MHF4U1-ASSIGNMENT CHAPTER 1A
MHF4U1-ASSIGNMENT CHAPTER 1A NAME:______________________
True/False
Indicate whether the statement is true or false.
____ 1. Even functions are symmetric about the x-axis.
____ 2. Odd-degree polynomials have at least one x-intercept.
____ 3. Even-degree polynomial functions always begin and end on the same side of the x-axis.
____ 4. The graph of a quartic function cannot have exactly three x-intercepts.
____ 5. The function y = x5 – 5x3 + 7 is symmetric about the origin.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 6. An equation representing a function that extends from quadrant 3 to quadrant 1 is
|a. |y = x3 |c. |y = 2x6 |
|b. |y = –2x5 |d. |y = –5x4 |
____ 7. An equation representing a function that extends from quadrant 3 to quadrant 4 is
|a. |y = x3 + 7x – 1 |c. |y = 2x6 – 4x3 |
|b. |y = –2x5 + x – 1 |d. |y = –5x4 – 2x2 – 1 |
____ 8. The degree of the polynomial function y = x3 – 2x2 + 5x – 1 is
|a. |3 |c. |5 |
|b. |4 |d. |6 |
____ 9. The graph of the polynomial function y = –2x(x –1)2(x – 2)2 extends from
|a. |quadrant 3 to quadrant 1 |c. |quadrant 2 to quadrant 1 |
|b. |quadrant 3 to quadrant 4 |d. |quadrant 2 to quadrant 4 |
____ 10. The function y = 6(x – 1)4(x – 2)2(x + 1) changes sign at
|a. |x = 1 |c. |x = –1 |
|b. |x = 2 |d. |it doesn’t change sign |
____ 11. Which of the following is a polynomial function?
|a. |y = sin x |c. |y = 3x |
|b. |y = cos x |d. |y = x3 |
____ 12. Which of the following is an even function?
|a. |y = 2x4 + x3 |c. |y = 2x4 – x |
|b. |y = 2x4 + 11 |d. |y = –x3 + x5 |
____ 13. Which of the following graphs represents an even function?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 14. Which of the following graphs represents an odd function?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 15. The number of times that the function y = (x – 1)3(x + 2)(x – 4)2 changes sign is
|a. |0 |c. |2 |
|b. |1 |d. |3 |
____ 16. The function y = (x – 4)2(x – 7)(x + 3)3 is negative on the intervals
|a. |x ∈ (–∞, –3) and x ∈ (4, 7) |c. |x ∈ (–3, 4) and x ∈ (7, ∞) |
|b. |x ∈ (–∞, 3) and x ∈ (7, ∞) |d. |x ∈ (–3, 4) and x ∈ (4, 7) |
____ 17. The table of values represents a polynomial function.
|x | y |
|–3 | |6 | |
|–2 | |2 | |
|–1 | |0 | |
|0 | |0 | |
|1 | |2 | |
|2 | |6 | |
The function is
|a. |linear |c. |cubic |
|b. |quadratic |d. |quartic |
____ 18. The table of values represents a polynomial function.
|x | y |
|–3 | |–7 | |
|–2 | |2 | |
|–1 | |–3 | |
|0 | |0 | |
|1 | |3 | |
|2 | |–2 | |
|3 | |7 | |
The function appears to be
|a. |not symmetric |c. |symmetric about the y-axis |
|b. |symmetric about the x-axis |d. |symmetric about the origin |
____ 19. The least possible degree of the polynomial function represented by the graph shown is
[pic]
|a. |2 |c. |4 |
|b. |3 |d. |5 |
____ 20. An equation for the graph shown is
[pic]
|a. |y = x(x – 3) |c. |y = x2(x – 3) |
|b. |y = x(x – 3)3 |d. |y = x2(x – 3)3 |
____ 21. The graph of the function y = x(x – 1)3(x + 2)2 would most closely resemble
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 22. Which of the following graphs represents the function y = 2x5 – 3x4 + 1?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 23. Given the function y = (x – 1)2(x + 1)2, which finite differences will be equal (or constant)?
|a. |first differences |c. |third differences |
|b. |second differences |d. |fourth differences |
____ 24. Given the function y = –3x2 – 5x + 1, the second differences will all equal
|a. |3 |c. |6 |
|b. |–3 |d. |–6 |
____ 25. An equation for a cubic function with zeros 1, –2, and 3 that passes through the point (2, 12) is
|a. |y = x(x + 2)(x – 3) |c. |y = –3(x – 1)(x + 2)(x – 3) |
|b. |y = (x – 1)(x + 2)(x – 3) |d. |[pic] |
____ 26. An equation for a quintic function with zeros 1, 0, and 2 that passes through the point (–1, 24) is
|a. |y = 2x(x – 1)(x + 2)3 |c. |y = –3(x – 1)2(x – 2)2x2 |
|b. |y = –2x2(x – 1)2(x – 2) |d. |[pic] |
____ 27. State the interval(s) for which the graph of the function is negative.
[pic]
|a. |x ∈ (–∞, 1) and x ∈ (2, ∞) |c. |x ∈ (–1, 0) and x ∈ (2, ∞) |
|b. |x ∈ (–1, 2) |d. |x ∈ (–1, 0) and x ∈ (0, 2) |
Matching
Match each item with its description below.
|a. |quartic |f. |extends from quadrant 2 to quadrant 1 |
|b. |cubic |g. |extends from quadrant 2 to quadrant 4 |
|c. |quintic |h. |instantaneous rate of change |
|d. |never changes sign |i. |average rate of change |
|e. |even function |j. |is symmetric about the origin |
____ 28. slope of the tangent
____ 29. y = –2x(x – 1)(x + 3)2
____ 30. y = 7x6 – 3x2 + 5x
____ 31. y = x(x2 – 1)
____ 32. has between 1 and 5 x-intercepts
____ 33. y = –(x – 1)2(x + 4)4
____ 34. an odd function
____ 35. y = –2x2(x – 1)(x + 3)2
____ 36. slope of the secant
____ 37. any function for which f(x) = f(–x)
Completion
Complete each statement.
38. The polynomial function y = x(x – 1)(x + 2)2 has _______________ x-intercepts.
39. The polynomial function y = 2x9(x2 – 1) is an example of a(n) _______________ function.
40. The graph of the function y = x4(x – 1)6(x + 2)2 changes sign _______________ times.
41. For the polynomial function y = x5 – 3x4 – x + 1, the _______________ differences will be constant (equal).
Short Answer
42. The table of values represents a polynomial function. Determine the value of the constant finite differences.
|x | y |
|–3 | |169 | |
|–2 | |35 | |
|–1 | |3 | |
|0 | |1 | |
|1 | |5 | |
|2 | |29 | |
|–3 | |175 | |
43. Determine the type of polynomial function (linear, quadratic, cubic, etc.) that the table of values represents.
|x | y |
|–3 | |34 | |
|–2 | |17 | |
|–1 | |6 | |
|0 | |1 | |
|1 | |2 | |
|2 | |9 | |
|3 | |22 | |
44. The table of values represents a polynomial function. Determine the value of the leading coefficient.
|x | y |
|–3 | |169 | |
|–2 | |35 | |
|–1 | |3 | |
|0 | |1 | |
|1 | |5 | |
|2 | |39 | |
|–3 | |175 | |
45. Determine an equation for a cubic polynomial function with zeros 1, 2, and 3.
46. Determine an equation for a polynomial function with zeros 0 (order 2), 5 (order 2), and [pic] .
47. Determine an equation for the graph of the polynomial function shown.
[pic]
Problem
48. Determine an equation for the polynomial function represented in the table of values.
|x | y |
|–3 | |0 | |
|–2 | |–4 | |
|–1 | |–6 | |
|0 | |–6 | |
|1 | |–4 | |
|2 | |0 | |
|3 | |6 | |
49. Determine an equation for the quartic polynomial function represented by the table of values.
|x | y |
|–3 | |91 | |
|–2 | |21 | |
|–1 | |3 | |
|0 | |1 | |
|1 | |3 | |
|2 | |21 | |
|3 | |91 | |
50. Determine an equation in factored form for a polynomial function with zeros –1 (order 2) and 3 (order 3) that passes through the point (4, 5).
MHF4U1-ASSIGNMENT CHAPTER 1A
Answer Section
TRUE/FALSE
1. ANS: F PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.9 TOP: Polynomial and Rational Functions
KEY: even function, symmetry
2. ANS: T PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.3 TOP: Polynomial and Rational Functions
KEY: intercepts
3. ANS: T PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.3 TOP: Polynomial and Rational Functions
KEY: end behaviour
4. ANS: F PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: intercepts
5. ANS: F PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.9 TOP: Polynomial and Rational Functions
KEY: symmetry
MULTIPLE CHOICE
6. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.3 TOP: Polynomial and Rational Functions
KEY: end behaviour
7. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.3 TOP: Polynomial and Rational Functions
KEY: end behaviour
8. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.1 LOC: C1.1 TOP: Polynomial and Rational Functions
KEY: degree
9. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.3 TOP: Polynomial and Rational Functions
KEY: end behaviour
10. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: zeros, order
11. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.1 LOC: C1.1, C1.4 TOP: Polynomial and Rational Functions
KEY: recognize polynomial functions
12. ANS: B PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.9 TOP: Polynomial and Rational Functions
KEY: even function
13. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.9 TOP: Polynomial and Rational Functions
KEY: even function, graph
14. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.9 TOP: Polynomial and Rational Functions
KEY: odd function, graph
15. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: zeros, order
16. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: intervals, negative
17. ANS: B PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences, degree
18. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Sections 1.2, 1.3 LOC: C1.2, C1.9 TOP: Polynomial and Rational Functions
KEY: symmetry
19. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.2, C1.3 TOP: Polynomial and Rational Functions
KEY: degree, graph
20. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation, graph
21. ANS: A PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: factored form, graph
22. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.2, C1.3 TOP: Polynomial and Rational Functions
KEY: end behaviour, graph
23. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.1, C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences, degree
24. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.1, C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences
25. ANS: C PTS: 1 DIF: 2
REF: Knowledge and Understanding; Application OBJ: Section 1.3
LOC: C1.7 TOP: Polynomial and Rational Functions KEY: equation, set of conditions
26. ANS: B PTS: 1 DIF: 3
REF: Knowledge and Understanding; Application OBJ: Section 1.3
LOC: C1.7 TOP: Polynomial and Rational Functions KEY: equation, set of conditions
27. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.3 TOP: Polynomial and Rational Functions
KEY: intervals, negative
MATCHING
28. ANS: H PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
29. ANS: A PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
30. ANS: F PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
31. ANS: B PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
32. ANS: C PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
33. ANS: D PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
34. ANS: J PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
35. ANS: G PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
36. ANS: I PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
37. ANS: E PTS: 1 DIF: 2
REF: Knowledge and Understanding; Thinking OBJ: Sections 1.2, 1.3, 1.5, 1.6
LOC: C1.3, C1.5, C1.9, D1.7
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: degree, end behaviour, even function, odd function, average rate of change, instantaneous rate of change
COMPLETION
38. ANS: 3
PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: intercepts
39. ANS: odd
PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.9 TOP: Polynomial and Rational Functions
KEY: odd function
40. ANS: 0
PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.3 LOC: C1.5 TOP: Polynomial and Rational Functions
KEY: intercepts
41. ANS: fifth
PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 1.2 LOC: C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences
SHORT ANSWER
42. ANS:
48
PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application
OBJ: Section 1.2 LOC: C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences
43. ANS:
quadratic
PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application
OBJ: Section 1.2 LOC: C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences
44. ANS:
2
PTS: 1 DIF: 3 REF: Knowledge and Understanding; Application
OBJ: Section 1.2 LOC: C1.2 TOP: Polynomial and Rational Functions
KEY: finite differences, leading coefficient
45. ANS:
y = (x – 1)(x – 2)(x – 3)
PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation, set of conditions
46. ANS:
y = x2(x – 5)2(2x – 1)
PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation, set of conditions
47. ANS:
y = 0.5(x + 2)2(x – 1)3
PTS: 1 DIF: 3 REF: Knowledge and Understanding; Application
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation, graph
PROBLEM
48. ANS:
Calculating first and second differences shows that the function is quadratic.
(Second differences = 2)
This means that the leading coefficient is
2 = a(2!)
a = 1
Since the zeros are evident at x = –3 and x = 2, the polynomial has the form y = a(x + 3)(x – 2). Substituting the value of a, the answer is y = (x + 3)(x – 2).
Many other solutions are possible.
PTS: 1 DIF: 3 REF: Knowledge and Understanding; Thinking; Application
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation, table of values
49. ANS:
Calculating first, second, third, and fourth differences shows that the function is quartic.
(Fourth differences = 24)
This means that the leading coefficient is
[pic]
Notice that this is an even function. Therefore, the polynomial must have the form
[pic]. Substituting the value of a, the equation becomes [pic].
Clearly, [pic], since the point (0, 1) in the table of values represents the y-intercept. So, now the equation is [pic].
Substituting the point (1, 3), for example, into this equation yields [pic].
The answer then is [pic].
Many other solutions are possible.
PTS: 1 DIF: 4 REF: Knowledge and Understanding; Thinking; Application
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation, table of values
50. ANS:
[pic]
PTS: 1 DIF: 2 REF: Knowledge and Understanding; Application
OBJ: Section 1.3 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: equation
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