For every number x in its domain, the number –x is also in the domain and f(x) = f(– x).

For every number x in its domain, the number –x is also in the domain and f(– x). = – f(x).

A function is Even if and only if its graph is symmetric with respect to the y-axis.

A function is Odd if and only if its graph is symmetric with respect to the origin.

f(– x) = –(– x)4 + 3(– x )2 – 5 f(– x) = –x4 + 3 x2 – 5 f(– x) = f(x)

f(x) is an Even function

g(– x) = 5(– x)3 – (– x ) g(– x) = – 5x3 + x g(– x) = – [5x3 – x] g(– x) = – g(x)

g(x) is an Odd function

h(– x) = (– x)3 – 1 h(– x) = – x3 – 1

xhxxh

xxh

1

13

3

h(x) is neither function

Even functions will always have even powers in polynomials. Constants are with the even powers!

10 x Odd functions will always have odd powers in polynomials.

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). Working defn. Reading a graph from left to right, as

x-coord. are increasing the y- coord. are increasing. A function f is decreasing on an open

interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2). Working defn. Reading a graph from left to right, as

x-coord. are increasing the y- coord. are decreasing. A function f is constant on an open

interval I if, for any choice of x in I, the values of f(x) are equal. Working defn. Reading a graph from left to right, as

x-coord. are increasing the y- coord. are constant. Forms a horizontal segment.

MUST BE OPEN INTERVALS BY DEFINITION! X’s ONLY!

-4 < x < -1 or (-4, -1)

-6 < x < -4 or (-6, -4)-1 < x < 0 or (-1, 0)3 < x < 6 or (3, 6)

0 < x < 3 or (0, 3)

Y’s ONLY!

A function f has a local minimum at c if there is an open interval I containing c so that, for all x in I, f(x) > f(c). We call f(c) a local min.

I

(-4, -4 )

I

( 0, 4 )

Relative Minimums -4 , 4

A function f has a local maximum at c if there is an open interval I containing c so that, for all x in I, f(x) < f(c). We call f(c) a local max.

I(-1, 6 )

I

( 3, 4 )

Relative Maximums 6 , 4 A function f is defined on some interval I. If the is a number u in I for which f(x) < f(u) for all x in I, then f(u) is the absolute max.

Absolute Minimum -5

A function f is defined on some interval I. If the is a number u in I for which f(x) > f(u) for all x in I, then f(u) is the absolute min.

Absolute Maximum 6

I

(6, -5 )

I

Relative Minimum

Relative Maximum -1.53197311.531973

Find the INTERVALS for where the graph is increasing and decreasing.

Decreasing

Increasing

Rounding errors with the x-coordinates.We will always write four decimal places.

(– 0.8165, 0.8165)

(– 2, – 0.8165) ( 0.8165, 2 )

ALWAYS OPEN INTERVALS ( __ , __ )

SLOPE12

12

xxyym

53 2 xxfy

1st pt. (1, 8) 2nd pt. (3, 32)

12224

13832

m

5126 3 xxxfy1st pt. (-1, 11) 2nd pt. (1, -1)

6212

11111

m