Transcript 2.2.4 even odd end behavior notes

2.2.4 Even & Odd Functions
Essential Question:
How do you identify even and odd functions?
Main Idea
• You have already learned about all types of
functions.
• We are going to extend our learning of
function families to understand behavior of
certain function types.
•
What are the types of functions?
•
Predict what an even function and odd
function look like.
Symmetric about the y axis
FUNCTIONS
Symmetric about the origin
Even functions have y-axis Symmetry
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7
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5
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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.
Even Functions
A function, f(x), is an even function if f(x) = f(-x) , for all x
in its domain.
An even function has the important property line symmetry
with the y-axis as it axis. This allows us to plot the right handside of the y-axis, then to complete the sketch, draw the mirror
image on the left-side of the y-axis.
Y-axis
Y-axis
y = f(x)
y = f(x)
Right-hand
side of f(x)
Y-axis is
an axis of
symmetry
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
Odd functions have origin Symmetry
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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.
Odd Functions
A function, f(x), is an odd function if f(-x) = –f(x) , for all x
in its domain.
An odd function has the important property point symmetry,
180° rotation about the origin. This allows us to plot the topside of the x-axis, then to complete the sketch, draw the 180°
rotated mirror image on the bottom-side of the x-axis.
Y-axis
Y-axis
y = f(x)
y = f(x)
Top-side of
y = f(x)
180° rotation
Bottom-side of y = f(x)
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
x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
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5
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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
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
Methods for Identifying
1) Plug in the value (-x)
2) Plug in a real number value
3) Use Desmos to see the type of symmetry
Exercise
Show whether the following functions are EVEN, ODD or NEITHRER.
(1)
f (x)  4 x 1
Answer: neither
 (2)
h(x)  x 2  5
Answer: even
(3)
x2  5
g(x) 
x
Answer: odd
 (4)
f (x)  3x
Answer: even
 (5)
h(x)  x 3  2x
Answer: odd


End Behavior
• End behavior: where a graph is heading (what
is happening on the ends)
– Helps with characteristics such as domain and
range
– Works for most function families
– Mostly for polynomial functions to compare
degrees
End Behavior
• 3 possibilities:
– Output approaches a certain value
– Output continues without bound (infinity)
– Output may oscillate (periodic) and fail to approach a
particular number
Notation
• In unit 2, we used arrows to indicate the end
behavior. We will develop the notation for
pre-calculus.
• (
,
)
Notation
• Read as “as f(x) approaches negative infinity, x approaches
negative infinity and as f(x) approaches positive infinity, x
approaches positive infinity”.
x  , f( x)  
x  , f( x)  
Even Functions
• Based on what you know about even
functions, make a prediction about
the pattern or basic rule for their end
behavior.
Even Functions
• Due to the line symmetry of an even function about the y-axis,
the end behavior will be the same on both sides (either both
increasing or decreasing).
x  , f( x)  
x  , f( x)  
x  , f( x)  
x  , f( x)  
Odd Functions
• Based on what you know about odd
functions, make a prediction about
the pattern or basic rule for their end
behavior.
Odd Functions
• Due to the line symmetry of an odd function about the origin,
the end behavior will be the opposite on each end.
x  , f( x)  
x  , f( x)  
x  , f( x)  
x  , f( x)  
Other Functions
• Based on what you know about
other functions, what can you
predict or conclude about their end
behavior?
Other Functions
• Since other functions don’t have even or odd symmetry, the
end behavior depends on the function (3 possibilities).
x  , f(x)  0
x  , f(x)  
x  , f( x)  2
x  , f(x)  2
Polynomial Functions
• Using what you know about a degree and leading coefficient
of a polynomial, it can help you know end behavior without a
graph.
Summary
• Write a paragraph, with detailed, complete
sentences.
• How do you identify even and odd functions?
• Write 3-5 study questions in the left column
to correspond with the notes.