1. Higher Maths
Graphs & Functions
Strategies
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2. Graphs & Functons The following questions are on
Graphs & Functions Higher
The following questions are on
Graphs & Functons
Non-calculator questions will be indicated
You will need a pencil, paper, ruler and rubber.
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3. a) b) 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).
b) On the same diagram, sketch the graph of y = 2f(-x) a)
Reflect across the y axis b)
Now scale by 2 in the y direction
Hint
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4. 2a a = 4 b = 2 c = 1 1 a is the amplitude: 1 in  2 in 2 
Graphs & Functions Higher
The diagram shows a sketch of part of
the graph of a trigonometric function
whose equation is of the form
Determine the values of a, b and c 2a 1
a = 4
1 in 
2 in 2 
a is the amplitude:
b = 2
b is the number of waves in 2
c = 1
c is where the wave is centred vertically
Hint
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5. Functions and are defined on suitable domains.
Graphs & Functions Higher
Functions and are defined on suitable domains.
a) Find an expression for h(x) where h(x) = f(g(x)).
b) Write down any restrictions on the domain of h. a) b)
Hint
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6. 10 - f(x) is positive for -1 < x < 5
Graphs & Functions Higher
a) Express in the form
b) On the same diagram sketch
i) the graph of
ii) the graph of
c) Find the range of values of x for
which is positive b) a) c)
Solve:
Hint
10 - f(x) is positive for < x < 5
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7. f(x) 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
m is +
m is +
m is -
Hint
f(x)
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8. a) b) Graphs & Functions Higher
Functions f and g are defined on suitable domains by
and
a) Find expressions for: i)
ii)
b) Solve a) b)
Hint
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9. y=f(x) y=g(x) Graphs & Functions Higher
The diagram shows the graphs of two quadratic
functions
Both graphs have a minimum turning point at (3, 2).
Sketch the graph of
and on the same diagram
sketch the graph of
y=f(x)
y=g(x)
Hint
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10. a) b) Graphs & Functions Higher Functions
are defined on a suitable set of real numbers.
a) Find expressions for
i) ii)
b) i) Show that
ii) Find a similar expression for
and hence solve the equation a) b)
Now use exact values
Repeat for ii)
Hint
equation reduces to
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11. a) b) c) from the graph, k  4 Graphs & Functions Higher Differentiate
A sketch of the graph of y = f(x) where 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
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
for SP, f(x) = 0
when x = 1
t.p. at A is: b)
Graph is
moved 2 units to the left, and 4 units up
t.p.’s are:
Hint c)
For 3 real roots, line y = k has to cut graph at 3 points
from the graph, k  4
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12. a) b) Graphs & Functions Higher a) Find
b) If find in its simplest form. a) b)
Hint
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13. graph moves to the left 1 unit
Graphs & Functions Higher
Part of the graph of is shown in the diagram.
On separate diagrams sketch the graph of
a) b)
Indicate on each graph the images of O, A, B, C, and D. a)
graph moves to the left 1 unit b)
graph is reflected in the x axis
graph is then scaled 2 units in the y direction
Hint
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14. a) b) c) Graphs & Functions Higher
Functions f and g are defined on the set of real numbers by
a) Find formulae for
i) ii)
b) The function h is defined by
Show that and sketch the graph of h.
c) Find the area enclosed between this graph and the x-axis. a) b) c)
Graph cuts x axis at 0 and 1
Now evaluate
Hint
Area
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15. a) b) Difference of 2 squares Simplify Graphs & Functions Higher
The functions f and g are defined on a suitable domain by
a) Find an expression for
b) Factorise a) b)
Difference of 2 squares
Simplify
Hint
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16. You have completed all 13 questions in this section
Graphs & Functions Higher
You have completed all 13 questions in this section
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17. 30° 45° 60° sin cos tan 1 Table of exact values
Graphs & Functions Higher
Table of exact values
30°
45°
60°
sin
cos
tan 1
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