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 cm 2 . 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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Unit 2 - Equivalent Expressions and Quadratic Functions 2013-2
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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 .
FMSS 2013 Page 2 of 16
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.
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?
FMSS 2013 Page 5 of 16
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?
FMSS 2013 Page 6 of 16
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
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 .
FMSS 2013 Page 7 of 16
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)