MPJNS-MHF4U1-ASSIGNMENT CHAPTER 2 B - Advanced Functions 12
MHF4U1-ASSIGNMENT CHAPTER 2 B
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. A family of quadratic functions has zeros –3 and 5. Which of the following is a member of this family?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 2. Which of the following represents a family of cubic polynomials with zeros –4, 2, and 6?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 3. Examine the graphs of the following polynomial functions. Which graph does not belong to the same family?
|a. | |c. | |
| |[pic] | |[pic] |
|b. | |d. | |
| |[pic] | |[pic] |
____ 4. What is an equation for the cubic function represented by this graph?
[pic]
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 5. A family of polynomials has equation y = k(x – 3)(x + 2)3. What is the value of k for the family member represented by this graph?
[pic]
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 6. How many cubic functions have zeros –10, –5, and 4?
|a. |1 |c. |3 |
|b. |2 |d. |infinitely many |
____ 7. What is an equation for the family of cubic functions with zeros –8, 3, and [pic]?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 8. Which statement corresponds to the values of x shown on the following number line?
[pic]
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 9. Which of the following number lines depicts the solution to 3x(x2 – 4) + 2x > 3x3 + 5(x + 1)?
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
____ 10. A graph of [pic] is shown. Use the graph to solve [pic].
[pic]
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 11. A graph of [pic] is shown. Use the graph to solve [pic].
[pic]
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 12. Toby is solving [pic] using a graphing calculator. He determines that the zeros of [pic] are 0 and approximately –2.8 and 1.8. Based on this information, what is the solution to [pic]?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |[pic] |
____ 13. The inequality [pic] can be solved by
|a. |considering all cases |c. |using intervals |
|b. |graphing |d. |all of the above |
____ 14. When solving [pic] by considering all cases, how many cases must be considered?
|a. |2 |c. |4 |
|b. |3 |d. |8 |
____ 15. Which inequality has [pic] as its solution?
|a. |[pic] |c. |[pic] |
|b. |[pic] |d. |all of the above |
____ 16. Which of the following is true about solving polynomial inequalities algebraically?
|a. |When solving by using intervals, cubic inequalities require fewer intervals to be tested than quadratic inequalities. |
|b. |When solving a quartic inequality by considering cases, eight cases must be considered. |
|c. |When solving by considering cases or by using intervals the polynomial must be factorable. |
|d. |When solving by using intervals, the number of zeros of the polynomial is the number |
| |of intervals that must be tested. |
Completion
Complete each statement.
17. Polynomial functions with the same zeros are said to belong to the same _______________.
18. When the equal sign in a polynomial equation is replaced with either [pic], [pic], [pic], or [pic], then a polynomial _______________ is created.
19. The solution to x2(x2 + 1) > 0 is _______________.
Short Answer
20. Solve by graphing using technology. Round answers to one decimal place.
a) x3 – 7 > 0
b) [pic]
21. Solve the following inequalities using an algebraic method.
a) x2 + x – 2 > 0
b) [pic]
c) [pic]
d) [pic]
Problem
22. Solve [pic] algebraically and graphically.
23. Solve [pic] algebraically. Display your result on a number line.
24. Create a cubic polynomial inequality for which x = 3 or [pic] is the solution. Explain your reasoning.
MPJNS-MHF4U1-ASSIGNMENT CHAPTER 2 B
Answer Section
MULTIPLE CHOICE
1. ANS: A PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: family of polynomials
2. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.8 TOP: Polynomial and Rational Functions
KEY: family of polynomials
3. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: family of polynomials
4. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: family of polynomials
5. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.8 TOP: Polynomial and Rational Functions
KEY: family of polynomials
6. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: family of polynomials
7. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.8 TOP: Polynomial and Rational Functions
KEY: family of polynomials
8. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Sections 2.5, 2.6 LOC: C4.1 TOP: Polynomial and Rational Functions
KEY: number line, polynomial inequality
9. ANS: A PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Sections 2.5, 2.6 LOC: C4.3 TOP: Polynomial and Rational Functions
KEY: number line, polynomial inequality
10. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Sections 2.5, 2.6 LOC: C4.2, C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
11. ANS: D PTS: 1 DIF: 3 REF: Knowledge and Understanding
OBJ: Sections 2.5, 2.6 LOC: C4.2, C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
12. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.5 LOC: C4.2 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
13. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Sections 2.5, 2.6 LOC: C4.2, C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
14. ANS: C PTS: 1 DIF: 3
REF: Knowledge and Understanding; Thinking OBJ: Section 2.6
LOC: C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality, consider all cases
15. ANS: D PTS: 1 DIF: 4
REF: Knowledge and Understanding; Thinking OBJ: Sections 2.5, 2.6
LOC: C1.7, C4.1, C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
16. ANS: C PTS: 1 DIF: 4
REF: Knowledge and Understanding; Thinking OBJ: Section 2.6
LOC: C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality, consider all cases, use intervals
COMPLETION
17. ANS: family
PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 2.4 LOC: C1.7 TOP: Polynomial and Rational Functions
KEY: family of polynomials
18. ANS: inequality
PTS: 1 DIF: 1 REF: Knowledge and Understanding
OBJ: Section 2.5 LOC: C4.1 TOP: Polynomial and Rational Functions
KEY: polynomial equation, polynomial inequality
19. ANS:
[pic]
all the real numbers
PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.6 LOC: C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
SHORT ANSWER
20. ANS:
a) x > 1.9
b) [pic]
PTS: 1 DIF: 2 REF: Knowledge and Understanding
OBJ: Section 2.5 LOC: C4.2 TOP: Polynomial and Rational Functions
KEY: polynomial inequality, technology NOT: Students will have to adjust the window for part b).
21. ANS:
a) [pic]
b) [pic]
c) [pic]
d) [pic]
PTS: 1 DIF: 3 REF: Knowledge and Understanding
OBJ: Section 2.6 LOC: C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
PROBLEM
22. ANS:
Factor [pic] using the factor theorem.
[pic] = –(x – 2)(x – 2)(x – 1)
So, the inequality becomes –(x – 2)(x – 2)(x – 1) [pic] 0.
The three factors are –(x – 2), x – 2, and x – 1. There are four cases to consider.
Case 1: –(x – 2) [pic] 0, x – 2 [pic] 0, x – 1 [pic] 0
So, x = 2 is a solution.
Case 2: –(x – 2) ≤ 0, x – 2 ≤ 0, x – 1 [pic] 0
So, x = 2 is a solution.
Case 3: –(x – 2) ≤ 0, x – 2 [pic] 0, x – 1 ≤ 0
No solution.
Case 4: –(x – 2) [pic] 0, x – 2 ≤ 0, x – 1 ≤ 0
So, x ≤ 1 is a solution.
Combining the results, the solution is x ≤ 1 or x = 2.
Use a graphing calculator to graph the corresponding polynomial function [pic]. Then, use the Zero operation.
The zeros are 1 and 2. From the graph, [pic] when x ≤ 1 or x = 2.
PTS: 1 DIF: 3 REF: Knowledge and Understanding
OBJ: Sections 2.5, 2.6 LOC: C4.2, C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
23. ANS:
[pic]
Factor [pic] using the factor theorem.
[pic] = –(x – 1)(x + 1)(x2 + 1)
So, the inequality becomes –(x – 1)(x + 1)(x2 + 1) < 0.
The three factors are –(x – 1), x + 1, and x2 + 1. There are four cases to consider.
Case 1: –(x – 1) < 0, x + 1 < 0, x2 + 1 < 0
No solution.
Case 2: –(x – 1) > 0, x + 1 > 0, x2 + 1 < 0
No solution.
Case 3: –(x – 1) > 0, x + 1 < 0, x2 + 1 > 0
So, x < –1 is a solution.
Case 4: –(x – 1) < 0, x + 1 > 0, x2 + 1 > 0
So, x > 1 is a solution.
Combining the results, the solution is x < –1 or x > 1.
[pic]
PTS: 1 DIF: 3 REF: Knowledge and Understanding
OBJ: Section 2.6 LOC: C4.3 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
24. ANS:
Answers may vary.
Example: [pic]
PTS: 1 DIF: 4 REF: Knowledge and Understanding; Thinking; Communication
OBJ: Sections 2.5, 2.6 LOC: C4.1, C4.2 TOP: Polynomial and Rational Functions
KEY: polynomial inequality
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