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END BEHAVIOR & SYMMETRY
Objective: describe the end behavior of a function; determine whether a function is “even, odd, or neither”
How do the exponents of a polynomial change the shape of its graph?
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Symmetry
EVEN FUNCTIONS
Symmetric about the y-axis
f(x) = f(-x)
All exponents are even numbers
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ODD FUNCTIONS
Symmetric about the origin
f(-x) = -f(x)
All exponents are odd numbers
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Example 1 – “Even”
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Example 2 – “Odd”
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Example 3 – “Even”
f(x) = 2x4 – 5x2
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Example 4 – “Odd”
f(x) = 2x3 – 5x
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Example 5 – “Even”
X Y-2 7-1 50 31 52 7
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Example 6 – “Odd”
X Y-2 -7-1 -50 31 52 7
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Examples of “neither”
Ex 1) f(x) = x3 + 5 Ex 2)
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Even, Odd, or Neither
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Even, Odd, or Neither
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Even, Odd, or Neither
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Even, Odd, or Neither
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Even, Odd, or Neither
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Even, Odd, or Neither
X Y
-2 1
-1 5
0 8
1 5
2 1
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Even, Odd, or Neither
X Y
-2 4
-1 8
0 12
1 16
2 20
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Even, Odd, or Neither
X Y
-2 1
-1 5
0 8
1 -5
2 -1
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Even, Odd, or Neither
f(x) = 3x4 – x2
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Even, Odd, or Neither
f(x) = -x6 + 5x2
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Even, Odd, or Neither
f(x) = -2x3 + x
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Even, Odd, or Neither
f(x) = 4x5 + 2x3 + x
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Even, Odd, or Neither
f(x) = 4x2 + 2x
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Even, Odd, or Neither
f(x) = 4x2 + 2
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End Behavior
The direction the graph is going on
the “left end” and “right end”
What does “y” do as “x” gets really BIG?
What does “y” do as “x” gets really SMALL?
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UP UP(∞) (∞)
Down Down(-∞) (-∞)
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UP(∞)Down(-∞)
UP(∞)Down(-∞)
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Example 1 – describe the end behavior of the function
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Example 2 – describe the end behavior of the function
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Example 3 – describe the end behavior of the function
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Example 4 – describe the end behavior of the function