1.5
Analyzing
Graphs of Functions
(2,4)
Find:
a. the domain
[-1,4)
b. the range
[-5,4]
c. f(-1) = -5
(4,0)
(-1,-5)
d. f(2) = 4
Vertical Line Test for Functions
Do the graphs represent y as a function of x?
no
yes
yes
Increasing and Decreasing Functions
-2 -1
1 2 3 4 5
1. The function is decreasing on the interval (-2,0).
2. The function is constant on the interval (0,3).
3. The function is increasing on the interval (3,5).
A function f is increasing on an interval if, for any
x1 and x2 in the interval,
x1 < x 2
implies
f(x1) < f(x2)
A function f is decreasing on an interval if, for any
x1 and x2 in the interval,
x1 < x 2
implies
f(x1) > f(x2)
A function f is constant on an interval if, for any
x1 and x2 in the interval,
f(x1) = f(x2)
go to page 57
Tests for Even and Odd Functions
A function is y = f(x) is even if, for each x in the
domain of f,
f(-x) = f(x)
An even function is symmetric about the y-axis.
A function is y = f(x) is odd if, for each x in the
domain of f,
f(-x) = -f(x)
An odd function is symmetric about the origin.
Ex. g(x) = x3 - x
g(-x) = (-x)3 – (-x) = -x3 + x = -(x3 – x)
Therefore, g(x) is odd because f(-x) = -f(x)
Ex. h(x) = x2 + 1
h(-x) = (-x)2 + 1 = x2 + 1
h(x) is even because f(-x) = f(x)
Summary of Graphs of Common Functions
f(x) = c
y=x
y x
y x
y=x3
y = x2
The Greatest Integer Function
y  [[ x]]
x y
0
.2
.5
.8
.99
0
0
0
0
0
-4 -3
x y
1.1
1.4
1.7
1.8
1.99
1
1
1
1
1
-2
-1
0
1
2
1
3
2
4 5
3
Average Rate of Change of a Function
Find the average rates of change of f(x) = x3 - 3x
from x1 = -2 to x2 = 0.
The average rate of change of f from x1 to x2
y f (x 2 )  f (x1)


x
x 2  x1
f (x 2 )  f (x1 ) f (0)  f (2)
0  (2)


1
x 2  
x1
0  2
2

Finding Average Speed
The average speed of s(t) from t1 to t2 is
s s(t 2 )  s(t1 )


t
t 2  t1
Ex. The distance (in feet) a moving car is from a
stoplight is given by the function s(t) = 20t3/2, where
t is thetime (in seconds). Find the average speed of
the car from t1 = 0 to t2 = 4 seconds.
s(t 2 )  s(t1 ) s(4)  s(0) 160 0



 40 ft /sec
t 2  t1
40
4
What’s the average
speed of the car from  540160  76ft /sec

5
4 to 9 seconds?