MHF 4U1- ASSIGNMENT CHAPTER 3 B
NAME:________________________
True/False
Indicate whether the statement is true or false.
____
1. The solution to
____
2. The solution to
is
.
is
.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
3. Solve the equation
.
a. x = –1
b. x = 5
____
4. Solve the equation
c. x = –5
d. no solution
.
a. x = 2
b.
x=
____
5. What are the x-intercepts of the graph of
a. –4, 5
b. –7, 3
____
c. 4, –5
d. 7, –3
have no x-intercepts?
c.
d.
7. Which of the following rational equations requires the use of technology to solve?
a.
c.
b.
____
?
6. For what values of k does the graph of
a.
b.
____
c. x = –2
d. no solution
d.
8. Which of the following rational inequalities requires the use of technology to solve?
a.
c.
b.
d.
____
9. Use the graph of
a.
b.
____ 10. Use the graph of
a.
b.
Short Answer
11. Solve.
a)
b)
c)
to solve the equation
.
c.
d. no solution
to solve the inequality
.
c.
d. no solution
d)
12. Solve each equation by graphing, using technology. Express your answers to two decimal places.
a)
b)
c)
d)
13. Solve each inequality by graphing, using technology. Express your answers to one decimal place.
a)
b)
c)
d)
Problem
14. A photographer uses a light meter to measure the intensity of light from a flash bulb. The intensity for the
flash bulb, I, in lux, is a function of the distance from the light, d, in metres, and can be represented by
.
a) Determine the following, to two decimal places.
i) the intensity of light 3 m from the flash bulb
ii) the average rate of change in the intensity of light for the interval
iii) the approximate instantaneous rate of change in the intensity of light at exactly 3 m from
flash bulb
b) What does the sign of your answer to part a), subpart iii), indicate about the light intensity?
the
15. Ingrid is docking a motorboat. She turns off the power and lets the boat coast toward the dock. The distance,
d, in metres, between the boat and the dock as a function of time, t, in seconds, is given by
.
a) What is the average velocity of the boat during the time interval from when Ingrid turns the boat off to
when it meets the dock?
b) Determine the velocity of the boat when Ingrid turns off the power, to one decimal place.
c) Determine the velocity of the boat when it meets the dock, to one decimal place.
d) Sketch a graph of the function. On the same set of axes, sketch a graph of the speed of the boat in relation
to the time.
16. Write an equation for the graph of the rational function shown. Explain your reasoning.
17. Determine an equation in the form
for a function that has asymptotes with equations
and
and a y-intercept of 2. Sketch a graph of your function.
18. Explain the difference between the solution to the equation
and the solution to
.
19. Write an equation for a rational function whose graph has all of the indicated features.
• vertical asymptote with equation x = 3
• horizontal asymptote with equation y = 2
• hole at x = 1
• no x-intercepts
20. a) Use the asymptotes and intercepts to make a quick sketch of the function
and its reciprocal,
, on the same set of axes.
b) Describe the symmetry in the graphs in part a).
c) Determine the equation of the mirror line in your graph from part a).
d) Determine intervals of increase and decrease for both f and fR. How do the sets of intervals compare?
e) Would the pattern from part d) occur for all pairs of functions
or why not.
and
? Explain why
MHF 4U1- ASSIGNMENT CHAPTER 3 B
Answer Section
TRUE/FALSE
1. ANS: F
The solution is x < –5 or x > 2.
PTS:
OBJ:
KEY:
2. ANS:
OBJ:
KEY:
1
DIF:
Section 3.4
LOC:
rational inequality
F
PTS:
Section 3.4
LOC:
rational inequality
3
C4.1, C4.2
REF: Knowledge and Understanding
TOP: Polynomial and Rational Functions
1
C4.1, C4.2
DIF: 2
REF: Knowledge and Understanding
TOP: Polynomial and Rational Functions
MULTIPLE CHOICE
3. ANS:
OBJ:
KEY:
4. ANS:
OBJ:
KEY:
5. ANS:
OBJ:
KEY:
6. ANS:
REF:
LOC:
7. ANS:
REF:
LOC:
TOP:
KEY:
8. ANS:
REF:
LOC:
TOP:
KEY:
9. ANS:
REF:
LOC:
KEY:
10. ANS:
REF:
LOC:
B
PTS: 1
DIF: 1
REF: Knowledge and Understanding
Section 3.4
LOC: C3.6
TOP: Polynomial and Rational Functions
rational equation
D
PTS: 1
DIF: 2
REF: Knowledge and Understanding
Section 3.4
LOC: C3.6
TOP: Polynomial and Rational Functions
rational equation
B
PTS: 1
DIF: 2
REF: Knowledge and Understanding
Section 3.4
LOC: C3.5
TOP: Polynomial and Rational Functions
rational function, x-intercept
A
PTS: 1
DIF: 3
Knowledge and Understanding; Thinking; Application
OBJ: Section 3.4
C3.5
TOP: Polynomial and Rational Functions KEY: rational function, x-intercept
C
PTS: 1
DIF: 2
Knowledge and Understanding; Thinking
OBJ: Section 3.4
C3.5, C3.6, D3.2
Polynomial and Rational Functions, Characteristics of Functions
rational equation, technology
D
PTS: 1
DIF: 3
Knowledge and Understanding; Thinking
OBJ: Section 3.4
C3.5, C3.6, C4.1, C4.2, D3.2
Polynomial and Rational Functions, Characteristics of Functions
rational equation, rational inequality, technology
B
PTS: 1
DIF: 1
Knowledge and Understanding; Application
OBJ: Section 3.4
C3.5, D3.2
TOP: Polynomial and Rational Functions, Characteristics of Functions
rational equation, graph
B
PTS: 1
DIF: 1
Knowledge and Understanding; Application
OBJ: Section 3.4
C4.1, C4.2
TOP: Polynomial and Rational Functions KEY: rational inequality, graph
SHORT ANSWER
11. ANS:
a)
b)
or x = –3
c)
d) No solution.
PTS: 1
DIF: 2
OBJ: Section 3.4
LOC: C3.6
KEY: rational equation
12. ANS:
a)
or
b)
or
c) No solution.
d)
or
REF: Knowledge and Understanding
TOP: Polynomial and Rational Functions
PTS:
OBJ:
TOP:
KEY:
13. ANS:
a)
b)
c)
d)
1
DIF: 2
REF: Knowledge and Understanding
Section 3.4
LOC: C3.6, D3.2
Polynomial and Rational Functions, Characteristics of Functions
rational equation, technology
PTS:
OBJ:
TOP:
KEY:
1
DIF: 2
REF: Knowledge and Understanding
Section 3.4
LOC: C3.6, D3.2
Polynomial and Rational Functions, Characteristics of Functions
rational inequality, technology
or
or
or
PROBLEM
14. ANS:
a) i) 1.11 lux
ii) –4.44 lux/m
iii) –0.74 lux/m
b) As the distance from the light increases, the intensity drops.
PTS: 1
DIF: 2
REF: Knowledge and Understanding; Application; Communication
OBJ: Sections 3.2, 3.5
LOC: C2.1, C3.7, D1.4, D1.5, D1.6, D1.7, D1.8, D1.9, D2.2
TOP: Polynomial and Rational Functions, Characteristics of Functions
KEY: word problem, reciprocal of quadratic function, rate of change
15. ANS:
a) 1.75 m/s toward the dock
(Ingrid turns off the boat at t = 0 and the boat hits the dock when d = 0. Setting d = 0 gives that t = 10 when
the boat hits the dock. To find the average velocity, determine
for
.)
b) 6.1m/s toward the dock
(This is the instantaneous velocity at t = 0.)
c) 0.5 m/s toward the dock
(This is the instantaneous velocity at t = 10.)
d)
The thicker line is the graph of the speed of the boat with respect to its starting position. It is decreasing
because the speed of the boat decreases once power has been cut. Students should not be expected to find this
graph precisely but should intuit its basic shape.
Note: Although
is negative, this derivative does not represent the speed of the boat, only the
rate of change in the distance between the boat and the dock, which is shrinking. In fact, the equation for the
speed of the boat with respect to its starting position is the exact opposite of this:
, which is
the equation of the thicker line graphed above. Again, students are not expected to be this precise.
PTS:
OBJ:
LOC:
TOP:
KEY:
16. ANS:
1
DIF: 3
REF: Knowledge and Understanding; Application
Sections 3.3, 3.5
C2.2, C2.3, D1.3, D1.4, D1.5, D1.6, D1.7, D1.8, D1.9, D2.2
Polynomial and Rational Functions, Characteristics of Functions
linear expressions in numerator and denominator, rate of change, graph
is one possible equation. Any function of the form
is a reasonable
candidate since it is difficult to tell from the graph how stretched the function is.
PTS:
OBJ:
TOP:
KEY:
17. ANS:
1
DIF: 3
REF: Knowledge and Understanding; Thinking; Application
Section 3.2
LOC: C2.1, C2.3, D3.1
Polynomial and Rational Functions, Characteristics of Functions
reciprocal of quadratic function
(Since there is a vertical asymptote at x = –1, the denominator must be a multiple of x + 1. Since there is a
horizontal asymptote at
, the ratio
= . If you pick c = 4, then d = 4 and a = 3. To ensure a y-intercept
of 2, d = 4 and b = 8.)
PTS: 1
DIF: 2
REF: Knowledge and Understanding; Thinking; Application
OBJ: Section 3.3
LOC: C2.2, C2.3
TOP: Polynomial and Rational Functions
KEY: linear expressions in numerator and denominator, graph
18. ANS:
The equation
has one solution, x = –6. However, there are infinitely many solutions to
,
with x = –6 being one of them, acting as a boundary on the interval of solutions for x.
PTS: 1
DIF: 2
REF: Knowledge and Understanding; Communication
OBJ: Section 3.4
LOC: C3.5, C4.1
TOP: Polynomial and Rational Functions
KEY: rational inequality, rational equation
19. ANS:
There are many possible answers. One is
PTS: 1
DIF: 4
OBJ: Section 3.5
LOC: C3.5, C3.7
KEY: rational function, hole
20. ANS:
a) For
For
: asymptotes
: asymptotes
.
REF: Knowledge and Understanding; Thinking; Application
TOP: Polynomial and Rational Functions
; intercepts
; intercepts
b) There is reflective symmetry in this graph, about a vertical mirror line that runs exactly halfway between
the two vertical asymptotes.
c) x = –1
d) For f: increasing for
, decreasing for no values of x
R
For f : increasing for no values of x, decreasing for
Where f increases, fR decreases, except at
and
, the locations of the vertical asymptotes.
e) Yes, this pattern would occur for all such pairs of functions. Values that are growing larger grow smaller
when reciprocated, and vice versa.
PTS:
REF:
OBJ:
TOP:
KEY:
NOT:
1
DIF: 4
Knowledge and Understanding; Application; Thinking; Communication
Section 3.3
LOC: C2.2, C2.3, D3.1
Polynomial and Rational Functions, Characteristics of Functions
linear expressions in numerator and denominator, graph, increasing, decreasing
Students can use a graphing calculator to confirm their graphs.