Symmetric about the y axis
FUNCTIONS
Symmetric about the origin
Even functions have y-axis Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an even function, for every point (x, y) on
the graph, the point (-x, y) is also on the graph.
f(-x) = f(x)
Odd functions have origin Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
f(-x) = - f(x)
x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
A function is even if f( -x) = f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
original function back again it is even.
f x   5 x  2 x  1
4
2
Is this function even?
YES
f  x   5( x)  2( x)  1  5x  2 x  1
4
2
4
2
f x   2 x  x Is this function even?
NO
3
3
f  x   2( x)  ( x)  2 x  x
3
A function is odd if f( -x) = - f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
negative of the function back again (all terms change signs)
it is odd.
f x   5 x  2 x  1
4
2
Is this function odd?
NO
f  x   5( x)  2( x)  1  5x  2 x  1
4
2
4
2
f x   2 x  x Is this function odd? YES
3
3
f  x   2( x)  ( x)  2 x  x
3
If a function is not even or odd we just say neither
(meaning neither even nor odd)
Determine if the following functions are even, odd or
neither.
Not the original and all
3
terms didn’t change
signs, so NEITHER.
f x   5 x  1
f  x   5 x   1  5 x  1
3
3
f x   3x  x  2
4
2
Got f(x) back so
EVEN.
f  x   3( x)  ( x)  2  3x  x  2
4
2
4
2
Increasing and Decreasing Functions
8
Increasing/Decreasing Functions
A function f is increasing on an interval if as x
increases, then f(x) increases.
A function f is decreasing on an interval if as x
increases, then f(x) decreases.
f(x) is decreasing
in the interval
(,1.5).
vertex
(1.5,-2)
y  x 2  3x
f(x) is increasing
in the interval
(1.5, ) .
Increasing, Decreasing,
Constant Intervals
A function f is constant on an interval if as
x increases, then f(x) remains the same.
Find the interval(s)
over which the
interval is increasing,
decreasing and
constant?
3
y  x  3x
Answer Now
Increasing, Decreasing,
Constant Intervals
Find the interval(s) over
which the interval is
increasing, decreasing
and constant?
Answer Now