Name:
Sha Tin College Mathematics Department
Key Stage 4 Extended Level Course
Unit 9 Assignment: Functions
Need to Know
Types of function
Linear
f ( x)  ax  b
Quadratic
Total /103
Formulae
Need to Know
Types of function
Exponential
f ( x)  ax  bx  c
or
2
f ( x )  a ( x  b)  c
Cubic
2
f ( x)  ax 3  bx 2  cx  d
Re ciprocal
a
f ( x) 
x
Transformations of function
f ( x)  a
a causes a vertical shift upwards
Transformations of function
f ( x  a)
a causes a horizontal shift of a to the left
Transformations of function
f ( x)
Reflection in y axis
f ( x)  a x
Absolute
Value
f ( x)  f ( x)
Trigonometric
f ( x)  a sin(bx  c)  d
or
f ( x)  a cos(bx  c)  d
Transformations of function
af ( x)
a causes a stretch, factor a
Transformations of function
 f ( x)
Reflection in x axis
Function notation
f 1 ( x)
Inverse function
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
1
Function notation
fg ( x)
Composite functions, do function g(x) and
then do function f(x) on the result
Log Laws
log x  log y  log xy
log x
log x  log y 
log y
a log x  log x a
Log Laws
If y  b x index statement
Then
x  logb y log statement
Solving log equations
If a x  b
Then
log b
x
log a
A: Types of function. Domain and Range. Symmetry, period,
asymptotes, vertices, intercepts.
1. (Non calc) State the domain and range of the following relation
{(2,4), (-2,4), (6, 36)}
Domain __________________________
Range ____________________________
[2]
2. (Non calc)
f(x) = x + 1
1
2
3
Domain
Range
Complete the mapping diagram to show the range of f(x)
[2]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
2
3. (non calc)
-2
2
3
-3
4
9
Domain
Range
Is this mapping diagram showing a function?
If so, describe the function fully.
[1]
[2]
4. Complete this table.
Graph sketch
Name
[21]
General Form
Domain
{x : x  R}
Range
{ y : y  R}
Quadratic
Cubic
Reciprocal
y
a
x
Exponential
Total for Section A
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
/28
3
B: Transformations of functions.
1.(no calc)
The above function is f(x) = cos x
On the above graph sketch each of the following – and label them clearly.
(a) f(x +2)
[2]
(b) f(-x)
[2]
(c) 3f(x)
[2]
2.(no calc)
(a) Sketch the graph of y = x2 + 1 on the axes above
[2]
2
(b) Sketch the graph of y = -x on the same axes above
[2]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
4
3. (no calc)
For a particular function f(x) = (x+3)(x-2)
Determine the following:
(a) The coordinates of the x intercepts
( ____, ____ ) and ( ___ , ___)
[2]
(b) The coordinates of the vertex
( ___ , ___ )
[2]
(c ) The coordinates of the y intercept
( ___ , ___ )
[2]
4. (no calc)
Find the quadratic equation in the form y = a(x – b)2 + c
When the vertex is (0, -2) and passing through the point (3, 10).
[3]
5. (no calc)
Find the quadratic equation with x intercepts (-2, 0) and (2, 0) and passing through the
point (1,3).
[3]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
5
6. (no calc)
Find the quadratic equation in the form y = a(x – b)2 + c
When the vertex is at (3, -1) and the value of a = 1
Sketch the graph of the function.
Equation y = _________________
Sketch
[3]
Total for Section B
/25
C: Trigonometric functions and their transformations
1. (no calc)
Above is a graph of y = sin x
(a) Label the x axis in degrees, marking off 0o, 90o, 180o, 270o, 360o.
[2]
(b) What is the range of this function?
[2]
(c) What is the period of this function?
[2]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
6
2. (no calc)
(a) Sketch the graph of f(x) = tan x for 0o < x < 360o
Include any asymptotes.
[3]
(b) Sketch the graph of f(x) = cos x for 0o < x < 360o on the same axes.
[2]
(c) Solve the equation tan x = cos x for 0o < x < 360o
[2]
Total for Section C
/13
D: Using function notation, composite and inverse functions.
1.(no calc)
(a)
Given the function
3  2x ,
f :x
find the value of f (2) ,
Answer (a) f (2) 
(b)
[1]
find the inverse function f 1 .
Answer (b) f 1 : x
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
[2]
7
f :x
2.
x 1
.
3
Find
(a)
(b)
1
f  ,
8
f ( x) 
(b)
[1]
Answer (b)
[2]
the inverse function, f 1 ( x) .
3.
(a)
Answer (a)
3x  2
,
x 1
 x  1 .
Find f (4) .
Answer (a)
[1]
Answer (b) x =
[2]
Solve f ( x)  4 .
Total for Section D
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
/9
8
E: The log function
1.(no calc) Sketch a graph of y = 3x and y = log3x on the same axes. Show clearly the
location of the intercepts and the asymptotes.
Total for Section E
/6
F: The Log laws, solving Log equations
1. (no calc)
Express the following as single logarithms
(a) Log 2 + Log 5
[1]
(a) Log 7 + Log 2
[1]
(c) 2Log 3 + Log 6
[2]
(d) Log 2 + ½ Log 16
[2]
2. Solve these Log equations, giving your answers to 2 dps
(a) 2x = 7
[3]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
9
(b) 5.3x = 11
[3]
3. David invested $1000 at a rate of 6% interest, compounded annually. The value of his
investment in dollars V, can be calculated using the following formula.
V = 1000(1.06)x
V: Value of investment
x: number of years
How many years would it take to have $2000?
[3]
4. In a similar investment situation,$1000 could be invested at a compound interest rate
of 14%. How many years would it take before the value of the investment reached
$2000?
[3]
5. If Logx64 = 3
(a) Write this log equation as an index equation
[2]
(b) Solve the index equation above
[2]
Total for Section F
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
/22
10
Unit 9 Assignment: Functions
CAN DO STATEMENTS
Syllabus
Main Learning Objectives
Tick
Reference
Here!!
3.1
NEW Understand the meaning of domain and range and be able
to illustrate these with the use of arrow diagrams (mappings)
3.2 3.5
NEW Be able to recognize the following function types from
3.7 3.6
the shape of their graphs.
7.7
Linear f(x) = ax + b
Quadratic f(x) = ax2 + bx + c
Cubic f(x) = ax3 + bx2 + cx + d
Reciprocal f(x) = a/x
Exponential f(x) = ax with 0<a<1 or a>1
Absolute value f(x) = ax + b
3.2
3.2 3.5
3.7 3.6
Paper 3
3.3 3.9
Paper 3
3.4
3.8
3.8
3.10
3.11.1
3.11.2
3.11.3
NEW Be able to recognize the following function types from
the shape of their graphs.
Trigonometric f(x) = asin(bx+c) + d or acos(bx + c) + d, tanx
NEW Sketch various graphs as above in order to solve word
problems
Modelling – drawing a graph and finding a model.
Interpolating and extrapolating. Evaluating the model.
NEW Transformations of these functions. Determine at most
two of a,b,c or d in simple cases of graphs listed above.
Describe and identify, using the language of transformations,
the changes to the graph of y = f(x) when y = f(x) + k, y =
kf(x), y = f (x+k) – where k is an integer
Modelling – modeling “real life” data by an appropriate
graph to find a model. Interpolating and extrapolating.
Evaluating the model.
NEW Find the quadratic equation given a) vertex and another
point
b) x –intercepts and a point
c) vertex or x-intercepts
with a = 1
NEW Use function notation and be able to substitute in for x eg
If f(x) = 3x find f(4). Simplify functions.
NEW Find composite functions
NEW Find inverse functions
NEW Logarithmic function as the inverse of the exponential
function.
y = ax equivalent to x = logay
NEW Know the rules of logarithms. Understand how they
relate to the rules of indices.
NEW Be able to solve index equations using logs. Recognise
that the solution to ax = b as x = logb/loga
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
11
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Functions
12