Higher Maths
Graphs & Functions
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Graphs & Functions
Higher
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Graphs & Functions
Higher
The diagram shows the graph of a function f.
f has a minimum turning point at (0, -3) and a
point of inflexion at (-4, 2).
a) sketch the graph of y = f(-x).
y = 2f(-x)
b) On the same diagram, sketch the graph of y = 2f(-x)
a)
y
Reflect across the y axis
y = f(-x)
4
2
-1
b)
3
Now scale by 2 in the y direction
4
x
-3
-6
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Graphs & Functions
Higher
The diagram shows a sketch of part of
the graph of a trigonometric function
whose equation is of the form y  a sin(bx)  c
Determine the values of a, b and c
a is the amplitude:
1
2a
a=4
b is the number of waves in 2
1 in 
2 in 2 
b=2
c is where the wave is centred vertically
c=1
Hint
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Graphs & Functions
Functions f ( x) 
Higher
1
and g ( x)  2 x  3 are defined on suitable domains.
x4
a)
Find an expression for h(x) where h(x) = f(g(x)).
b)
Write down any restrictions on the domain of h.
a)
f ( g ( x))  f (2 x  3)
b)
2 x 1  0
 x

1
2x  3  4
 h( x ) 
1
2x 1
1
2
Hint
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Graphs & Functions
Higher
2
in the form ( x  a)  b
a) Express f ( x)  x  4 x  5
b) On the same diagram sketch
2
c)
i)
the graph of
ii)
the graph of
(2, 1)
y  f ( x)
y=f(x)
y  10  f ( x)
b)
5
Find the range of values of x for
which
10  f ( x)
is positive
y= 10 - f(x)
(2, 1)
a)
( x  2)2  4  5  ( x  2)2  4  5
(2, -1)
 ( x  2)2  1
c)
Solve:
10  ( x  2)2  1  0
 ( x  2)2  9
 ( x  2)  3
y= -f(x)
 x  1 or 5
Hint
10 - f(x) is positive for -1 < x < 5
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Graphs & Functions
Higher
The graph of a function f intersects the x-axis at (–a, 0)
and (e, 0) as shown.
There is a point of inflexion at (0, b) and a maximum turning
point at (c, d).
Sketch the graph of the derived function f 
m is +
m is +
m is -
f(x)
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Graphs & Functions
Higher
Functions f and g are defined on suitable domains by
f ( x)  sin( x) and g ( x )  2 x
a)
Find expressions for:
i)
f ( g ( x))
ii) g ( f ( x))
b)
Solve 2 f ( g ( x))  g ( f ( x)) for 0  x  360
a)
f ( g ( x))  f (2 x)  sin 2x
b)
2sin 2x  2sin x
 sin 2 x  sin x  0
 2sin x cos x  sin x  0
 sin x  0
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or
g ( f ( x))  g (sin x)  2sin x
cos x 
1
2
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 sin x(2 cos x  1)  0
x  0, 180, 360
x  60, 300
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Graphs & Functions
Higher
The diagram shows the graphs of two quadratic
functions y  f ( x) and y  g ( x)
Both graphs have a minimum turning point at (3, 2).
Sketch the graph of y  f ( x)
and on the same diagram
sketch the graph of y  g ( x )
y=f(x)
y=g(x)
Hint
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Graphs & Functions
Higher

Functions f ( x)  sin x, g ( x)  cos x and h( x)  x  4
are defined on a suitable set of real numbers.
a) Find expressions for
g (h( x))
i)
ii)
f ( h( x ))
b) i)
Show that
ii)
f (h( x)) 
1
2
sin x 
1
2
cos x
Find a similar expression for g (h( x))
and hence solve the equation f (h( x))  g (h( x))  1 for 0  x  2

a)
f (h( x))  f ( x  )
b)
sin( x  )  sin x cos
4


 sin( x  )
g (h( x))  cos( x  )
4


4
4
 sin

4
cos x
4
Now use exact values
Repeat for ii)
equation reduces to
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2
sin x  1
2
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sin x 
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2
1

2
2
x
 3
4
,
4
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Graphs & Functions
Higher
A sketch of the graph of y = f(x) where f ( x)  x3  6 x 2  9 x is shown.
The graph has a maximum at A and a minimum at B(3, 0)
a) Find the co-ordinates of the turning point at A.
b) Hence, sketch the graph of g ( x)  f ( x  2)  4
Indicate the co-ordinates of the turning points. There is no need to
calculate the co-ordinates of the points of intersection with the axes.
c) Write down the range of values of k for which g(x) = k has 3 real roots.
a)
Differentiate
f ( x)  3x 2  12 x  9
when x = 1
y4
t.p. at A is:
for SP, f(x) = 0
x  1 or x  3
(1, 4)
moved 2 units to the left, and 4 units up
b)
Graph is
c)
For 3 real roots, line y = k has to cut graph at 3 points
t.p.’s are:
(3, 0)  (1, 4)
(1, 4)  (1, 8)
from the graph, k  4
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Graphs & Functions
f ( x)  3  x
a) Find
3
x
g ( x)  ,
and
3
3 x
, x3
find
p( x)  f ( g ( x))  f
p(q( x)) 
b)

x0
p( x) where p( x)  f ( g ( x))
b) If q ( x) 
a)
Higher
p  3 
 3 x 

3
 
 x

 3

3
1
 3 x 
3
3 x
 9  3(3  x)  3  x


 3 x  3
in its simplest form.
p ( q ( x ))

3
3
x

3 x 3
x
3( x 1)
x

 9
 3

 3 
 3 x
 3 x
3x 3  x

3 x
3

x
Hint
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Graphs & Functions
Higher
Part of the graph of y  f ( x) is shown in the diagram.
On separate diagrams sketch the graph of
a) y  f ( x  1)
b) y  2 f ( x)
Indicate on each graph the images of O, A, B, C, and D.
a)
b)
graph moves to the left 1 unit
graph is reflected in the x axis
graph is then scaled 2 units in the y direction
Hint
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Graphs & Functions
Higher
Functions f and g are defined on the set of real numbers
by f ( x)  x  1 and g ( x)  x 2
a) Find formulae for
i)
ii) g ( f ( x))
f ( g ( x))
b) The function h is defined by h( x)  f ( g ( x))  g ( f ( x))
Show that h( x)  2 x 2  2 x and sketch the graph of h.
c) Find the area enclosed between this graph and the x-axis.
a)
f ( g ( x))  f ( x 2 )  x 2  1
b)
h( x)  x 2  1   x  1
c)
2
g ( f ( x))  g ( x  1)   x  1
h( x)  x 2  1  x 2  2 x  1
Graph cuts x axis at 0 and 1
Area
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
1
3
unit
Now evaluate

1
0
2
 2 x2  2 x
2 x2  2 x dx
Hint
2
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Graphs & Functions
Higher
The functions f and g are defined on a suitable domain by
f ( x)  x 2  1 and g ( x)  x 2  2
a) Find an expression for f ( g ( x))
b) Factorise f ( g ( x))
a)
f ( g ( x))  f ( x  2)   x  2   1
b)
Difference of 2 squares
2
Simplify
2
2

 x
2
  x
 2  1
2

 2 1
  x 2  3 x 2  1
Hint
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Graphs & Functions
Higher
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Graphs & Functions
Higher
Table of exact values
sin
cos
tan
Previous
30°
45°
60°

6
1
2

4

3
1
2
1
2
3
2
3
2
1
3
1
1
2
3