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### Paper 1

FUNCTIONS
(a) Demonstrate the ability to work with various types of functions including those listed in the following Assessment Standard.
(b) Recognise relationships between variables in terms of numerical, graphical, verbal and symbolic representations and convert flexibly between these representations (tables, graphs, words and formulae).

Generate as many graphs as necessary, initially by means of point-by-point plotting, supported by available technology, to make and test conjectures about the effect of the parameters
k, p, a and q for functions including:
y = a(x + p)2 + q

y = sinkx
y = coskx
y = tankx
y = sin(x + p)
y = cos(x + p)
y = tan(x + p)

Identify characteristics as listed below and hence use applicable characteristics to sketch graphs of functions including those listed above:
(a) domain and range;
(b) intercepts with the axes;
(c) turning points, minima and maxima;
(d) asymptotes;
(e) shape and symmetry;
(f) periodicity and amplitude;
(g) average gradient (average rate of change);
(h) intervals on which the function increases/decreases;
(i) the discrete or continuous nature of the graph.

Clarification

• Use and interpret functional notation.
In the teaching process learners must be able to understand how f(x) has been transformed to generate
f(-x)
-f(x)
f(x+a)
f(x) + a
f(ax)
af(x)
• The above necessitates an understanding of how transformations and functions are integrated.
• Note:
Trigonometric functions are tested in Paper 1 with respect to the characteristics of the functions as outlined above.
• Not more than TWO variations will be examined on one function simultaneously.
For example:
Sketch the graphs of
y = sin 2x
and
y = 2cos(x + 30°)
on the same system of axes.