Higher Mathematics Unit 1.2
Functions
Contents
1.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Sketch Graphs of Common Functions.
f ( g ( x)) .
Inverse Functions.
Graphical Interpretation.
Logarithmic/Exponential Function.
Trigonometry Revision
Exact Values
Radians
Equations
Trig Graphs
References/Resources.
Sketching the Graphs of Common Functions
(a)
(b)
Pupils should be familiar with, and be able to sketch, the graphs of the following
y  x , y   x , y  kx , y  x 2 , y  x 3 , y  x
functions:
1
y3 x , y
.
x
1
Be careful to discuss the domain of x and .
x
Pupils must be able to sketch graphs such as y  sin x , y  cos 2 x etc.
Some pupils will be good at this from “C” level, while others will need the revision.
Leave the harder examples such as y  10 cos(3x  60) until Unit 3.
(c)
Pupils must be able to draw sketches of graphs of quadratic functions.
At this stage deal only with functions whose graphs meet the x-axis.
The procedure is:
(i)
Decide on shape of graph. (“happy” or “sad”?)
(ii)
Find coords of y intercept.
(iii) Solve equation to find coords of x intercept – emphasise
roots
(iv)
Draw sketch.
It is essential that pupils have a good graphical awareness, so spend sufficient time
reinforcing “C” level Knowledge of the quadratic function.
(d)
Domain and Range
At this stage, all that needs to be known is
“Domain” means “set of x values”.
“Range” means “set of y values”.
Practice should be given at stating a suitable domain and range:
Pupils should know that
1
is not defined when u  0 .
u
u is defined only when u  0 .
1
. f is not defined when x  3 (div. by zero).
x3
So a suitable domain is x  R , x  3 .
For example, consider f ( x) 
1
. Now x  1 is defined when x  1 0 , i.e. when x  1 .
x 1
But when x  1 we have a division by zero.
Therefore the domain here is x  1 .
Completing the Square
Also, consider g ( x) 
(e)
By far the best approach is to Equate Coefficients. In most exam questions
nowadays the candidates are told what to do, so they just have to follow instructions.
For example, “Express 3x 2  12 x  37 in the form a( x  b) 2  c .”
3x 2  12 x  37  a( x  b) 2  c
 ax 2  2abx  ab 2  c
Equating coefficients, it can be seen that a  3. Also, 2ab  12  6b  12  b  2.
And ab 2  c  37  12  c  37  c  25.
 3x 2  12 x  37  3( x  2) 2  25.
This method works well for all quadratic functions including examples such as
3  x  4x .
2
Notes
(a)
Use “completing the square” to find the max/min value of a quadratic or related function.
1
.
For example, the minimum of x 2  4 x  9 or the maximum of 2
x  4x  9
But, when dealing with fractions, make sure that the denominator is always strictly
positive.
(b)
The form “ a( x  b) 2  c ” always works, but abler pupils should be encouraged to choose
their own “form”.
For example, x 2  7 x  23 is best written in the form ( x  b) 2  c.
16  3 x  2 x 2 is best written in the form a  b( x  c) 2 .
(c)
Pupils should state reasons for results.
For example, “Find the minimum value of x 2  3 x  7. ”
Using the methods outlined above, it will be found that x 2  3x  7  ( x  32 ) 2  194 .
The working to demonstrate the minimum may be done as follows:
( x  32 ) 2  0 for all real x. (square of a real number).
 ( x  32 ) 2  194  194 for all real x.
Hence the minimum value is
d)
2.
19
4
and this occurs when x  32  0 , i.e. when x   32 .
Use completing the square to help graph the quadratic function.
y  x 2  4x  9
Eg
 ( x  2) 2  5
Relate back to the graph y = x2
f(g(x)
(a)
Revise the ideas behind a function, e.g. f ( x)  3 x  2 :
x
3
+2
Or, say , y  ( x  4) 2 :
f (x )
x
+4
“all squared”
( x  4)2 .
Arrow diagrams are a good way of illustrating “composition of functions”:
g
x
f
g(x)
f(g(x))
F
In the above diagram, F ( x)  f ( g ( x)) .
In a typical question we are required to find a formula for f ( g ( x)) , simplifying the
answer, if necessary.
For example: “ f ( x)  2 x  3 and g ( x)  x 2 . Find a formula for f ( g ( x)) .”
Some pupils will have difficulty here with f ( x 2 ) . For such pupils it is worthwhile
considering, for example, f ()  ? ; f ()  ? ; f ( )  ? and so on. Numerical
examples can help to overcome problems.
Eg f ( x)  x 2 , g ( x)  x  1; f ( g (3)) etc.
1
Examples such as f ( x)  , g ( x)  x  1; f ( g ( x)) could be used before any harder
x
examples of type.
Also consideration of examples of type f(f(x)) should also be given.
[i.e. get them used to the idea that it is f that is special, not x; they must understand that
function f is a rule, which in this case, when applied to a number, tells us to “multiply the
number by 2 and then add 3 to the result.”]
Do examples involving three functions, e.g. f ( g (h( x))).
Example:
Functions f(x) = 3x – 1 and g(x) = x2+7 are defined on the set of real numbers.
a) Find h(x) where h(x) = g(f(x))
b) (i) Write down the coordinates of the minimum turning point of y=h(x)
(ii) Hence state the range of the function h.




3.
1
2
3
4
(2)
(2)
g(3x – 1)
(3x – 1)2 + 7
(1/3, 7)
y  7, accept h(x)  7
7
Inverse Functions
Calculating a formula for f 1 is not in the syllabus.
Pupils only have to know that if f and g are such that
inverses.
An arrow diagram is useful to illustrate this:
f ( g ( x))  x then f and g are
g
x
g(x)
f
If this is the case, given the graph of f(x), then the graph of g(x) can be found by reflecting
f(x) in the line y = x.
Examples of simple inverse functions such as the following should be gone over.
“Add 4”
“Subtract 4”
“Multiply by 5”
“Divide by 5”
“square”
“Square root”, and so on.
4.
Graphical Interpretation
This is best illustrated by example:
y
(1,3)
f(x)
-2
O
3
7
x
Pupils might be asked to sketch the following types of related graphs:
y  2 f ( x) , y   f ( x) , y  3 f ( x) , y  4  f ( x) , y  4  f ( x)
y  f ( x  3) , y  f ( x  3) , y  f (2 x) , y  f ( 12 x) , y  f ( x)
The best way to consider 4  f ( x ) is to do it as  f ( x)  4 .
,
.
The pupils might be required to sketch, for example, the graph of g(x), where the graph of
g(x) is got by reflecting the graph of f(x) in the line with equation x  7 .
It is important that whatever the level of information given in the original graph (e.g.
coords of points), the same level of information should be displayed in the pupils’
answers.
Example:
The diagram shows the graph of y = g(x).
a) Sketch the graph of y = -g(x)
b) On the same diagram, sketch the
graph of y = 3 – g(x)
(2)
(2)
5.

1


2
3

4
reflection in the x-axis and any
one from (0, -1), (a, 2), (b, -3) clearly annotated.
the remaining two from the above
translation and any one from
(0, 2), (a, 5), (b, 0) clearly annotated
the remaining two from the above list
Logarithmic and Exponential Function
(a)
The Exponential Function
A brief introduction is required at this stage.
Pupils should be able to sketch the graph of a x
(a  0) .
(Note: if a > 1 then a growth function, if a < 1 a decay function)
Best to start with 2 x , 3x , 10 x .
No need to introduce e x here; this will be done in Unit 3.
x
1
1
Pupils should know that (e.g.)    x  2  x .
2
2
y
x
2
2x
1
O
x
Pupils should know that a x  0 for positive a and x  R .
Pupils should know that the graph of an exponential function intersects the y-axis at the
point (0,1).
Links should be made with the previous section eg y = 2x + 1
(b)
Introducing the Log Function
The log function may be introduced as the inverse of the exponential function.
For example,
23  8  log 2 8  3
26  64  log 2 64  6
24  161  log 2 161  4
Pupils should get a great deal of practice at these, with different number bases.
From the above relationships it can be seen that, for example,
(3,8) lies on the graph of 2 x
(8,3) lies on the graph of log 2 x .
x
And in general, (a,b) on graph of 2
(b,a) on graph of log 2 x .
Thus the graph of log 2 x is got by reflecting the graph of 2 x in the straight line with
equation y  x .
Most candidates will need a great deal of practice with the above ideas.
y
2x
(a,b)
log 2 x
(0,1)
O
(b,a)
(1,0)
x
yx
Reinforce log a 1  0 and log a a  1 .
Pupils will need much practice with basic log graphs before moving onto typical exam
type questions.
Example:
The function f is of the form f ( x)  log b ( x  a) .
The graph of y = f(x) is also shown in the diagram.
a) Write down the values of a and b.
(2)
b) State the domain of f.
(1)



6.
1
2
3
a=4
b=5
domain x > a
Some Revision of “C” Work
(a)
cos 2   sin 2   1 and tan  
Identities
sin 
cos 
Use reference triangles to prove these:
1

sin 
cos
Apply P.T. to this triangle to obtain cos 2   sin 2   1.
Also, by definition of the tangent ratio, tan  
sin 
.
cos 
Candidates must be able to rearrange the formula c 2  s 2  1 to get c 2  1  s 2 and
s2  1 c2.
(b)
Sine Rule/Cosine Rule/Area of Triangle
The time spent on these will depend on the ability level and knowledge of the class.
(c)
Graphs
These are considered in the chapter on waves, but it is worthwhile to do some graphical
work at this stage.
7.
Exact-Values
Use standard reference triangles, viz:
1
2
3
2
45
60 
1
1
These triangles are got by “halving” the unit square along a diagonal, and by “halving” an
equilateral triangle of side two units down its altitude.
Pupils should be given much practice at drawing these triangles and obtaining the values of the
sine, cosine and tangent ratios of 30  , 45 and 60 .
For multiples of 90  , the graphs may be used to obtain exact-values of sine and cosine.
Candidates should be able to obtain the exact-values of trig ratios of related angles, e.g.
45
135 
sin 135   sin 45 
SA
TC
1
2
cos135   cos 45  
,
1
2
,
tan 135    tan 45   1.
330 
30 
1
sin 330    sin 30    ,
2
cos 330   cos 30  
3
,
2
tan 330    tan 30   
1
3
.
sin( 90   A)  cos A
Candidates should know that
cos(90   A)  sin A
sin( 180   A)  sin A
cos(180   A)   cos A
Sin(- A) = - Sin A
Cos(- A) = Cos A
The first two formulae may be got from, e.g.
B
r
A

( B  90  A).
x
sin A  r ; cos B  r .
 sin A  cos B, showing that sin A  cos(90   A).
y
y
The third and fourth can be got by considering
8.
y
S A
T C
and
Radians
To introduce this topic, consider the unit circle:
B
1

1
1
A
When the length of arc AB is equal to 1 unit then angle  is equal to 1 radian.
Now, since circumference of unit circle is equal to 2 units, we have
2 radians  360  ,
which gives the key relation, which must be known:
 radians  180 
Give pupils practice at converting (both ways) radians
degrees.
Give much practice at obtaining exact values of trig ratios, e.g. sin

, cos
2
7
, tan
, etc.
3
4
4
7
, we use proportion, CAST, a diagram and a reference triangle to get:
To obtain, e.g., tan
4
7
tan
 tan 315    tan 45   1.
4
Give pupils practice at obtaining decimal approximations using calculator, e.g. sin 1.2  0.932 .
9.
Equations
(a)
Revision of “C” Level Work
Candidates should be proficient at equations of the type
3 cos x   1  0
(0  x  360)
2 sin 2 x   1
(0  x  360)
Working for the second equation may be set out as follows:
1
2
1
sin x  
2
1
sin x  
2
sin 2 x  
sin x  
1
2
(0  x  360) S
A
T
C
x  45, 135
or 
or 
1
2
1
2
sin x   
1
2
x  225, 315
(0  x  360) S
A
T
C
And the solution set is {45, 135, 225, 315}
N.B. some people will like to draw sketches of the angles beside the S A diagrams.
TC
For example, to obtain the first two solutions to the above equation the sketches would be
For weaker candidates these sketches are very helpful.
(b)
Equations Involving a Single Ratio
In theory, these equations could come up at “C” level, but in practice they don’t nowadays.
However, our “C” level pupils should have had extensive practice at these equations in S4.
Examples of the above are:
1
(0  x  180)
2
1
(ii ) tan 2 x 
(0  x  2 )
3
(iii ) cos 2 2 x   1 (0  x  180)
(i ) sin 3 x  
Working (for(i)) may be set out as follows:
1
(0  3 x  540)
2
3x  30, 150, 390, 510
x  10, 50, 130, 170
sin 3 x  
10.
S A
T C
Preliminaries/Revision of S4 Work
a)
(i)
(ii)
(iii)
(iv)
(v)
Pupils must be familiar with degreee/radian measure.
Pupils must be familiar with basic sine/cosine graphs, both in degrees and radians.
Pupils must be able to use reference triangles to obtain exact values of ratios of simple
angles.
Pupils must be able to obtain exact values of ratios using CAST diagram and related
1
3
angles. e.g. tan 210   tan 30  
;
cos 210    cos 30   
. And so on.
2
3
Pupil must be familiar with graphs of sines/cosines of multiple angles, i.e. graphs of
functions such as f ( x)  3 cos 2 x  .
Abler pupils should be proficient in these, but weaker pupils might need some revision.
b)
Sketching Graphs
Given sufficient practice, most candidates should become highly proficient here.
The approach is best illustrated by example:
Sketch the graph of y  10 sin( 6 x  30) 
This is best done in the following steps:
(0  x  60).
(i)
(ii)
Write down maximum and minimum values. (10 and –10 here.)
Draw a set of axes on which to draw the graph:
y
10
O
15
30
45
60
x
-10
(iii)
Draw “basic” sine/cosine graph. (sine here):
(iv)
Now find maximum/minimum turning points and the intersection points with the axes.
When doing this, make use of the basic sine graph, above.
Also, as each point is obtained, mark it on the diagram that has been drawn in (ii), above.
For a maximum, 6x  30  90
x  20
Hence the max T.P. has coords (20,10).
For a minimum, 6x  30  270
x  50
Hence the min T.P. has coords (50,-10).
On the x axis, 6 x  30  0, 180, 360
x  5, 35, 65
Hence the points on x-axis are (5,0), (35,0).
[Note that (65,0) is outside the range here.]
Now obtain the y-intercept: On y axis, x  0  y  10 sin( 30)   5 .
Hence the point of intersection with the y axis is (0,-5).
(v)
Now, having been plotting the points as they have been found, “join the dots” to obtain a
fully annotated graph.
Notes
(a)
Consider at first only a single cycle of the graph. Once achieved, go on to consider
several cycles.
(b)
Do initial examples in degree measure. Once achieved, do examples in radian measure
also.
(c)
Find and plot the y-intercept. Mark schemes for old papers indicate that a mark is awarded
for this.
(d)
Graphs must be annotated, as indicated, above.
(e)
It is important to consider periods of graphs. For example, in radians, we have:
sin x  period 2
2
sin 2 x  period

2
2
sin 3 x  period
3
2
sin x  period
2

(f)
c)
Do examples that include vertical translations, such as y  100  50 sin( 2t  60)  , for
0  t  180.
The approach is to sketch the graph of y  50 sin( 2t  60)  first, then apply the vertical
translation.
Waves on the Graphic Calculator (for more able pupils at this stage)
The graphic calculator is an ideal resource for introducing the Wave Function.
Get the class to obtain the graphs of functions such as:
y  3 sin x   4 cos x  ,
y  5 cos x   12 sin x 
Help will be given so that the appropriate scale is chosen.
The pupils should be encouraged to make conjectures about the answers to the following
questions:
(i)
Max/min values?
(ii)
Period of graph?
(iii) Is the shape of the graph familiar?
The conclusion is that we should try to write functions such as the above ones as single sine or
cosine functions.
References/Resources
(1)
(2)
H.H.M. pages 22-51
MIA p19 – 57