Warm up
Find the average rate of change for the following
functions on the interval -3 < x < 4.
1. f(x) = 2x - 5
2. f(x) = 3x + 2
More Characteristics of Functions:
Recall:
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Extrema
Domain
Range
Asymptote
x-intercept
y-intercept
Negative function
Positive function
• Even/Odd/Neither
• End Behavior
Even and Odd Functions
and
End Behavior
EQ: How can a function have symmetry?
EQ: How can I determine what a function
does as it goes infinitely down the x-axis in
either direction?
Even Function:
Symmetric about y-axis
f(x) = f(-x)
If polynomial, all exponents are even
Odd Function:
Symmetric about origin
f(-x) = -f(x)
If polynomial, all exponents are odd
Even and Odd Functions
A. From graphs:
1) even functions are symmetric to the y­axis
2) odd functions have rotational symmetry about the origin
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 1
f(x) = x2 ­ 7
Ex. 2
f(x) = 3x3 ­ 7x
Ex. 3
f(x) = 2x2 + 4x ­ 8
Ex. 4
f(x) = x + 5
End Behavior
What does the function do as we travel up and down the xaxis forever in both directions?
End behavior on right:
As x ⇒ ∞,
y ⇒ ____
End behavior on left:
As x ⇒ -∞,
y ⇒ ____
Example 1
Type of Function?
Max/Min? Domain:
Range:
Asymptote:
x­intercept:
y­intercept:
When is the function negative:
When is the function positive:
Interval of Increase:
Interval of Decrease:
Constant Interval: Even/Odd/Neither?
End Behavior: Example 2
Type of Function?
Max/Min? Domain:
Range:
Asymptote:
x­intercept:
y­intercept:
When is the function negative:
When is the function positive:
Interval of Increase:
Interval of Decrease:
Constant Interval: Even/Odd/Neither?
End Behavior: Example 3
Type of Function?
Max/Min? Domain:
Range:
Asymptote:
x­intercept:
y­intercept:
When is the function negative:
When is the function positive:
Interval of Increase:
Interval of Decrease:
Constant Interval: Even/Odd/Neither?
End Behavior: Example 4
Type of Function?
Max/Min? Domain:
Range:
Asymptote:
x­intercept:
y­intercept:
When is the function negative:
When is the function positive:
Interval of Increase:
Interval of Decrease:
Constant Interval: Even/Odd/Neither?
End Behavior: