end behavior & symmetry objective: describe the end behavior of a function; determine whether a...
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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?
Symmetry
EVEN FUNCTIONS
Symmetric about the y-axis
f(x) = f(-x)
All exponents are even numbers
ODD FUNCTIONS
Symmetric about the origin
f(-x) = -f(x)
All exponents are odd numbers
Example 1 – “Even”
Example 2 – “Odd”
Example 3 – “Even”
f(x) = 2x4 – 5x2
Example 4 – “Odd”
f(x) = 2x3 – 5x
Example 5 – “Even”
X Y-2 7-1 50 31 52 7
Example 6 – “Odd”
X Y-2 -7-1 -50 31 52 7
Examples of “neither”
Ex 1) f(x) = x3 + 5 Ex 2)
Even, Odd, or Neither
Even, Odd, or Neither
Even, Odd, or Neither
Even, Odd, or Neither
Even, Odd, or Neither
Even, Odd, or Neither
X Y
-2 1
-1 5
0 8
1 5
2 1
Even, Odd, or Neither
X Y
-2 4
-1 8
0 12
1 16
2 20
Even, Odd, or Neither
X Y
-2 1
-1 5
0 8
1 -5
2 -1
Even, Odd, or Neither
f(x) = 3x4 – x2
Even, Odd, or Neither
f(x) = -x6 + 5x2
Even, Odd, or Neither
f(x) = -2x3 + x
Even, Odd, or Neither
f(x) = 4x5 + 2x3 + x
Even, Odd, or Neither
f(x) = 4x2 + 2x
Even, Odd, or Neither
f(x) = 4x2 + 2
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?
UP UP(∞) (∞)
Down Down(-∞) (-∞)
UP(∞)Down(-∞)
UP(∞)Down(-∞)
Example 1 – describe the end behavior of the function
Example 2 – describe the end behavior of the function
Example 3 – describe the end behavior of the function
Example 4 – describe the end behavior of the function
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