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The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.
Example Find where the graph is increasing? Where is it decreasing? Where is it constant?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
Example
Find where the graph is increasing? Where is it decreasing? Where is it constant?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
Example
Find where the graph is increasing? Where is it decreasing? Where is it constant?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
Example Where are the relative minimums? Where are the relative maximums?
Why are the maximums and minimums called relative or local?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
A graph is symmetric with respect to the
y-axis if, for every point (x,y) on the graph,
the point (-x,y) is also on the graph. All even
functions have graphs with this kind of symmetry.
A graph is symmetric with respect to the origin if,
for every point (x,y) on the graph, the point (-x,-y)
is also on the graph. Observe that the first- and third-
quadrant portions of odd functions are reflections of
one another with respect to the origin. Notice that f(x)
and f(-x) have opposite signs, so that f(-x)=-f(x). All
odd functions have graphs with origin symmetry.
Example
Is this an even or odd function?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
Example
Is this an even or odd function?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
Example
Is this an even or odd function?
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
A function that is defined by two or more equations over
a specified domain is called a piecewise function. Many
cellular phone plans can be represented with piecewise
functions. See the piecewise function below:
A cellular phone company offers the following plan:
$20 per month buys 60 minutes
Additional time costs $0.40 per minute.
( )
C t =20 if 0 t 60
20 0.40( 60) if t>60t
≤ ≤+ −
Example
Find and interpret each of the following.
( )
C t =20 if 0 t 60
20 0.40( 60) if t>60t
≤ ≤+ −
( )( )( )
45
60
90
C
C
C
Example
Graph the following piecewise function.
( )
f x =3 if - x 3
2 3 if x>3x
∞ ≤ ≤−
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
2
2
2 2
f(x+h)-f(x) for f(x)=x 2 5
h f(x+h)
f(x+h)=(x+h) 2(x+h)-5
x 2 2 2 5
Find x
First find
hx h x h
− −
−+ + − − −
Continued on the next slide.
( )
2
2 2 2
2 2 2
f(x+h)-f(x) for f(x)=x 2 5
h f(x+h) from the previous slide
f(x+h)-f(x) find
h
x 2 2 2 5 x 2 5f(x+h)-f(x)
h
x 2 2 2 5 2 5
2
Find x
Use
Second
hx h x h x
h
hx h x h x x
h
− −
+ + − − − − − −=
+ + − − − − + +
( )
2 2
2 2
2x+h-2
hx h h
hh x h
h
+ −
+ −
Example
Find and simplify the expressions if
f(x+h)-f(x)Find f(x+h) Find , h 0
h≠
2( ) 2 1f x x x= − +
Some piecewise functions are called step functions
because their graphs form discontinuous steps. One such
function is called the greatest integer function, symbolized
by int(x) or [x], where
int(x)= the greatest integer that is less than or equal to x.
For example,
int(1)=1, int(1.3)=1, int(1.5)=1, int(1.9)=1
int(2)=2, int(2.3)=2, int(2.5)=2, int(2.9)=2
Example
The USPS charges $ .42 for letters 1 oz. or less. For letters
2 oz. or less they charge $ .59, and 3 oz. or less, they charge $ . 76.
Graph this function and then find the following charges.
a. The charge for a letter that weights 1.5 oz.
b. The charge for a letter that weights 2.3 oz.
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
$1.00
$ .75
$ .50
$ .25
(a)
(b)
(c)
(d)
There is a relative minimum at x=?
4
3
2
0
−
−−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y