FMSS 2013 Page 1 of 16

MCR3U0: Unit 2 – Equivalent Expressions and

Quadratic Functions

Radical Expressions

1) Express as a mixed radical in simplest form.

a) c) e)

b) d)

f)

2) Simplify.

a) d)

b) e)

c) f)

3) Simplify.

a) d)

b) e)

c) f)

4) Simplify.

a) d)

b) e)

c) f)

For questions 5 to 9, calculate the exact values and express your answers in simplest radical form.

5) Calculate the length of the diagonal of a square with side length 4 cm.

6) A square has an area of 450 cm2. Calculate the side length.

7) Determine the length of the diagonal of a rectangle with dimensions 3 cm 9 cm.

8) Determine the length of the line segment from A(-2, 7) to B(4, 1).

9) Calculate the perimeter and area of the triangle to the right.

10) If and , which is greatest, or ?

11) Express each radical in simplest form.

a) c)

b) d)

12) Simplify .

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Solutions

1a) 1b) 1c) 1d) 1e) 1f)

2a) 2b) 2c) 32 2d) 2e) 2f) -140

3a) 3b) 3c) 3d) 3e) 3f)

4a) 4b) 4c) 4d)

4e) 4f) 5) cm 6) cm 7) cm

8) 9) Perimeter = units, Area = 12 square units 10)

11a) 11b) 11c) 11d) – 12)

Polynomial Expressions

13) Expand and Simplify

a) d) b) e)

c) f)

14) Expand and Simplify

a) d) b) e)

c) f)

15) Expand and Simplify

a) d) b) e)

16) Factor

a) d)

b) e)

c) f)

17) Factor

a) d)

b) e)

c) f)

18) Factor

a) d)

b) e)

c) f)

19) Show that and are equivalent.

20) Show that and are not equivalent.

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21) a) Is equivalent to ? Justify your decision.

b) Write a simplified expression that is equivalent to .

22) Show that the expressions and are not equivalent.

23) Determine whether the functions in each given pair are equivalent.

a) and b) and c) and

e) and

f) and

g) and

h) and

24) The two equal sides of an isosceles triangle each have a length of . The perimeter of the triangle is

. Determine the length of the third side.

25) For each pair of functions, label the pairs as equivalent, non-equivalent, or cannot be determined.

a) c) e) for all values of in the domain

b) d)

26) Halla used her graphing calculator to graph three different polynomial functions on the same axes. The equations

of the functions appeared to be different, but her calculator showed only two different graphs. She concluded that

two of her functions were equivalent.

a) Is her conclusion correct? Explain.

b) How could she determine which, if any, of the functions were equivalent without using her graphing

calculator?

27) a) Consider the linear functions and . Suppose that , and

. Show that the functions must be equivalent.

b) Consider the two quadratic functions and . Suppose that

, , . Show that the functions must be equivalent.

28) Is the equation true for all, some, or no real numbers? Explain.

29) a) If has two terms and has three terms, how many terms will the product of and have

before like terms are collected?

b) In general, if two or more polynomials are to be multiplied, how can you determine how many terms the

product will have before like terms are collected? Explain and illustrate with an example.

Solutions 13a) 25x

3 + 15x

2 – 20x 13b) 2x

2 – 7x – 30 13c) 16x

2 – 53 x + 33 13d) n

2 – 13n + 72 13e) -68x

2 – 52x – 2 13f) 5a

2 – 26a – 37

14a) 4x3 – 100x 14b) -2a

3 – 16a

2 – 32a 14c) x

3 – 5x

2 – 4x + 20 14d) -6x

3 + 31x

2 – 23x – 20 14e) 729a

3 – 1215a

2 + 675a – 125

14f) a2 – 2ad – b

2 + 2bc – c

2 + d

2

15a) x4 + 4x

3 + 2x

2 – 4x + 1 15b) 8 – 12a + 6a

2 – a

3 15c) x

6 – x

4 – 2x

3 – 3x

2 – 2x – 1 15d) -16x

2 + 43x – 13

16a) (x -7)(x + 2) 16b) (x +5y)(x - y) 16c) 6(m -6)(m – 9) 16d) (2y +7)(y – 1) 16e) (4a – 7b)(2a + 3b) 16f) 2(2x + 5)(4x + 9)

17a) (x -3)(x + 3) 17b) (2n -7)(2n + 7) 17c) (x4 + 1)(x

2 + 1)(x – 1)(x + 1) 17d) (3y – 8)(3y + 2) 17e) -12(2x – 3)(x – 3)

17f) –(pq+ 9)(pq – 9)

18a) (x -3)(2x – 7) 18b) (x + 5)(y + 6) 18c) (x -1)(x + 2)(x – 2) 18d) (y – x + 7) (y + x – 7) 18e) 3(2x – 7)(x – 2)

18f) (2m2 - 5)(6m - 7)

19. ;

20. Answer may vary. For example, ;

21. a) No; for , left side is 25, right side is 13 b)

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22. i) Answers may vary. For example,

ii) Answers will vary. For example, if , but

23. a) e.g., if , then and . .

b) . c) d) e) yes f) no g) yes

24.

25. a) cannot be determined b) cannot be determined c) not equivalent

d) cannot be determined e) equivalent 26. a) Yes b) Replace variables with numbers and simplify.

27. a) Answers may vary. For example, both functions are linear; a pair of linear functions intersect at only one point,

unless they are equivalent; since the functions are equal at two values, they must be equivalent. b) Answers may vary. For example, both functions are quadratic; a pair of quadratic functions intersect at most in two points,

unless they are equivalent; since the functions are equal at 3 values, they must be equivalent. 28. All real numbers. Expressions are equivalent. So the equation is an identity.

29. a) 6; Answers may vary. For example, , has 6 terms

b) Answers may vary. For example, will have terms.

Zeros of a Quadratic Function

30) Solve (all answers must be exact)

a) 3x2 12x 0 b) 2x2 4x 6 0 c) 3x2 5x 2 0

d) 4x2 11x 8 0 e) f)

g) h) i)

j) k) l)

m) n)

31) Determine the value(s) of k for which the expression x2 4x k 0 will have

a) two equal real roots b) two real distinct roots

32) a) Graph the function y 3x2 2x for 3 x 3.

b) On the same set of axes, graph the function y 1. c) Use your graph to determine the points of intersection of the two functions. d) Verify the solutions algebraically.

33) What value(s) of k, where k is an integer, will allow each expression to be solved by factoring?

a)x2 6 kx b)x2 kx 4 c)2x2 x k 0 d)6x2 kx 6 0

34) The width of a rectangle is 4 cm less than the length. To the nearest tenth of a centimetre, what length and width will result in a total area of 48 cm2?

35) Three lengths of pipe measuring 24 cm, 31 cm, and 38 cm will be used to create a right triangle. The

same length of pipe will be cut off each of the three pipes to allow a right triangle to be created. What is that length?

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36) A garden measuring 4 m by 5 m is to be extended on each side by the same amount to create a rectangular garden of area 25 m2. What amount, to the nearest tenth of a metre, must be added to each side to achieve this?

37) A picture measuring 20 cm by 16 cm is to be centred on a mat before it is framed. The mat width on

each of the four sides of the picture is to be equal. To the nearest tenth of a centimetre, what width of mat is needed so that the area of the mat and the area of the picture are equal?

Solutions

30. a) x 0 and x 4 b) x 1 and x 3 c) x 1

3 and x 2 d) x

11 249

8

e)

f) g)

h)

i)

j)

k)

l)

m) n)

31. a) k 4 b) k 4

32. a) and b) c) x 1 and x 1

3 d) verified algebraically

33. a) k 1, 1, 5, 5 b) k 3, 0, 3 c) k 3 d) k 13, 13 34. length 9.2 cm; width 5.2 cm 35. 3 cm 36. 0.5 m 37. 3.7 cm

Maximum and Minimum of a Quadratic Function

38) Determine the maximum or minimum value for each algebraically.

a) c) b) d)

39) Determine the maximum or minimum value.

a) d)

b) e)

c) f)

40) Determine the vertex for each quadratic function. State if the vertex is a minimum or maximum.

a) d)

b) e)

c) f)

41) Find the maximum or minimum value of the function and the value of x when it occurs.

a) e)

b) f)

c)

g)

d)

42) Show that the value of cannot be less than 1.

43) Find the minimum product of two numbers whose difference is 12. What are the two numbers?

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44) Find the maximum product of two numbers whose sum is 23. What are the two numbers?

45) Two numbers have a sum of 13.

a) Find the minimum of the sum of their squares.

b) What are the two numbers?

46) Determine the maximum area of a triangle, in square centimetres, if the sum of its base and its height is

13 cm.

47) The profit P(x) of a cosmetics company, in thousands of dollars, is given by P(x) = -5x2 + 400x – 2550,

where x is the amount spent on advertising, in thousands of dollars.

a) Determine the maximum profit the company can make.

b) Determine the amount spent on advertising that will result in the maximum profit.

c) What amount must be spent on advertising to obtain a profit of at least $4 000 000?

48) If y = x2 + kx + 3, determine the value(s) of k for which the minimum value of the function is an integer.

Explain your reasoning.

49) If y = -4x2 + kx – 1, determine the value(s) of k for which the maximum value of the function is an

integer. Explain your reasoning.

Solutions 38a) maximum: 6 38b) minimum: 0 38c) maximum: 8 38d) minimum: -7 39a) -5 39b) -4 39c) -18 39d) 27 39e) 2 39f) -5 40a) (-5, -19); minimum 40b) (-3, -2); minimum 40c) (1, 4) maximum

40d) (6, 31); maximum 40e) (-1, 2); maximum 40f)

; minimum 41a) minimum of

at

41b) minimum of

at

41c) maximum of at 41d) maximum of

at

41e) maximum of at 41f) maximum of

at

41g) maximum of at 42) minimum value is 2, therefore 3x

2 – 6x + 5 cannot be less than 1.

43) -36; -6, 6 44) 132.25; 11.5, 11.5 45a) 84.5 b) 6.5, 6.5 46) 21.125 cm2 47a) $5450000 47b) Maximum profit occurs

when $40000 is spent on advertising. 47c) $22 971 48) k must be an even integer 49) k must be divisible by 4

Families of Quadratic Functions

50) What characteristics will two parabolas in the family )4)(3()( xxaxf share?

51) How are the parabolas 4)2(3)( 2 xxf and 4)2(6)( 2 xxg the same? How are they

different?

52) Write an equation that describes the family of functions with a) Zeroes of 2 and -6 b) A vertex of (-1, 2)

c) x-intercepts of 2 and 2

53) Determine the equation of the parabolas that meet the given conditions a) x-intercepts -4 and 3, and passes through (2,7) b) vertex of (-2, 5) and passes through (4, -8) c) vertex of (1, 6) and passes through (0, -7) d) x-intercepts 0 and 8, and passes through (-3, -6)

e) x-intercepts of 7 and 7 and that passes through (-5, 3)

f) passes through the point (2, 4) and has x-intercepts 21 and 21

g) x-intercepts of 4 and passing through the point (3,6)

54) Determine the equation of the quadratic function that passes through (-4, 5) if its zeros are 32 and

32 .

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55) A projective is launched off the top of a platform. The table gives the height of the projectile at different times during its flight.

Time (s) 0 1 2 3 4 5 6

Height (m) 11 36 51 56 51 36 11

a) Draw a scatter plot of the data and a curve of best fit. b) Determine an equation that will model this set of data.

56) What is the equation of the parabola at the right if the point

(-4, -9) is on the graph?

Solutions 50. zeroes of 3 and -4 51. Both have vertex of (2, -4)

52a. 0,),6)(2()( aRaxxaxf 52b. 0,,2)1()( 2 aRaxaxf

52c. 0,),2()( 2 aRaxaxf

53a. )3)(4(6

7)( xxxf 53b. 5)2(

36

13)( 2 xxf

53c. 6)1(13)( 2 xxf 53d. )8(33

6)( xxxf 53e. )7(

6

1)( 2 xxf

53f. 2)4(

49

8)( xxf 54. )14(

33

5)( 2 xxxf 55b. 56)3(5)( 2 tth

56. )1)(3(3)( xxxf

Linear Quadratic Systems

57) Find the point(s) of intersection by graphing. a) ,

b) ,

c) ,

58) Determine the point(s) of intersection algebraically. a) ,

b) ,

c) ,

59) Determine the number of point(s) of intersection of and without

solving.

60) Determine the point(s) of intersection of each pair of functions. a) ,

b) ,

c) ,

d) ,

61) An integer is two more than another integer. Twice the larger integer is one more than the square of the

smaller integer. Find the two integers.

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62) The revenue function for a production by a theatre group is , where t is the ticket price

in dollars. The cost function for the production is . Determine the ticket price that will

allow the production to break even.

63) a) Copy the graph of . Then draw lines with slope that

intersects the parabola at (i) one point, (ii) two points, and (iii) no points.

b) Write the equations of the lines from part (a).

c) How are all the lines with slope that do not intersect the parabola related?

64) Determine the value of such that intersects the quadratic function

at exactly one point.

65) A daredevil jumps off the CN Tower and falls freely for several seconds before

releasing his parachute. His height, , in metres, seconds after jumping can be

modeled by:

before he released his parachute; and

after he released his parachute.

How long after jumping did the daredevil release his parachute?

66) Determine the coordinates of any points of intersection of the functions and .

67) Determine the equation of the line that passes through the points of intersection of the graphs of the quadratic

functions and .

68) In how many ways could the graph of two parabolas intersect? Draw a sketch to illustrate each possibility.

Solutions: 57a) (3, 9) (-2, 4) 57b) (0, 3) (-0.25, 2.875) 57c) no solutions 58a) (4, 3) (6, -5) 58b) (2, 7) (-0.5, -0.5) 58c) no solutions 59. One solution 60a) (1.5, 8) (-7, -43) 60b) (1.91, 8.91) (-1.57, 5.43) 60c) no solutions 60d) (-0.16, 3.2) (-1.59, -3.95) 61) 3 and 5 or -1 and 1 62) $3.00 63a)

63b) y = -4x – 6, y = -4x + 1, y = -4x + 5

63c) y-intercepts are all less than 1

64) k = -5 65) 7.20 seconds

66) (0, -2) ,

67) y =

68)

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Introducing Rational Expressions

69) Simplify. State any restrictions on the variables.

a)

b)

c)

70) Simplify. State any restrictions on the variables.

a)

c)

e)

b)

d)

f)

71) Simplify. State any restrictions on the variables.

a)

c)

e)

b)

d)

f)

72) Determine which pairs of functions are equivalent. Explain your reasoning.

a) and

b) and

73) Simplify. State any restrictions on the variables.

a)

c)

b)

d)

74) Simplify. State any restrictions on the variables.

a)

b)

75) Write rational expressions in one variable so that the restrictions on the variables are as follows.

a) b) c)

d)

Solutions

69a)

69b)

69c)

70a)

70b)

70c)

70d)

70e)

70f)

71a)

71b)

71c)

71d)

71e)

71f)

72a) yes 72b) no, not the same domain

73a) 73b)

73c)

73d)

74a)

, no restrictions 74b)

75) Answers will vary

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Multiplying and Dividing Rational Expressions

For all questions below, state any restrictions on the variables.

76) Simplify

77) Simplify.

78) Simplify.

79) Simplify

80) Simplify

81) Simplify

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82) Simplify

83) Simplify

84) Write two different pairs of rational expressions with a product of

.

85) Write two different pairs of rational expressions with a product of

.

Solutions

76a)

76b)

76c)

76d)

77a)

77b)

77c)

77d)

77e)

77f)

78a)

78b)

78c)

78d)

78e)

78f)

79a)

79b)

79c)

79d) 16a

2

79e) 79f)

80a)

80b)

80c)

80d) 80e)

80f)

81a)

81b)

81c)

81d)

81e)

81f)

81g)

81h)

82a)

82b)

82c)

82d)

82e)

82f)

83a)

83b)

83c)

83d)

84)

85)

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Adding and Subtracting Rational Expressions

For all questions below, state any restrictions on the variables.

86) Simplify

87) Find the lowest common multiple of each pair of expressions

88) Find the lowest common multiple of each of the following.

89) State the lowest common multiple in factored form.

90) Simplify

91) Simplify

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92) Simplify.

93) Write two rational expressions with binomial denominators for each of the following sums.

SOLUTIONS

86a)

86b)

86c)

86d)

86e)

86f)

86g)

87a) 20a

2b

3

87b) 6m2n

2 87c) 12x

3y

3 87d) 60s

2t2 88a) 6(m+2) 88b) 15(y – 1)(y + 2) 88c) 12(m – 2)(m – 3)

88d) 20(2x – 3) 89a) (x + 2)2 89b) (y – 2)(y + 2)(y + 4) 89c) (t + 3)(t – 4)(t + 1) 89d) 2(x – 2)(x + 1)(x – 4)

89e) (m + 3)2(m – 5) 90a)

90b)

90c)

90d)

90e)

90f)

90g)

90h)

91a)

91b)

91c)

91d)

91e)

91f)

91g)

91h)

91i)

92a)

92b)

92c)

92d)

92e)

92f)

92g)

92h)

92i)

15) Answers may vary 15a)

15b)

15c)

15d)

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Mixed Algebra Practice

94) Simplify each of the following. State restrictions, if necessary.

95) Simplify

SOLUTIONS

94a) 2m2 – 6m – 7 94b)

94c)

94d)

94e)

94f)

94g)

94h)

94i)

95a)

95b)

95c)

95d)

95e)

95f)

Mixed Bag Applications – Equivalent Expressions and Quadratic Functions

96) The height, h, in metres, above the ground of a football t seconds after it is thrown can be modelled by the

function h(t) = –4.9t2 + 19.6t + 2. Determine how long the football will be in the air, to the nearest tenth of a

second.

97) A parachutist jumps from an airplane and immediately opens his parachute. His altitude, y, in metres after t

seconds is modelled by the equation y = –4t + 300. A second parachutist jumps 5 seconds later and free falls

for a few seconds. Her altitude, in metres, during this time, is modelled by the equation

y = –4.9(t – 5)2 + 300. When does she reach the same altitude as the first parachute?

98) A rectangle has an area of 330 m2. One side is 7 m longer than the other. What are the dimensions of the

rectangle?

99) Is it possible for n2 + 25 to equal –8n? Explain.

100) The polynomial x4 – 5x

2 + 4 is not a quadratic expression, but it is factorable. Explain how you could use

what you know about factoring quadratic expressions to factor this expression.

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101) A rectangle is six times as long as it is wide. Determine the ratio of its area to its perimeter, in simplest

form, if its width is w.

102) Arnold has 24 m of fencing to surround a garden, bounded on one side by the wall of his house. What are

the dimensions of the largest rectangular garden that he can enclose?

103) If the function f(x) = ax2 + 5x + c has only one x-intercept, what is the mathematical relationship between

a and c?

104) Is it possible to determine the defining equation of a function given the following information? If so,

justify your answer and provide an example.

a) the vertex and one intercept

b) the vertex of a parabola and another point.

c) any two points on the parabola

105) The sum of two numbers is 10. What is the maximum product of these

numbers?

106) A sheet of metal that is 30 cm wide and 6 m long is to be used to make a

rectangular eavestrough by bending the sheet along the dotted lines. What

values of x maximizes the capacity of the eavestrough?

107) Are the expressions

and

equivalent for all values of x where both expressions are

defined? If they are, prove it.

108) In a nutrient medium, the rate of increase in the suraface area of a cell culture can be modelled by the

quadratic function

where S is the rate of increase in the surface area, in square millimetres per hour, and t is the time, in

hours, since the culture began growing. Find the maximum rate of increase in the surface area and the

time taken to reach this maximum.

109) Alice is in a 20-km running race. She alwasy runs the first half at an average speed of 2 km/h faster than

the second half.

a) Let x represent her speed in the first half. Determine a simplified expression in terms of x for the total

time needed for the race.

b) If Alice runs the first half at 10 km/h, how long will it take her to run the race?

110) An RCMP patrol boat left Goderich and travelled for 45 km along the coast of Lake Huron at a speed of s

kilometers per hour.

a) Write an expression that represents the time taken, in hours.

b) The boat returned to Goderich at a speed of 2s kilometres per hour. Write an expression that represents

the time taken, in hours.

c) Write and simplify an expression that represents the total time, in hours, the boat was travelling.

d) If s represents 10 km/h, for how many hours was the boat travelling?

111) A large dealership has been seeling new cars at $6000 over the factory price. Sales have been averaging

80 cars per month. Because of inflation, the $6000 markup is going to be increased. The marketing

manager has determined that, for every $100 increase, there will be one less car sold each month. What

should the new markup be in order to maximize revenue?

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112) The diameter of the smaller circle is d. The diameter of the larger circle is d + 1.

a) Write an expression that represents the area of the smaller circle in terms of d.

b) Write an expression that represents the area of the larger circle in terms of d.

c) Write and simplify an expression that represents the area of the shaded part of the diagram in terms of d.

d) If d represents 10 cm, find the area of the sahded part of the diagram, to the nearest thenth of a square

centimetre.

113) The difference between the length of the hypotenuse and the length of the next longest side of a right

trangle is 3 cm. The difference between the lengths of the two perpendicular sides is 3 cm. Find the three

side lenghts.

114) The UV index on a sunny day can be modelled by the function , where x

represents the time of da on a 24-hour clock and f(x) represents the UV index. Between what hours was

the UV index greater than 7?

115) Pat has 30 m of fencing to enclose there identical stalls behind the

barn, as shown.

a) What dimensions will produce a maximum area for each stall?

b) What is the maximum area of each stall.

Solutions 96. 4.1 seconds 97. 7.5 seconds 98. 15 m by 22 m 99. No 100. (x - 2)(x + 2)(x - 1)(x + 1)

101.

102. 12 m by 6 m 103.

104a. yes b. yes c. no

105. 25 106. 7.5 cm 107. Yes 108. 0.05 mm2/h; 2.5 h 109a.

109b) 2.25 hours 110a)

110b)

110c)

110d) 6.75 hours 111) $7000 112a)

112b)

112c)

112d)16.5 cm

2 113) 9 cm, 12 cm,

15, cm 114) 11:00 < t < 15:00, or between 11 a.m. and 3 p.m. 115a) 3.75 m by 5 m 115b) 18.75 m2