TOPIC 6:
MATHEMATICAL MODELS
EXPONENTIAL FUNCTIONS
In this course you need to deal with two forms of exponential
function:
A relation is any set of points on the Cartesian plane.
² f (x) = kax + c which is increasing for all x
A function is a relation in which no two different ordered points
have the same x-coordinate or first member.
² f (x) = ka¡x + c which is decreasing for all x.
In each case:
We use a vertical line test to see if a relation is a function.
² a and k control the steepness of the curve
Consider a function y = f (x).
² y = c is the equation of the horizontal asymptote.
² the domain of f is the set of permissible values that x may
take
² the range of f is the set of permissible values that y may
take
² the x-intercepts are the values of x which make f (x) = 0
y
f(x) = kax + c
k+c
y=c
² the y-intercept is f (0).
x
LINEAR FUNCTIONS
Exponential functions are commonly used to model growth and
decay problems.
A linear function has the form f (x) = mx + c
where m and c are constants.
Exponential equations are equations where the variable appears
in an index or exponent.
y
The graph of y = f (x) is a
straight line with gradient m
and y-intercept c.
m
You should be able to solve exponential equations using
technology.
1
c
PROPERTIES OF FUNCTIONS
x
y = f(x)
You should be able to use a graphics calculator to:
² graph a function
² find axes intercepts
² find asymptotes
QUADRATIC FUNCTIONS
A quadratic function has the form f (x) = ax2 + bx + c
where a, b, and c are constants, a 6= 0.
An asymptote is a line which a function gets closer and closer
to but never meets.
The graph of y = f (x) is a parabola.
The graph has a vertical axis of symmetry x = ¡
³
vertex or turning point at
¡
³
b
b
,f ¡
2a
2a
´´
We observe horizontal asymptotes in the exponential functions.
b
and
2a
We observe vertical asymptotes in power functions such as
1
1
f (x) =
and f (x) = 2 .
x
x
.
If a > 0, the graph opens upwards
and the vertex is a local minimum.
SKILL BUILDER - SHORT QUESTIONS
If a < 0, the graph opens downwards
and the vertex is a local maximum.
1 The graph below shows the cost $C of a telephone call which
lasts t minutes.
Cost ($)
4
3
2
1
If the graph has x-intercepts, the axis of symmetry lies midway
between them.
y
c
b
x = ¡ 2a
1
y = f(x)
x-intercepts
¡
b
b
¡ 2a
, f ¡ 2a
2
3
4
t (min)
a Find the equation of the line C(t).
b Using your equation, calculate the cost for a call lasting
23 minutes.
c Determine the length of a phone call which costs $18:31.
Round your answer to the nearest minute.
x
¡
² find domain and range
² find turning points
² find where functions meet.
¢¢
2 Consider the function h(t) = 1 ¡ 2t2 .
You should be able to use technology to find points at which:
a Find h(0).
² a linear function meets a quadratic
b Find the values of t for which h(t) = 0.
² two quadratic functions meet.
c Write down the domain of h.
d Write down the range of h.
41
Mathematical Studies SL – Exam Preparation & Practice Guide (3rd edition)
9 The sum of the first n terms of an arithmetic sequence can be
n
found using the formula Sn = (2a + (n ¡ 1) £ d), where a
2
is the first term of the sequence and d is the common difference.
3 The graph below shows the percentage P of radioactive
Carbon-14 remaining in an organism t thousands of years after
it dies.
100
P (% of carbon-14)
a The first term of an arithmetic sequence is 7 and its
common difference is ¡5. If the sum of the first n terms
of the sequence is ¡1001, show that n can be found by
solving the equation 5n2 ¡ 19n ¡ 2002 = 0.
80
60
40
b Solve the equation in a to find n.
20
t (thousand years)
8
12
4
10 The number of people N on a small island t years after
settlement, increases according to the formula
N = 120 £ (1:04)t . Use this formula to calculate:
a Use the graph to estimate:
a the number of people who started the settlement
i the percentage of Carbon-14 remaining after
4 thousand years
ii the number of years for the percentage of Carbon-14
to fall to 50%.
b the number of people present after 4 years
c the number of years it will take for the number of people
to double from the initial settlement.
p
11 A function f is defined as f (x) = x + 4 for
¡4 6 x 6 12, x 2 R .
b The equation of the graph is P = 100 £ (1:1318)¡t for
all t > 0.
Calculate the percentage remaining after 19 thousand
years.
c Write down the equation of the asymptote to the curve.
b Calculate g(¡2).
a Determine the value of k if 810 ball bearings with
diameter 2 mm can be cast.
b Find the number of ball bearings with diameter 3 mm
which could be cast from this quantity of metal.
5 The data below shows the price P that each bag of rice can
be purchased for at a wholesale market in Jakarta if b bags are
bought.
30
35
40
45
50
P rupiah
38 000
36 000
34 000
32 000
30 000
iii f (12).
12 The number of ball bearings of diameter d mm which can be
cast from a given quantity of metal is given by N = kd¡3 .
c Find the value of x for which g(x) = f (x).
b bags
ii f (0)
c Hence, write down the range of f (x).
4 Consider the functions f (x) = 15 ¡ 2x and g(x) = 2x + 1.
a Calculate f (2).
i f (¡4)
a Calculate:
b Sketch y = f (x).
c If 23 ball bearings of radius r could be cast, find the value
of r.
13 The population growth of a hive of bees is given by
t
P (t) = 120(2:25) 3 , where t represents time in weeks.
a Determine the function P (b).
a Sketch P (t) for 0 6 t 6 20.
b Hence find the total cost of purchasing 60 bags of rice.
b Find the population of bees in the hive after 10 weeks.
c Find how long it takes for the hive to number more than
5000 bees.
6 The graph of a quadratic function has x-intercepts 2 and ¡1,
and passes through the point (3, 12).
14 A graph of the quadratic
y = ax2 + bx + c
is shown alongside,
including the vertex V
and y-intercept.
a Find the equation of the function in the form
y = ax2 + bx + c.
b Using an algebraic method, find the vertex of the graph of
the function.
y
9
V(1, 7)
x
x
7 The exponential function y = a £ 2 + b passes through the
following points:
x
0
1
2
3
y 20 p 35 q
a Determine the value of c.
b Use the axis of symmetry to write an equation involving
a and b.
c Use the point (1, 7) to write another equation involving a
and b.
d Find a and b.
a Write down two linear equations which could be used to
determine the values of a and b.
b Solve the linear equations simultaneously to find a and b.
15 Consider the function f (x) = x3 ¡ 3x2 ¡ x + 3, where f is
defined on the domain ¡2 6 x 6 3, x 2 R .
c Hence, find the values of p and q.
a Sketch the graph of y = f (x), showing any axes
intercepts and turning points.
8 Consider the function k(t) = 2t ¡ 4 for 0 6 t < 4, t 2 Z .
b Determine the range of f .
a List the elements of the domain of k(t).
b List the elements of the range of k(t).
16 An object thrown vertically has height (relative to the ground)
given by H(t) = 19:6t ¡ 4:9t2 metres after time t seconds.
c Sketch the function k on a set of axes, showing all elements
in the domain and range.
Mathematical Studies SL – Exam Preparation & Practice Guide (3rd edition)
a Determine H(0) and interpret its meaning.
42
22 The diagram shows the graph of the quadratic function
f (x) = x2 + mx + n, including the vertex V.
y
b Find the time when the object returns to ground level.
c Write down the domain of H(t) using interval notation.
d Determine the maximum height reached by the object.
17 A part of the graph of the function f (x) = x2 + 3x ¡ 28 is
shown below.
y
V(1, 3)
x
A
a Determine the values of m and n.
b Find k given that the graph passes through the point
(3, k).
x
B
C
c Find the domain and range of f (x).
a Expand and simplify the expression (x + 7)(x ¡ 4).
1
.
x¡2
a Sketch a graph of the function on a grid like the one below.
y
23 Consider the function y = 3 +
b Hence find the coordinates of A and B.
c Determine the equation of the axis of symmetry.
d Write down the coordinates of C, the vertex of the
parabola.
8
4
18 Consider the function f (x) = 8x ¡ 2x2 .
a Factorise fully f (x).
-3
3
x
6
-4
b Determine the x-intercepts for the graph of y = f (x).
c Write down the equation of the axis of symmetry.
b Write down the equations of the vertical and horizontal
asymptotes.
d Determine the coordinates of the vertex.
19 The function g(x) = 4 £ 2x ¡ 3 is defined for all x 2 R .
Write down the range of g.
24 Before it is turned on, a refrigerator has an internal temperature
of 27± C. Three hours later it has cooled to 6± C.
The internal temperature T (in ± C) is given by the function
T (t) = A £ B ¡t + 3, where A and B are constants and t is
measured in hours.
20 The number of ants in a colony is given by
N (t) = 30 ¡ 27 £ 3¡t thousand, where t is the time in months
after the beginning of the colony.
a Given that T (0) = 27, determine the value of A.
a Calculate the initial number of ants in the colony.
b Hence determine the value of B.
c Find the internal temperature of the refrigerator 5 hours
after being turned on.
b Calculate the population of ants after 2 months.
c Find the time taken for the colony to reach 20 000 ants.
d Determine the equation of the horizontal asymptote of
N (t).
d Write down the minimum temperature that the refrigerator
could be expected to reach.
e According to the function N (t), what is the smallest
number of ants the colony will never reach?
21 Consider the function f (x) =
25 The diagrams below are sketches of four out of the following
five functions. Match each diagram with the correct function.
2x
.
x¡1
A y = ax
B y =a¡x
a Find the y-intercept.
D y = x2 ¡ a
E y =x¡a
b Determine the minimum value of f (x) for x > 1.
a
C y=
1
x
b
c Write down the equation of the vertical asymptote.
d Calculate f (5).
1
e Sketch the graph of y = f (x) for ¡4 6 x 6 7 on a
set of axes like those given, showing all the features found
above.
12
c
y
d
8
4
-3
3
6
26 A quadratic function has the form f (x) = ax2 + bx + 7.
It is known that f (2) = 7 and f (4) = 23.
x
-4
a Construct a set of simultaneous equations involving a
and b.
b Find a and b.
c Hence calculate f (¡1).
-8
43
Mathematical Studies SL – Exam Preparation & Practice Guide (3rd edition)
30 A graph of the form y = a(x ¡ 1)(x ¡ 5)2 is shown below.
y
P
27 The daily profit made by a business depends on the number of
workers employed. It can be modelled by the function
P (x) = ¡50x2 + 1000x ¡ 2000 euros, where x is the number
of workers employed on any given day.
a Using a set of axes like those below, sketch the graph of
P (x) = ¡50x2 + 1000x ¡ 2000.
3000
2000
1000
P
8
4
-1000
-2000
c Find the coordinates of P, which is a local maximum.
31
c Write down the value(s) of x for which
x4 ¡ 2x3 ¡ 3x2 + 8x ¡ 4 = 0.
28 Identify the graphs which best represent the functions f (x),
g(x), and h(x).
a f (x) = x2 ¡ 3x
32 Let f (x) = 3 ¡ 4¡x .
b g(x) = x2 + 3x
c h(x) = x2 ¡ 9
y
B
3
y
b For the graph of y = f (x), determine the:
3
D
y
9
a Points A(2, p) and B(¡2, q) lie on y = f (x).
Determine p and q.
-3
x
a Sketch the function y = x4 ¡ 2x3 ¡ 3x2 + 8x ¡ 4 for
¡3 6 x 6 3 using an appropriate scale.
b Write down the coordinates of any turning points, and add
these points to your graph in a.
c Find the maximum possible profit.
C
x
a Determine the values of b and c.
b Given that the y-intercept is ¡50, calculate the value of a.
20 x
16
12
b Determine the number of workers required to maximise
the profit.
A
c
b
x
i x and y intercepts
ii equation of the horizontal asymptote.
c Sketch the graph of y = f (x), showing all detail from
above.
d Write down the range of f (x).
y
33 Consider h(x) = x2 ¡ 2¡x +
x
x
-3
1
.
x
a Determine h(¡2).
b Solve h(x) = 2.
c Write down the equation of the vertical asymptote.
E
y
d Sketch y = h(x), illustrating your results from a, b,
and c.
e Determine the range of y = h(x).
3
x
SKILL BUILDER - LONG QUESTIONS
1 A company manufactures and sells DVD players.
If x DVD players are made and sold each week, the weekly
cost to the company is C(x) = x2 + 400 dollars.
The weekly income obtained by selling x DVD players is
I(x) = 50x dollars.
29 Consider the graph of y = f (x), where
f (x) = x3 (x ¡ 2)(x ¡ 3).
a Use your graphics calculator to find:
i all x-intercepts
ii the coordinates of any turning points.
a Find:
i the weekly cost for producing 20 DVD players
ii the weekly income when 20 DVD players are sold
iii the profit or loss incurred when 20 DVD players are
made and sold.
b Sketch the graph of y = f (x) using a grid like the one
given. Show all of the features found above.
4
y
b Find an expression for the weekly profit
P (x) = I(x) ¡ C(x) when x DVD players are produced
and sold. Do not simplify or factorise your answer.
2
1
2
3
x
c The maximum weekly profit occurs at the vertex of the
function P (x).
Determine the number of DVD players which must be
made and sold each week to gain the maximum profit.
-2
-4
Mathematical Studies SL – Exam Preparation & Practice Guide (3rd edition)
44
a Write down the equation of the:
i vertical asymptote
ii horizontal asymptote.
d Calculate the profit made on each DVD player if the
company does maximise the profit.
e Given that P (x) can be written as
P (x) = (x ¡ 10)(40 ¡ x), find the largest number of
DVD players the company could produce each week and
still make a positive profit.
2 Consider the graph of y = f (x) where f (x) = 2 +
a Find the x-intercept.
b
4
.
x+1
c Determine the value of a such that y = ax intersects
4
y=
when x = 2.
2x ¡ 3
d For the value of a calculated in c, sketch the graph of
y = ax on a set of axes like those given above.
b Find the y-intercept.
c Calculate f (¡2).
d Determine the equation of the:
i horizontal asymptote
ii vertical asymptote.
4
for ¡5 6 x 6 3.
e Sketch the graph of y = 2 +
x+1
Label the axes intercepts and asymptotes clearly.
5
¡ 1 for x 2 R .
x
a For what value of x is f (x) undefined?
6 Consider the function f (x) =
b Find the asymptotes of y = f (x).
3 A rectangular field with one side x m is enclosed by a 160 m
long fence.
c The points A(1, m) and B(n, 0) lie on the graph of
y = f (x). Calculate the values of m and n.
a Write down the width of the field in terms of x.
b Write a function A(x) for the area of the field.
d Sketch the graph of y = f (x) for 0 6 x 6 8. Clearly
show all features found in parts a, b, and c.
5
¡ 1 = 5 ¡ x can be rearranged into
e
i Show that
x
the form (x ¡ 1)(x ¡ 5) = 0.
ii Hence solve the equation.
c Sketch the graph of y = A(x).
d Find the dimensions of the field which has the largest
possible area for the given length of fence.
e The actual area of the field is 1200 m2 .
i Find the dimensions of the field.
ii The average production yield for this field is
6:5 kg m¡2 . Determine the amount of production lost
by not using the best possible dimensions for the
fence.
f The linear function h(x) = cx + d passes through the
points A and B.
i Write down the values of c and d.
ii Add the graph of y = h(x) to your sketch in d.
g Write down the positive values of x for which
h(x) > f (x):
4 A population of insects grows according to the rule
N = 1200 £ kt , where N is the number of insects and t
is the time in months since the population was first measured.
h A quadratic function g(x) passes through the origin and
the points A and B. Determine the values of p, q, and r
given that g(x) = px2 + qx + r:
a Given that N = 4800 when t = 4, find the value of k.
b Complete the following table of values, giving your
answers correct to the nearest 10 insects.
t
0
1
2
3
4
5
7 The monthly cost $C for a mobile phone depends on the total
outgoing call time t minutes for the month. Jane compares three
different monthly mobile phone plans:
Plan 1: Fixed fee of $50
Plan 2: Fixed fee of $20 plus $0.60 per minute of
outgoing calls.
Plan 3: No fixed fee but $2.00 per minute of
outgoing calls.
6
4800
N
c Draw the graph of N for 0 6 t 6 6. Use 2 cm to
represent 1 month on the horizontal axis, and 1 cm to
represent 1000 insects on the vertical axis.
d Use your graph to find the number of insects present after
2 12 months. Give your answer correct to the nearest 100
insects.
e Find the number of months it takes for the population to
reach 20 000 insects.
f Calculate the percentage change in the number of insects
in the sixth month.
5 The graph of y =
a Use a graph to illustrate the costs of the three monthly
plans. Your axes should show 0 6 t 6 60 and
0 6 C 6 120. Let 1 cm represent 5 minutes on the t-axis,
and 1 cm represent $10 on the C-axis. Clearly label each
plan.
b Calculate the monthly cost of each plan if Jane were to
use her mobile phone for outgoing calls totalling:
i 30 minutes
ii 60 minutes.
c In terms of C and t, write down the equation of:
i Plan 1
ii Plan 2
iii Plan 3
d Determine the coordinates of all points of intersection
between the three graphs. Clearly mark these coordinates
on your graph in a.
4
is shown below.
2x ¡ 3
y
16
12
4
8
y=
4
2x ¡ 3
-3 -2 -1
-4
-8
-12
-16
1
2
3
4
i The graph of y = ax where a > 0, is to be drawn
on the axes above. Determine the y-intercept for this
graph.
ii Determine the equation of the horizontal asymptote
for y = ax .
e Complete the following statements:
5 x
i “Plan 1 is the least expensive provided ...... minutes
of outgoing calls are made.”
ii “Plan 2 is the least expensive provided ...... minutes
of outgoing calls are made.”
45
Mathematical Studies SL – Exam Preparation & Practice Guide (3rd edition)
e Using your graph, or otherwise, write down the:
iii “Plan 3 is the least expensive provided ...... minutes
of outgoing calls are made.”
i minimum possible surface area of the prism
ii dimensions which minimise the surface area.
8 The cost of producing x bicycles in a factory is given by
C(x) = 6000 + 40x pounds. The revenue earned from the
sale of these bicycles is given by R(x) = 100x pounds.
f Write down the range of A(w).
11 Consider the four graphs A, B, C, and D below.
a Graph each function on a set of axes like those below.
y (1, 5)
A
y
B
16 000
(1, 4)
1
x
x
-2
12 000
8000
4000
-20
y
C
y
D
(0, 2)
20 40 60 80 100 120 140 160 180 x
x
x
-4
(2, -2)
b Determine the initial setup cost before any bicycles are
produced.
c Calculate the number of bicycles which must be produced
and sold in order for the factory to break even.
a Which graphs, if any, have the same domains? Explain
your answer.
d Calculate the revenue earned per bicycle.
b Write down the range of each graph.
e Write down the profit function P (x) for all x > 0.
c Copy and complete the table below, matching each graph
above to one of the equations listed below.
f Find the profit resulting from the sale of 400 bicycles.
x3
.
2x
a Use your graphics calculator to find:
9 Consider the function f (x) =
Graph
y = a(x ¡ h)2 + k
i the y-intercept
ii the maximum value of f (x) for ¡2 6 x 6 12
iii f (2) and f (¡1).
y = mx + c
y = p £ qx + r
b Sketch the graph of y = f (x) for ¡2 6 x 6 12 on
a set of axes like those shown below, showing all of the
features found above.
y
4
2
-2
-2
-4
-6
2
4
6
Equation
8
10
12
y=
v
x
d Using the information shown on each graph, and the
equations from c, determine the value of each unknown
constant a, h, k, m, c, p, q, r, and v.
x
TOPIC 7:
CALCULUS
RATES OF CHANGE
c On the same axes as above, sketch the function
g(x) = (x ¡ 5)2 ¡ 7.
Clearly show the coordinates of the minimum for g(x),
and any intersection points with f (x).
d Hence, or otherwise,
find all solutions to
A rate is a comparison between two quantities of different kinds.
The average rate of change between two points on a graph is
the gradient of the chord between them.
x3
+ 7 = (x ¡ 5)2 .
2x
The instantaneous rate of change at a particular instant is the
gradient of the tangent to the graph at that point.
10 The length l, width w, and height h of a rectangular prism are
all measured in centimetres. The volume of the prism is fixed
and equal to 2000 cm3 , and the length is double the width.
APPROXIMATING THE GRADIENT OF A TANGENT
TO y = f (x)
1000
.
w2
b Hence, show that the surface area A of the prism is given
6000
by A(w) = 4w2 +
cm2 .
w
c Write down the domain of A(w).
For the graph of a function y = f (x), consider two points
A(x, f (x)) and B(x + h, f (x + h)).
y
y = f(x)
a Show that h =
A(x, f(x))
d Draw the graph of A(w) for 0 6 w 6 30 and
0 6 A 6 5000. Clearly show the coordinates of any
turning points.
Mathematical Studies SL – Exam Preparation & Practice Guide (3rd edition)
x
B(x + h, f(x + h))
46