Continuity and End Behavior
Section 3-5
Learning Target
• I can determine whether a
function is continuous or
discontinuous
• I can identify the end
behavior of functions
• I can determine whether a
function is increasing or
decreasing on an interval
Continuous Function-the graph of this function is smooth or
continuous curves. (Linear and Quadratic Function)
Discontinuous function- the graph of this function cannot be
trace without lifting your pencil (step wise and absolute value).
The chart below shows the different types of discontinuous
functions.
Function
1
y 2
x
Name
Infinite
Discontinuity
Graph
Characteristics
The graph becomes
greater and greater
as it approaches a
given x-value.
Function
Name
Graph
Characteristics
The graph stops at a
given value of the
domain and then
begins again at a
different range value
for the same value
of the domain.
x 1
f ( x) 
x 1
2
Jump
Discontinuity
 x 2  1 if x  0
f ( x)  
if x  0
x
Point
Discontinuity
When there is a value in
the domain for which
the function is
undefined, but the
pieces of the graph
match up. There is
a hole in the graph.
Continuity Test
A function is continuous at x = c if it satisfies the
following conditions:
1. the function is defined at c; or f(c) exists
2. the function approaches the same yvalue on the left and right sides of x = c
3. the y-value that the function approaches
from each side is f(c).
Example: Determine whether the function is continuous at the
given x-value.
x2  4
f ( x) 
; x  2
x2
Determine whether each function is continuous at the given x-value.
a. f(x) = x3 + 2x - 1; x = 2
x2 + 5x + 6
b. f(x) =
; x = -3
x+3
c.
 x2
if x  0
f ( x) = 
;x  0
x

2
if
x

0

Another tool that is used for analyzing functions is end behavior. End behavior
is just a way to describe what happens to the graph at the ends. In the table
below “leading coefficient” refers to the coefficient in the term of the polynomial
with the highest power of x. For instance, 3 is the leading coefficient in the
polynomial f ( x )  3x 4  2 x 3  4 x  7 .
Polynomial Properties
End Behavior
Leading coefficient is
positive; highest power
is even
As x gets very large, y
gets very large; as x
gets very small, y gets
very large.
Example
f ( x )  3x 2
f ( x )   as x  
f ( x )   as x  
f ( x )  3x 2
f ( x )   as x  
f ( x )   as x  
Leading coefficient is
negative; highest power
is even
As x gets very large, y
gets very small; as x
gets very small, y gets
very small.
Polynomial Properties
End Behavior
Example
f ( x)  2 x3
f ( x )   as x  
f ( x )   as x  
Leading coefficient is
positive; highest power is
odd
As x gets very large, y
gets very large; as x gets
very small, y gets very
small.
f ( x )  2 x 3
f ( x )   as x  
f ( x )   as x  
Leading coefficient is
negative; highest power
is odd
As x gets very large, y
gets very small; as x gets
very small, y gets very
large.
Another characteristic of functions used for analysis is the
monotonicity of the function. This means that on an
interval, the function is increasing or decreasing on that
particular interval. Whether a graph is increasing or
decreasing is always judged by viewing a graph from left to
right.
Function
Increasing
Decreasing
Constant
Characteristics
A function f is increasing on an interval
if and only if for every a and b
contained in the interval, f(a)<f(b)
whenever a<b
A function f is increasing on an interval
if and only if for every a and b
contained in the interval, f(a)>f(b)
whenever a<b
A function f is increasing on an interval
if and only if for every a and b
contained in the interval, f(a)=f(b)
whenever a<b
Example: Describe the end behavior of the function. Graph the function and
determine the intervals on which the function is increasing and the intervals on
which the function is decreasing. f ( x )  5x 3  x 2  x  4
The leading coefficient is positive and the highest power is odd. Therefore, as
x  , y   and as x  , y   .
Use your graphing calculator to find the relative max and relative min to find your
boundaries for the increasing and decreasing intervals.
Describe the end behavior of f(x) = 3x2 + 4x + 1 and g(x) = -2x2 + 2x.
Use your calculator to create a table of function values so you can investigate the end behavior of
the y-values.
f(x) = 3x2 + 4x + 1
f(x)
x
-10,000
299,960,001
-1000
2,996,001
-100
29,601
-10
261
0
1
10
341
100
30,401
1000
3,004,001
10,000
300,040,001
g(x) = -2x2 + 2x
g(x)
x
-10,000
-200,020,000
-1000
-2,002,000
-100
-20,200
-10
-220
0
0
10
-180
100
-19,800
1000
-1,998,000
10,000
-199,980,000
Notice that both polynomial functions have y-values that become very large in absolute value as x
gets very large in absolute value. The end behavior of f(x) can be summarized by stating that as x
 , f(x)   and as x  -, f(x)  . The end behavior of g(x) can be summarized by stating
that as x  , g(x)  - and as x  - , g(x)  -. You may wish to graph these functions on a
graphing calculator to verify this summary.
Graph each function. Determine the interval(s) on which the function is increasing and the
interval(s) on which the function is decreasing.
a. f(x) = (x + 1)2 - 4
The graph of this function is obtained
by transforming the parent graph
p(x) = x2. The parent graph has been
translated 1 unit to the left, and
translated down 4 units. The function
is decreasing for x < -1 and increasing
for x > -1. At x = -1, there is a critical
point.
b. f(x) = |x - 3| + 2
The graph of this function is obtained
by transforming the parent graph
p(x) = |x|. The parent graph has been
reflected about the x-axis, translated
3 units right, and translated up two
units. This function is increasing for
x < 3 and decreasing for x > 3. There
is a critical point when x = 3.
c. f(x) = -x3 - x2 + x + 2
This function has more than one
critical point. It changes direction at
1
x = -1 and x = 3. The function is
decreasing for x < -1. The function is
1
also decreasing for x > . When
3
1
-1 < x < 3, the function is increasing.
Helpful Websites
Discontinuity:
http://www.sparknotes.com/math/precalc/continuityandlimits/prob
lems3.rhtml
http://math.usask.ca/~maclean/101/Limits/Printables/BW/Continuit
y.pdf
End behavior: http://www.purplemath.com/modules/polyends.htm
3-5 Self Check Quiz:
http://www.glencoe.com/sec/math/studytools/cgibin/msgQuiz.php4?isbn=0-07-8608619&chapter=3&lesson=5&quizType=1&headerFile=4&state=