This learning activity looks at using simple functional relations to describe symmetry in the graphs of power and polynomial functions. The types of functions be considered for this task are those specified in the study design.
A function f is said to be an even function if f (x ) = f ( x ) or equivalently f ( x ) - f (x ) = 0 for all x in it domain. a. State the symmetry property this characterises, and give several examples of functions and their graphs that satisfy this functional relation.
A function f is said to be an odd function if f (x ) = f ( x ) or equivalently f ( x ) + f (x ) = 0 for all x in it domain. b. State the symmetry property this characterises, and give several examples of functions and their graphs that satisfy this functional relation. c. Give some examples of functions that are neither even nor odd, and show their graphs. d. Is there a function that is both even and odd?
Consider graphs of functions of the form non-zero integers.
over their natural domain, where p and q are a. Systematically vary p and q and determine whether the corresponding graphs indicate even or odd functions. b. Investigate the behaviour of the function around the origin, and indicate whether the graph of the function has a unique tangent at the origin or not.
Consider graphs of polynomial functions over domain R.
a. Express the function as the sum of an even function and an odd function. Show how the graph of f can be constructed from the graph of the even function and the graph of the odd function. b. Use a range of polynomial functions to explore what type of function each of the following combinations produces: even + even, even + odd, odd + odd. c. Show that the graph of every cubic polynomial function contains a point about which it has half-turn symmetry.
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VCAA

Mathematical Methods Unit 3
The following content from the areas of study is addressed through this learning activity.
Area of study
Functions and graphs
Content dot point
1, 2, 5
Algebra
Calculus
Probability and statistics -
2
3, 5
The following outcomes, key knowledge and key skills are addressed through this task.
2
3
Outcome
1
Key knowledge dot point
1, 4, 5, 7
1, 2, 3, 4
2, 4
Key skill dot point
1, 6, 11, 13
1, 2, 3, 5
2, 3, 4, 6, 9
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VCAA Page 2