Download Determine Rates of Change from Data or an Equation

D63
Average Rate of Change
From Data:
1.
The average maximum daily temperature in Toronto in February is 2C . The average
maximum daily temperature increases approximately 6C per month during March,
April, May and June. Estimate the average maximum daily temperature in Toronto in
each month.
a) March
b) April
c) June
2.
Given the following data:
x
y
0
5
2
7
4
13
6
23
8
37
calculate the average rate of change of y over each interval:
a) x  0 to x  10
b) x  0 to x  4
c) x  0 to x  2
x
y
4.
d) x  6 to x  10
Given the following data, calculate the average rate of change of y between each
consecutive pair of values of x.
50
7.0
100
6.0
150
4.6
200
2.9
250
0.9
300
-1.3
350
-3.8
A Bunsen burner is used to heat water in a beaker. The table shows the temperature
every five seconds. (from Addison-Wesley Advanced Functions and Introductory Calculus)
Time, t
(s)
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Temperature, T
( C )
20.0
40.0
53.3
62.9
70.0
75.6
80.0
83.6
86.7
89.2
91.4
93.3
95.0
96.5
97.8
98.9
100.0
Temperature of Water Being Heated
120
100
Temperature (C)
3.
10
55
80
60
40
20
0
0
20
40
60
80
100
Time (s)
a) Determine the average rate of change in temperature over 80 seconds. (include units)
b) Determine the average rate of change in temperature over the time interval from
t=0 s to t=30 s. (include units)
c) Determine the average rate of change in temperature over the time interval from
t=30 s to t=50 s. (include units)
d) Determine the average rate of change in temperature, accurate to two decimal
places, over the time interval from t=50 s to t=80 s. (include units)
D64
From an Equation:
5.
A pebble falls from the top of a cliff. The pebble’s height above the ground is modeled
by h  t   5t2  5t  180 , where h is the height in metres at t seconds since the pebble
started to fall. Include units.
a) Determine the average rate of change of height between 1 second and 4 seconds.
b) Determine the average rate of change of height between 3 seconds and 4 seconds.
c) Determine the average rate of change of height between 3 seconds and 3.1 seconds.
d) Determine the average rate of change of height between 3 seconds and 3.01 seconds.
e) Estimate the rate at which the height is changing at 3 seconds.
6.
The population of a town is modeled by P  t   6t2  110t  3000 , where P is the population
and t is the number of years since 1990. Include units.
a) Find the average rate of change in population between 1995 and 2005.
b) Find the average rate of change in population between 2000 and 2005.
c) Find the average rate of change in population between 2004 and 2005.
d) Estimate the rate at which the population is changing in 2005.
7.
t
An object moves along a line so that at time t its position is s  t   3sin   , where
4
s is in metres. Provide answers accurate to two decimal places. Include units.
a) Find the average rate of change in position between 0 seconds and 1 second.
b) Find the average rate of change in position between 1 second and 2 seconds.
c) Find the average rate of change in position between 2 seconds and 3 seconds.
d) Find the average rate of change in position between 2.5 seconds and 3 seconds.
e) Find the average rate of change in position between 2.9 seconds and 3 seconds.
f) Find the average rate of change in position between 2.99 seconds and 3 seconds.
g) Estimate the rate at which the position is changing at 3 seconds.
Answers:
1. a) 4C b) 10C c) 22C
2. a) 5 b) 2 c) 1 d) 8 3. a) -0.02, -0.028, -0.034, -0.04, -0.044, -0.05
4. a) 1C/s b) 2C / s c) 0.57C / s d) 0.29C / s
5. a) 30m/s b) 40m/s c) 35.5m/ s d) 35.05m/s
5. e) 35m/s
6. a) 230 people/year b) 260 people/year c) 284 people/year d) 285 people/year
7. a) 0.74m/ s b) 0.70m/ s c) 0.61m/ s d) 0.58m/ s e) 0.56m/ s f) 0.55m/ s g) 0.55m/ s
Estimating Instantaneous Rates of Change
1.
D65
The height of a shotput can be modeled by the function h  t   4.9t2  8t  1.5 , where h
is the height in metres and t is the time in seconds. The graph of the function is
provided below.
Height vs Time
Height (m)
Time (s)
a)
By referring to the graph, at what point do you think the shotput was traveling
the fastest?
b) Determine the average rate of change of height with respect to time on a short
interval near the point you chose in part a).
c) Estimate the instantaneous rate of change at the point you chose in part a).
d) Was your answer to the average rate of change the same as the instantaneous
rate of change? If not, why not?
e) Estimate the instantaneous rate of change for each of the following points,
accurate to one decimal place. Show your calculations.
i) t  0s
ii) t  0.25s
iii) t  0.5s
iv) t  0.8s
v) t  1s
vi) t  1.5s
vii) t  1.8s
2.
Describe the graphical feature (max/min point, increasing/decreasing interval) and the
sign of the slope of the tangent lines for the intervals given in the chart below, by
referring to the graph and the instantaneous rates of change values in question #1.
sign of the slope of the
Interval
Graphical feature
tangent lines
0s to 0.8s
at 0.8s
0.8s to 1.8s
3.
The thickness of the ice on a lake for one week is modelled by function
t  d  0.1d3  1.2d2  4.4d  14.8 , where t is the thickness in centimeters and d is the
number of days after December 31st. The graph of the function is provided below.
Thickness vs Time
Thickness
(cm)
Time (days)
a) Refer to the graph: when do you think the warmest day occurred during the week?
b) Determine the average rate of change of the thickness of the ice with respect to
time on a short interval near the point you chose in part a).
c) Estimate the instantaneous rate of change at the point you chose in part a).
d) Was your answer to the average rate of change the same as the instantaneous
rate of change? If not, why not?
e) Estimate the instantaneous rate of change for each of the following points,
accurate to one decimal place. Show your calculations.
i) t  0d
ii) t  1d
iii) t  2d
iv) t  3d
v) t  4d
vi) t  5d
vii) t  6d
viii) t  7d
4. Describe the graphical feature (max/min point, increasing/decreasing interval) and the
sign of the slope of the tangent lines for the intervals given in the chart below, by
referring to the graph and the instantaneous rates of change values in question #3.
sign of the slope of
Interval
Graphical feature
the tangent lines
0 d to 3 d
at 3 d
3 d to 5 d
at 5 d
5 d to 7d
5. Describe the context where the slope of the tangent is zero. What does it mean?
6. Given the function f  x   3 2
x 1
 12 ,
a) sketch the function
b) is the function increasing or decreasing?
c) draw the secant through the points where x  1 and x  0
d) find the average rate of change of the function f  x  from x  1 to x  0
e) describe how to find the instantaneous rate of change of f  x  at x  1
Estimate the instantaneous rate of change of f  x  at x  1 .
(accurate to three decimal places)
7. Given the function f  x   2 3
x4
 10 ,
a) sketch the function
b) is the function increasing or decreasing?
c) draw the secant through the points where x  4 and x  5
d) find the average rate of change of the function f  x  from x  4 and x  5
e) estimate the instantaneous rate of change of f  x  at x  4 .
(accurate to three decimal places)
1
8. Given the function f  x   3  
x 5
2
 11 ,
a) sketch the function
b) is the function increasing or decreasing?
c) find the average rate of change of the function f  x  from x  3 to x  4
d) estimate the instantaneous rate of change of f  x  at x  3 .
(accurate to three decimal places)
D66
1
9. Given the function f  x     
x 3
2
D67
9,
a) sketch the function
b) is the function increasing or decreasing?
c) find the average rate of change of the function f  x  from x  3 to x  2
d) Estimate the instantaneous rate of change of f  x  at x  3 .
(accurate to three decimal places)
10. Given the function f  x   cos x ,
a)
sketch the function
b)

is the function increasing or decreasing over the interval x   0,  ?


c)
find the average rate of change of f  x  from x 
d)
estimate the instantaneous rate of change of f  x  at x 
4
to x 


3
4
2
(three decimal places)
(three decimal places)

11. Given the function f  x   3sin  x   ,
4

a) sketch the function
5 7
b) is the function increasing or decreasing over the interval x   ,  ?
 4 4 
17
3
c) find the average rate of change of f  x  from x 
to x 
(three decimal places)
12
17
d) estimate the instantaneous rate of change of f  x  at x 
12
12. Given the function f  x   2cos  x 

2
(three decimal places)
 
 1 ,
12 
a) sketch the function
7 13 
,
?
 12 12 
2
3
c) find the average rate of change of f  x  from x 
to x 
(three decimal places)
3
4
2
d) estimate the instantaneous rate of change of f  x  at x 
(three decimal places)
3
b) is the function increasing or decreasing over the interval x  
13. The vertical position of a person on a ferris wheel can be modeled by the

equation h  t   10cos  t   11 . Find the average rate of change as t goes
 15 
from 24 s to 24.5 s . (accurate to two decimal places) Find the instantaneous rate of
change at t  24 s . (accurate to two decimal places)
14. The depth of water, D, in metres, at the end of a pier in Vacation Village varies
with the tides throughout the day and can be modeled by the equation
D  t   1.5cos 0.575  t  3.5    3.8 , where t is time in hours. Find the average rate
of change as t goes from 2 h to 2.5 h . Find the instantaneous rate of change at
t  2 h . (accurate to two decimal places)
15.
A population of algae in a swimming pool starts at 2500. After 15 minutes, the
population is 5000. This population can be modeled by the equation
t
P  t   2500 16 , where P is the population after t hours.
a) Find the algae population after ten minutes.
b) Estimate the instantaneous rate of change of the algae population at one
hour.
c) Estimate the instantaneous rate of change of the algae population at three
hours.
Answers:
1. a) & b) & c) answers may vary d) answers were close but not exactly the same
1. e)i) 8.0 m/ s ii) 5.5 m/ s e)iii) 3.1 m/ s iv) 0.2 m/ s v) 1.8 m/ s vi) 6.7 m/ s
vii) 9.6 m/ s
2. increasing, positive slope; max. point, zero slope; decreasing, negative slope
3. a) & b) & c) answers may vary d) answers were close but not exactly the same e)i) 4.4 cm/d
3. e)ii) 2.3 cm/d
iii) 0.8 cm/d
iv) 0.1 cm/d
v) 0.4 cm/d
vi) 0.1 cm/d
vii) 0.8 cm/d
viii) 2.3 cm/d
4. decreasing, negative slope; min. point, zero slope; increasing, positive slope; max. point, zero slope;
decreasing, negative slope
5. slope is zero when there is a maximum or minimum point; the tangent is a horizontal line
6. a)
7. a)
8. a)
12
12
-12 -11 -10 -9
-8
-7
-6
-5
-4
-3
-2
-1
12
11
11
11
10
10
10
9
9
9
8
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
1
1
-1
1
2
3
4
5
6
7
8
9
10
11
-12 -11 -10 -9
12
-8
-7
-6
-5
-4
-3
-2
-1
-1
2
1
1
2
3
4
5
6
7
8
9
10
11
12
-12 -11 -10 -9
-8
-7
-6
-5
-4
-3
-2
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-9
-10
-11
-12
6. b) increasing d)3
9. a)
e) 2.080
10. a)
2
3
4
5
6
7
8
9
10
11
12
-7
-8
-8
1
-8
-9
-9
-10
-10
-11
-11
-12
-12
7. b) decreasing d) -4 e) -2.198 8. b) decreasing c) -6 d) -8.315
11. a)
12
11






10
9
8
7
6
5
4
3
2
1
-12 -11 -10 -9
-8
-7
-6
-5
-4
-3
-2
-1
-1

1
2
3
4
5
6
7
8
9
10
11













12
-2






-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
9. b) increasing c) 0.5 d) 0.693 10. b) decreasing c) -0.791 d) -0.707 11.b) decreasing c) -2.373 d) -2.597
12. a)
12. b) increasing c) 1.842 d) 1.932
13. 1.95 m/s ; 1.99 m/s
14. 0.57 m/h ; 0.65 m/h


15.











15. a) 3969
b) 111 057 algae/hour
c) 28 430 703 algae/hour
D68
D69
UNIT #9 Test Review
1.
Given the following data, calculate the average rate of change of y between each
consecutive pair of values of x.
x
y
2.
-2
-27
0
1
2
5
4
81

An object moves along a line so that at time t its position is s  t   2cos  t    3 ,

6
where s is in metres. Provide answers accurate to two decimal places.
a) find the average rate of change in position between 3 and 4 seconds.
b) estimate the instantaneous rate of change in position at 3 seconds.
DO NOT USE LIMITS.
3.
Given the function f  x   3 2 
a)
x 7
8,
find the average rate of change of the function f  x  from x  4 to x  3
b) estimate the instantaneous rate of change in position at x  4
(answer accurate to two decimal places)
DO NOT USE LIMITS.
Answers:
1. 14 , 2 , 38
2. a) 0.32 m/s
b) 1.23m/s
3. a) -24 b) -16.64
D70
MHF4UI Exam Review
Number Systems
1.
Simplify.
a)
b)
45
e) 2i 3i 
f)
6  15 5
9
c)
 4i   5i 
g)
2.
Express 4 3 as an entire radical.
3.
Solve for x, x  C .
2
a) 15x  11x  12  0
b)
d)
27  2 75
50
10  32
8
2
25x  30x  11
4.
Check one solution from 3b.
5.
Determine a possible quadratic equation in the form ax  bx  c  0 with the
given roots.
2
a)
3
and 2
5
b)
d) 2  5i and 2  5i
5  2 3 and 5  2 3
e)
c)
3  2
3  2
and
4
4
5  2 7i
5  2 7i
and
3
3
Polynomial Functions
1.
3
Given the function f  x   3x  5x  6 , state:
a) the degree
b) the end behaviours
2.
State the zeroes of f  x   x  x  6 x  1 x  2 x  3 . Sketch this function.
3.
The maximum number of turning points for a function of degree 10 is ___.
4.
The minimum number of zeroes for a function of degree 12 is ___.
5.
Sketch each of the following polynomial functions with the given properties.
Clearly label x-intercepts for parts a and b.
2
a) f  x   2  x  6 x  1 x  2
b) f  x   3  x  4   x  2  x  2 
c) degree 4, positive leading coefficient, three turning points and two zeroes
d) degree 5, negative leading coefficient, three equal real roots and two complex
roots
e) degree 4, negative leading coefficient, one zero and three turning points
f) degree 3, positive leading coefficient, one real root and two complex roots
g) degree 5, positive leading coefficient, two equal real roots, two complex roots
and one distinct real root
6.
Determine the equation of the cubic function, in standard form,
with zeroes -2, -1 and 4 which passes through the point 2, 72 .
7.
Divide 3x  8x  4x  3 by x  2 . Write the division statement in the
form P  x  D x   Q  x   R  x  .
4
2
3
D71
2
8. Without performing long division, determine the remainder when x  3x  39x  10 is divided by
x 5.
3
2
9. When the polynomial 4x  kx  mx  2 is divided by x  1 , the remainder is 4. When the polynomial
is divided by x  2 , the remainder is 40. Determine the values of k and m.
10. Factor fully.
3
2
x  2x  11x  12
a)
3
27x  8
d)
11. Solve for x, x  C .
3
2
6x  11x  3x  2  0
a)
3
64m  27  0
d)
3
2
b)
e)
c)
f)
16  250m
b)
x  3x  3x  1  0
x  2x  7x  2
3
3
2
3
2
12x  8x  3x  2
3
2x  1
c)
3
3
 3x  2 
3
g) 2  16  x  1 
2
x  5x  5x  11  0
12. Solve for x, x  R . Write your answer using interval notation.
Also graph each solution on a real number line.
3
2
3
2
x  5x  2x  8  0
a)
b) 2x  x  6x  0
3
2
3
2
2x  3x  7x  10
c)
d) 4x  14x  16x  5  0
13. Given the following base function and transformations:
a)
state the function notation for the transformations
b)
state the resulting function equation
c)
graph the function
3
i)
y  x ; vertically stretched by a factor of 3, shift left 2 units and up 4 units
4
1
,
2
ii)
y  x ; reflected in the y-axis, horizontally stretched by a factor of
iii)
shift right one unit
3
y  x ; horizontally stretched by a factor of 2, reflected in the x-axis,
shift down 5 units
Exponential and Logarithmic Functions and Equations
1. For each of the following functions, state the properties:
a)
x
1
y 
4
b)
y 3
x
c)
d)
y  log x
5
y  log1 x
6
i) domain
ii) range
iii) equation of the asymptote
v) y-intercept
vi) increasing or decreasing function?

2. Solve.
3x 1
a) 3 4
  192
x 2
3
b)
x
 3  216
c)
iv) x-intercept
vii) end behaviours
x 3
x 6
4
4
d)
 25  2


 4 
 65
3. Evaluate, without a calculator.
1
2
a)
36
f)
 1 
log  
4 16
 
k)
log 4  log 9
6
6
o) log8 128
l)
b)
27
g)
log1000
c)
h)
log 100  log 20
5
5
p)
3
3
4
3
7
 1 4
 
 16 
log7 43
i)
7
m) log6 36
 1 
log 
9 243 


e)
j)
log 1
3
n)
log 32
16
log 49
7
log
64
4
D72
4.
Evaluate log7 19 , accurate to four decimal places.
5.
State the equation of the inverse of y  log4 x .
6.
For each function, describe the transformations, graph and state domain and range.
a) y  3 2 
x4
d) y  2log3  x  3  1
7.
1
y  3 
2
b)
6
e)
y  2 3
c)
2
y  3log  x  4   5
f)
2
y  log
1
2

7 4

b)
 3 m7
log9  8
 n





   x  1  3
Solve, accurate to four decimal places.
2x 3
5x
2x 1
 31
a) 5
b) 4  7
9.
Solve, stating necessary restrictions.
a) log6  x  3  1  log6  x  2 
Rational Functions
Given the function f  x  
 a4c 
log 

3

 d
c)
8.
log  x  2   log  x  1   1
b)
4
4
3x  8
, determine
x 2
a) the equation of the vertical asymptote and the horizontal asymptote
b) a sketch of f  x  . Show necessary calculations.
c) positive intervals
e) increasing intervals
g) domain
2.
d)
f)
negative intervals
decreasing intervals
h) range
Sketch the following functions. Label all key features. Show necessary
calculations. State the domain of each function.
a) f  x  
3.
2x  4
Rewrite each expression with no logarithms of products, quotients or powers.
a) log3 m n
1.
x 1
2x  10
2
x  x  20
b)
g x 
2
x  3x  4
2
x x6
Given the sketch of f  x  , sketch the reciprocal function of f  x  .
10
y
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
x
10
4
4. Graph f  x  
1
2
x  6x  9
D73
.
5. Create a function that has a graph with:
a)
vertical asymptotes at x  4 and x  2 , horizontal asymptote at y  3 ,
b)
c)
x-intercepts of 3 and 2
vertical asymptote at x  1 , no horizontal asymptote, no linear oblique
asymptote, x-intercepts of 4 and 5
no horizontal, vertical or linear oblique asymptotes, hole at x  4 ,
y-intercept of 9
6. Solve for x , x  R .
a)
4
 2
x 1
b)
4
2

x 1 x 2
Trigonometry
1. Convert the following to degrees.
r
2
r
a)
b) 2.34 , accurate to one decimal place
3
2. Convert 135 to radians.
3. Determine the arc length when the radius is 8 cm and the central angle is 165 . State the exact
answer and the answer accurate to three decimal places.
4. P  3, 6 is a point on the terminal arm of an angle  in standard position where 0    2 .
Determine the exact value(s) of cos  .
5. Determine the exact value of each of the following. Include a clearly labeled sketch. Do not use a
calculator.
 5 
 7 
a)
b) cot  
c) cos 
tan 


 3 
 4 
6. Solve for x , where 0  x  2 .
a)
b)


3sin  x    2  0 , accurate to two decimal places
2

2cos x  3  4 , state exact answers
7. Solve for x, where 0  x  2 . State exact answers. Do not use a calculator.
2 

a)
b) tan x  1
2sin  x 
  1

3 
8. For each of the following functions, state the properties:
y  csc x
a)
b) y  sec x
c) y  tanx
d) y  cotx
i)
iv)
vii)
domain
ii) range
iii) equation(s) of vertical asymptote(s)
period
v) zeroes
vi) y-intercept
provide a clearly labeled sketch of the function
9. Sketch one period of each of the following functions. For each of the following, state the
amplitude, period, phase shift, and vertical shift. Also state the domain and range.
1 
2  

y  2sin    
a)
b) y  3cos     2
  4
2 
3 

6
10. Determine the equation of a sine function with maximum value 12, amplitude 4,
2

period
, and phase shift
.
3
11.
4

A cosine function on the interval x  0 has a maximum point at  ,7  and a
6

5
minimum point at  , 1  . Determine a possible equation.
 12

12. The depth of the water around the harbour changes from 2 m at low tide at
03:00 h to 16 m at high tide at 09:00 h. One cycle is completed every 12 hours.
The changes in the depth of the water over time can be modeled by a sine function.
a) Use the information to sketch the function over a 24-hour period showing
water depth, d, as a function of the time, t, with t  0 being midnight. Clearly
label the graph.
b) Determine an equation which expresses the depth of water, d, as a function of
time, t, of the graph above.
c) Determine the depth of water in the harbour at 05:30 h, correct to one
decimal place.
d) It is safe to enter the harbour when there is at least 4 m of water. During
what times is it unsafe to enter the harbour? Answer accurate to the nearest
minute.
13. A ferris wheel with radius of 15 m rotates once every 30 seconds. Passengers
got on at the lowest point which is 2 m above the ground.
a) draw a graph showing the height above the ground during the first two cycles
b) write an equation which expresses the height as a function of time on the ride
c) calculate the height above the ground after 20 seconds
d) at what times in the first rotation will the rider be 28 m above the ground?
Answer accurate to one decimal place.
14. Express as a single sine or cosine function.
a) 7 sin 4x cos 4x
b)
2
2
c)
cos 7x  sin 7x
2
3  6sin 5x
15. Write each of the following in terms of the cofunction identity.
 3 
 2 
a) cos  
b) cot  
 8 
16. Expand sin 
 3 

4


 , then evaluate. State the exact answer.
6
17. Solve for x, where x  0,2  . State exact answers where possible; otherwise,
round correctly to two decimal places.
a) sin2xcos x  cos2x sin x 
3
2


b)
cos
c) sin2x  sin x  0
d)
2sin x  5sinx  3  0
e) 10cos x  cosx  2  0
2
f)
2cos x  3sinx  0
g)
h)
3sin x  4sinx  3  0
2 sin2x  cos2x
4
cos x  1  sin
4
2
2
2
sin x
D74
18. Prove the following identities:
D75
a)
sin x  2cos x  cos x  1
b)
sin x cos x

1
csc x sec x
c)
1
 secx  tan x
secx  tan x
d)
cot x
cot x

 2sec x
csc x  1 csc x  1
e)
sin x  tan x
 sin x
1  sec x
4
2
4
Combination of Functions
1. For y  2  x  1   x  4  , determine the following:
3
a)
c)
x-intercept(s)
b) y-intercepts
If you were to complete a finite differences chart, describe the pattern for
constant x .
2. For the following functions:
x 1
y  2 3  4
i)
ii)
y  3log  5x  1   6
4
determine the following:
a)
x-intercept(s)
b) y-intercept
c) end behaviours
d)
If you were to complete a finite differences chart, describe the pattern you
would see.

3. For y  2sin 3  x 
 
a)
 
   3 , determine the following:
6 
y-intercept
b) x-intercept(s) for one period
c)
all zeroes
2
4. For f  x   x  9 and g  x   4  x , determine the following:
a)
domain of f  x 
c)
 f  g  x 
b) domain of g  x 
and state the domain
d)
 f  g  5
5. For f  x  sinx and g  x   tanx , determine the following:
a)
domain of f  x 
c)
f
   x  and state the domain
g
b) domain of g  x 
d)
 f   3 
 
 (exact answer)
 g  4 
6. For f  x   x  4 and g  x   log2  x  1  , determine the following:
a)
domain of f  x 
c)
 f g  x 
7. For y 
and state the domain
b) domain of g  x 
d)
 f g  9
2
, determine functions f and g such that y  f g  x   .
3x  1
D76
Answers (Number Systems):
2  5 5
b)
3
1. a) 3 5
c) 7 3
 3  2 i 3  2 i 


,

5
5




e) -6 f) 20
2
3. b) x  
2
d) 5 2 i
5 2 2i
g)
4
2.
2
5. a) 5x  7x  6  0
 3 4 
3. a) x   , 
 5 3
48
2
b) x  10x  13  0
c) 16x  24x  7  0
2
5. d) x  4x  9  0
e) 9x  30x  53  0
Answers (Polynomial Functions):
1. a)3 b) as x  , f x   , as x  , f x  
2. -6, -1, 0, 2, 3
3. 9
4. 0
5. possible sketches
a)
b)
y
c)
y
OR
y
OR
y
x


















-10
d)
OR
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
8
9
OR
f)
y
x
x
3
2

10. a)  x  1   x  4   x  3

2
11. b) x  1,2  3,2  3
 3 
,0 
 2



3

2
c) 2x  13x  22x  1
d) x  
 1  41 1  41 
x
,


4
4


9. k  3, m  3



x
2
d) 3x  2  9x  6x  4
g) 2 1  2x  4x  10x  7
2


 1 1 
, ,2
2 3 
11. a) x  
 3 3  3 3 i 3  3 3 i 


,
,

4
8
8




c) x  1, 3  2 i, 3  2 i
12. a) x   , 1 2, 4 
1

x   ,  
2

2,   d)
b) y   2  x  1 
ii) a) y  f  2  x  1 
3
8. 15

2
f) 7 5x  1  x  x  1
b) y  3 x  2  4
13. i) a) y  3f  x  2  4



2,  c)
12. b) x  
2
b)  x  2 x  4x  1
10. e) 2 2  5m  4  10m  25m


2
y
x
x
x
4
g)
y
y
7. 3x  8x  4x  3   x  2 3x  6x  4x  12  21
6. y  3x  3x  30x  24
x
x
10
e)
y
y
7
y
x
x
x

OR
y
1 
2 
4
iii) a) y   f  x   5
3
1



13. iii) b) y    x   5
2
13. c) i)
ii)
y





































iii)
y
y











x
x








































x































Answers (Exponential and Logarithmic Functions and Equations):
as x   , f  x   0
1. a) i) D  x  R
ii) R  y y  0, y  R
iii) y  0
iv) none v) 1 vi) increasing vii)
1. b) i) D  x  R
ii) R  y y  0, y  R
iii) y  0
iv) none v) 1 vi) decreasing vii)
1. c) i) D  x x  0, x  R ii) R  y  R
iii) x  0
iv) 1 v) none vi) increasing vii)

as x   , f  x    
as x   , f  x    
as x   , f  x   0

as x  0 , f  x    
as x   , f  x    

1. d) i) D  x x  0, x  R ii) R  y  R
iii) x  0
4
2. a) x 
3
1
c)
8
2. l) 1
5. y  4
6. a)
1
b) x  3 c) x  6
3. a) 6 b)
81
5
7
5
m) 14 n)
o)
p)
4. 1.5131
4
3
2
8
d)
125
10 y
9
9
3
8
8
2
7
7
-3
-2
-1
1
2
3
4
5
6
7
8
10
-1
4
-2
3
3
y 2
2
1
1
-4
-5
y  6
-6
-7
-5
-4
-3
-2
-1
1
-1
2
3
4
5
6
7
8
-5
x
9
-4
-3
-2
-1
10
2
3
4
5
6
7
8
9
x
10
-3
-3
-4
-4
-9
1
-1
-2
-2
-8
y4
4
2
-3
c)
5
5
x
9
k) 2
6
6
1
-4
1
6
e) 2 f) -2 g) 3 h) 43 i) 0 j)
10 y
4
-5
as x   , f  x    
x
b)
5y
D77

as x  0 , f  x    
iv) 1 v) none vi) decreasing vii)
-5
-5
-10
vertical stretch factor of 3
vertical stretch factor of 3
vertical stretch factor of 2
shift right 4 units
shift left 1 unit
horizontal stretch factor of
shift down 6 units
shift up 2 units
D  x  R
D  x  R
reflections in x-axis & y-axis
shift left 2 units, shift up 4 units



e)
10 y
9
7
6
5
4
1
-1
-1
7y
6
5
5
4
4
3
3
2
1
1
2
-2
f)
6
2
3
-3

R  y y  4, y  R
7y
8
-4

R  y y  2, y  R
6. d)
-5
D  x  R

R  y y  6, y  R
-1
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
11
12
13
14
-10
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
10
1
-3
-3
-4
-4
-5
-5
-6
-4
-5
-6
-7
x  1
vertical stretch factor of 3
reflection in x-axis
shift right 4 units
shift up 5 units
reflection in y-axis
shift left 1 unit
shift down 3 units
D  x x  3, x  R
D  x x  4, x  R
D  x x  1, x  R
R  y  R
R  y  R
R  y  R


9. a) x  2, x  3
x
5
-3
x4
7
log m  8log9 n
3 9
9. b) x  1, x  2
4
-2
-7

1
2
b)
3
-2
-2
vertical stretch factor of 2
shift left 3 units
shift down 1 unit
7. a) 7log3 m  4log3 n
2
-1
15
-1
x
x  3

1
2
c) 4log3 a  log3 c  log3 d


8. a) -0.4332 b) -0.6402
Answers (Rational Functions):
1. a) x  2 , y  3 b)
 8
4
3

8 
d) x   , 
3 

2,   e)
f) none
2
1
-5 -4 -3 -2 -1
-1

c) x   ,2 
 3 
5y
1
2
3
-2
4
5
6
7
8
9
x
10
y  3
-3
g) D  x x  2, x  R
-4
-5
-6
-7
h) R  y y  3, y  R
-8
-9
-10
x 2
x   ,2
2, 
2. a) D  x x  4, x  5, x  R
9y
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
D78
b) D  x x  3, x  2, x  R
9y
3
4
5
6
x
7
y 1
1
y0
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
1
3
4
5
x
6
7
x 2
x  3
x  4
2
3.
10
y
8
6
4
y0
2
-10
-8
-6
-4
-2
2
4
6
8
x
10
-2
-4
-6
-8
-10
x  1
x  6
4.
5. possible answers:
10 y
a) f  x  
8
b) f  x  
6
4
c) f  x 
2
-4
-2
2
4
6
x
8
10
3  x  3 x  2 
 x  4   x  2
 x  4   x  52
x 1
 x  4   x  32

x4
y0
-2
-4
x 3
6 a)
b)
10
10
y
8
6
6
4
-8
-6
-4
-2
2
4
6
8
2
x
10
-10
-8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
x  1
x   , 3
4
y 2
2
-10
y
8
 1, 
x  1
4
6
8
x
10
y0
x 2
x   1,2
5, 
D79
Answers (Trigonometry):
1. a) 120
b) 134.1

r
r
6. a) x  2.30 ,3.98
2.

3
4
r
 2 4 
b) x   ,

 3 3 
8. a) i) D  x x   k,k  I, x  R



2



2
8. b) i) D  x x 
8. c) i) D  x x 
22
cm , 23.038 cm
3
3.
4. cos 
  11 
7. a) x   ,

2 6 
iii) x   k, k  I

  k,k  I, x  R 

ii) R  y y  1, y  1, y  R

  k,k  I, x  R 

ii) R  y  R
ii) R  y  R
iii) x 
3


2
iii) x 

iv) 2
  k, k  I
2
iv) 
  k, k  I
iii) x   k, k  I
For sketches for 8abcd part vii) – refer to your notes
2
9. a) amplitude=2, period= 4 , ps=
, vs=-4,
5. a)  3
5
b) undefined c)
1
2
  5 
b) x   , 
4 4 
ii) R  y y  1, y  1, y  R
8. d) i) D  x x   k, k  I, x  R
1
iv) 
v)

2
v) none vi) none
iv) 2
v)  k, k  I

vi) 0
vi) none
  k,k  I
b) amplitude=3, period= 2 , ps=

D    R , R  y 6  y  2, y  R
v) none vi) 1

, vs=2,
6

D    R , R  y 1  y  5, y  R












10. y  4sin 3  
 
12. b) d  7 sin 

6
 
 
 
  8 11. y  4cos 4  x    3 12. a)
4 
6 
 
a)
 t  6  9 c) 7.2 m
10

10
20
b) for sine function: h  15sin 
 15
35
25
20
 
t   17
15


10
OR
5
5
10
15
20
25
30
13. c) 24.5m d) 11.4s and 18.6s
35
14. a)
40
45
50
7
sin8x
2
  2 7 8 13 14 
17. a) x   , ,
,
,
,

9 9
5 

17. e) x   ,1.98, 4.30, 
3

for cosine function: h  15cos 
15
6 2
4
 t  7.5  17

OR h  15sin   t  7.5   17
15


30
16.
30

40
-5

20
12. d) between 01:31h and 04:29h and
between 13:31h and 16:29h
13.


3 
9
9
9
 7 11 
,

 6 6 
f) x  
9 
55
60
b) cos14x
 3 

4 
b) x  
c) 3cos10x

c) x  0,

g) x  0.31, 1.88, 3.45, 5.02


h  15cos   t  15   17
 15


15. a) sin  
8
2
4

,,
,2 
3
3


b) tan  
 6 
  5 

6 6 
d) x   ,
h) x  0.56,2.58
D80
Answers (Combination of Functions):
1. a) -1 and 4 b) 8 c) For constant x , the 4th differences in y are constant.
2. i) a)
log2
1
log3
ii) a) 3
3. a) 2  3
c) as x   , f  x   4
b) -2
b) -6
c)

as x   , f  x    
1 
, f x   
5
as x   , f  x    
as x 
b) one period:
4. a) Df  x x  3, x  3, x  R
5 7
,
18 18
d) For constant x , there is a constant ratio in
the 1st differences in y.
d) For constant y , there is a constant ratio in the
1st differences in x.
7 2

 5 2

k,

k,k  I
3
18
3
 18

c) all zeroes: x  
b) Dg  x x  4,x  R
c)  f  g   x   x  9  4  x ; D
 x x  3,3  x  4,x  R
fg 
2
5. a) Df  x  R



2
b) Dg  x x 

  k,k  I,x  R 

d)  f  g   5   7
f


g
 f   3  1
d)     
2
 g  4 
6. a) Df  x  R
b) Dg  x x  1,x  R
c)  f g   x   log2  x  1   4 ; Dg  x x  1,x  R
d)  f g   9   1
7. answers may vary
f x 
2
, g  x   3x  1
x
OR

c)    x   cos x ; D  x x  k, k  I, x  R 
f
2


g
f x 
2
, g  x   3x
x 1