2.
Symmetry
Let’s
start with a simple exploration. Open the Pretty Excel pictures worksheet
below. Spend some time experimenting and reflecting on the relationship
between f(x,
y)
and g(x,
y) geometrically (i.e. what visual patterns are there?),
numerically (i.e. what patterns are there in the tables?) and algebraically
(how can you generalise these patterns?). Can you describe the relationships
in terms of reflections and symmetry?
A
function f is even if its graph is symmetric
with respect to the Y-axis. This criterion can be stated algebraically
as follows: f is even if f(-x)
= f(x)
for all x in the domain of
f. For example, if you evaluate f at 3 and at -3, then you
will get the same value if f is even.
A function f is odd if its graph is symmetric
with respect to the origin. This criterion can be stated algebraically
as follows: f is odd if f(-x)
= -f(x)
for all x in the domain of
f. For example, if you evaluate f at 3, you get the negative
of f(-3) when f is odd. |
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Let's
check these conditions. The two applets below show the graphs of y
= xn
vs. y
= (-x)n
and y
= xn
vs. y
= -xn.
Click
on the sliders to change the value of n (make sure that you also choose
negative values). Deduce from the graphs and check it numerically:
1. For which values of n is y
= xn
an even function?
2. For which values of n is y
= xn
an odd function?
3. For which values of n is y
= xn
and y
= -xn symmetrical in the X-axis?
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