at least one solution to this equation. The ball will reach a height of 15 m. 31. a) Find two numbers whose product is 3 and whose sum is. The numbers are 1 and 3. b) c) Substitute: in: or Sample response: I prefer factoring because it takes less time and less space. 32. Sample responses: a) I can take the square root of each side.

24 b) I cannot find two numbers whose product is and whose sum is 3. So, I ll use the quadratic formula. Substitute: in: c) I can factor by decomposition. or d) I cannot find two numbers whose product is 3 and whose sum is. So, I ll use the quadratic formula. Substitute: in: 33. a) Since is not a real number, this equation has no real roots. Substitute: in: b) Substitute: in:

25 3. For an equation to have two real roots, Substitute: For to have two real roots, k must be less than 25. Sample response: A possible value of k is. So, an equation with two real roots is: 35. For an equation to have exactly one real root, Substitute: For to have exactly one real root, k must be equal to. Sample response: A possible value of k is 6. So, an equation with exactly one real root is:

26 Chapter : Analyzing Quadratic Functions Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. For a quadratic function, which characteristic of its graph is equivalent to the zero of the function? A. minimum point C. x-intercept B. maximum point D. y-intercept 2. Which statement below is NOT true for the graph of a quadratic function? A. The vertex of a parabola is its highest or lowest point. B. When the coefficient of is positive, the vertex of the parabola is a minimum point. C. The axis of symmetry intersects the parabola at the vertex. D. The parabola is symmetrical about the y-axis. 3. Which graph represents the quadratic function? A. 6 y C. 6 y x x 2 6 B. 6 y D. 6 y x x 2 6

27 . Which graph represents the quadratic function? A. 6 y C. 6 y x x B. 6 y D. 6 y x x Identify the y-intercept of the graph of this quadratic function: A. 37 B. 0 C. 39 D. 6. Use the graph of to determine the roots of. 12 y x 8 12 A. and C. and B. and D. and

28 7. Which of the following describes the translation that would be applied to the graph of to get the graph of? A. Translate 5 units left C. Translate 5 units down B. Translate 5 units up D. Translate 5 units right 8. Identify the coordinates of the vertex, the axis of symmetry, and the coordinates of points P and Q on the graph of this quadratic function. y Q P x 2 A. vertex: ( 1, 0); line of symmetry: x = 1; P(0, 1), Q(0, 1) B. vertex: ( 1, 0); line of symmetry: x = 1; P( 2, 1), Q(1, ) C. vertex: (0, 1); line of symmetry: x = 0; P( 2, 1), Q(, 1) D. vertex: (0, 1); line of symmetry: x = 0; P(0, 1), Q(1, 0)

29 9. Match the quadratic function to a graph below. A. y C. 6 6 y x x 2 6 y = f(x) 6 y = g(x) B. 6 y D. y = h(x) 6 y y = k(x) x x Determine an equation of a quadratic function with the given characteristics of its graph: coordinates of the vertex: V(0, 2); passes through A( 2, 18) A. C. B. D. 11. Which equation represents the same quadratic function as? A. C. B. D. 12. Write this equation in standard form: A. C. B. D. 13. Write in standard form, then identify the coordinates of the vertex. A. ; vertex: (3, ) B. ; vertex: (3, ) C. ; vertex: (3, ) D. ; vertex: (3, )

30 1. Determine the x-intercepts of the graph of this quadratic function: A. 8 3 and 6 B. 8 3 and 6 C. 8 3 and 6 D. 8 3 and Two numbers have a difference of 10. The sum of their squares is a minimum. Determine the numbers. A. 5 and 15 B. 2 and 8 C. 0 and 10 D. 5 and A ball is thrown upward with a speed of 16 m/s. Its height, h metres, after t seconds is modelled by the equation. What is the maximum height of the ball, to the nearest tenth? A m B. 1 m C m D. 1. m Short Answer 17. a) Use a table of values to graph the quadratic function for. y x b) Determine the x-intercepts. 18. A ball was thrown into the air with an upward velocity of 8 m/s. Its height, h metres, after t seconds is modelled by the equation. Use a graphing calculator to answer the questions below. Give your answers to the nearest tenth, if necessary. a) After how many seconds did the ball reach its maximum height? b) What was the ball s maximum height?

31 19. The graph of a quadratic function is shown. What can you say about the discriminant of the corresponding quadratic equation? 12 y x Use the graph of to determine the roots of. 12 y x How many x-intercepts does the graph have? a) b) c) 22. Determine an equation of a quadratic function with x-intercepts of 3 and 5, that passes through the point A(, 21). 23. Write this equation in standard form: 2. Determine the x-intercepts of the graph of. 25. Determine the x- and y-intercepts, the equation of the axis of symmetry, and the coordinates of the vertex of the graph of.

32 26. The graph of a quadratic function passes through A(3, 12) and has x-intercepts 1 and 5. Write an equation of the graph in factored form. 27. The graph of a quadratic function passes through B(, 6), and the zeros of the function are 5 and 6. Write an equation of the graph in general form. 28. The weekly profit of a manufacturer, P hundreds of dollars, is modelled by the equation, where x is the number of units produced per week, in thousands. a) How many units should the manufacturer produce per week to maximize profit? b) What is the maximum weekly profit? 29. A science museum wants to build an outdoor patio. The patio will be bordered on one side by a wall of the museum and the other 3 sides by 8 m of fencing. Determine the area of the largest patio possible. 30. A coffee shop sells coffee for $1.0 a cup. At this price, the store sells approximately 800 cups per day. Research indicates that for every $0.05 increase in price, the store will sell 0 fewer cups. Determine the price of a cup of coffee that will maximize the revenue. Problem 31. Graph the quadratic function. y x Determine: a) the intercepts b) the coordinates of the vertex c) the equation of the axis of symmetry d) the domain of the function e) the range of the function

33 Chapter : Analyzing Quadratic Functions Exam Review Answer Section MULTIPLE CHOICE 1. C 2. D 3. A. D 5. C 6. B 7. B 8. B 9. D 10. D 11. D 12. C 13. C 1. D 15. D 16. A SHORT ANSWER 17. a) x y x y

34 2 y x 8 b) The equation has 1 x-intercept: a) The ball reached its maximum height after 0.8 s. b) The ball s maximum height was 5.2 m. 19. The graph intersects the x-axis at 2 points, so the related quadratic equation has 2 real roots. This means that the discriminant is greater than a) The graph has 2 x-intercepts. b) The graph has 1 x-intercept. c) The graph has 0 x-intercepts x-intercepts: and 25. y-intercept: 6 x-intercepts: 1 and 3 equation of the axis of symmetry: coordinates of the vertex: (2, 2) a) 2000 b) $ m $1.83 PROBLEM 31. x

35 y y x a) x-intercepts: 1, 3 y-intercept: 3 b) vertex: (2, 1) c) axis of symmetry: a) domain: b) range:,