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Unit 3 Functions
SM3H 3.1 Notes Functions
Relation: A correspondence between two sets, called the domain and the range.
Domain: The set of inputs (the x-values) of a relation.
Range: The set of outputs (the y-values) of a relation.
Function: A relation in which for each input there is exactly one output. Each element of the
domain corresponds to exactly one element of the range.
Function Machine Rules: 1. The machine only accepts inputs that are part of the
domain.
2. The machine gets confused if there is more than one
possible output for any one input. It only works if there
is only one output for each input.
Examples: Express the relation shown in each map as a set of ordered pairs. Decide whether
each relation is a function.
a) b)
Examples: For each relation, write the domain and range and determine whether the relation is a
function.
a) 1,4 3,6 5,8 6,9
b) 2,5 2,6 4,7 6,9
c) 4,9 6,7 8,9 10,11
Examples: Determine whether the equation defines y as a function of x.
a) 2 2x y b) 2 1x y c) 3 1
2
xy
x
Function Notation
We often use the letters , , and f g h to represent functions. The function machine to the right
represents the function .y f x
f is the name of the function. It is the rule that relates x and .y
x represents the input, called the independent variable or
argument.
or y f x represents the output, called the dependent variable
(because the value of y depends on the value of x that is used as
an input).
f x is read “ of ,f x ” and means “the value (output) of the function f when the input is .x ”
f x DOES NOT mean f times !x
Example: For the function 9,f x x evaluate the following:
a) 2f b) 6f x f c) 3 f x
d) 3f x e) f x f) f x
g) 2f x h) 2f x i) f x h
Example: For the function 23 5 ,f x x x evaluate the following:
a) 2f b) 6f x f c) 3 f x
d) 3f x e) f x f) f x
g) 2f x h) 2f x i) f x h
Domain of a Function
The domain of a function f x is the set of all inputs x.
If the function is listed in a table or as a set of ordered pairs, the domain is the set of all
first coordinates.
If the function is described by a graph, the domain is the set of all x-coordinates of the
points on the graph.
If the function is described by an equation, the domain is the set of all real numbers for
which f x is a real number. Figure out if there are any x-values that cause “problems”
(zero in a denominator, square root of a negative, etc.) when you plug them into the
function. If so, these numbers are not part of the domain.
If the function is used in an application, the domain is the set of all numbers that make
sense in the problem.
Tips for finding domain: (domain restrictions) 1. If the equation has fractions, exclude any numbers that give a zero in a denominator.
2. If the equation has an even root, exclude any numbers that cause the expression under the
root to be negative.
Examples: Determine the domain of .f x
a) 2f x x x b) 2
4
9
xf x
x
c) 5f x x
d) 2f x x e) 1
2f x
x
f) 5 7f x x
Example: Express the area of a circle as a function of its radius. Find the domain of the function.
Sums, Differences, Products, and Quotients of Two Functions
The sum f g is defined by f g x f x g x
The difference f g is defined by f g x f x g x
The product f g is defined by f g x f x g x
The quotient f
g is defined by
, 0f xf
x g xg g x
The domain of , , or f g f g f g consists of all the numbers x that are in the domains of both
f and .g The domain of f g consists of all the numbers x for which 0g x that are in the
domains of both f and .g
Examples: Find the following and determine the domain given the functions
3
xf x
x
and
8.
3
xg x
x
a) f g x b) f g x c) f g x d) f
xg
SM3H 3.2 Notes The Graph of a Function Not every collection of points in the xy-plane represents a function. Remember, for a function,
each number x in the domain has exactly one image y in the range. The graph of the function
must satisfy the vertical line test.
Vertical Line Test
If it is possible for a vertical line to cross a graph more than once, then the
graph is not the graph of a function.
The graph at left is not a function because three y-values correspond to one
x-value.
Examples: Decide whether each graph represents a function. Then find the domain and range.
Discuss Symmetry.
a) b)
c) d)
Example (#9 on homework):
a) Find 0 and 6 .f f
b) Find 6 and 11 .f f
c) Is 3f positive or negative?
d) Is 4f positive or negative?
e) For what values of x is 0?f x
f) For what values of x is 0?f x
g) What is the domain of ?f
h) What is the range of ?f
i) What are the x-intercepts?
j) What is the y-intercept?
k) How often does the line 1 2y intersect the
graph?
l) How often does the line 5x intersect the graph?
m) For what values of x does 3?f x
n) For what values of x does 2?f x
Example: 2 2
4
xf x
x
a) Is the point 1,3 5 on the graph of f ?
b) If 0,x what is ?f x What point is on the graph of ?f
c) If 1 2,f x what is x ? What point(s) are on the graph of ?f
d) What is the domain of f ?
e) List the x-intercepts, if any, of the graph of .f
f) List the y-intercept, if there is one, of the graph of .f
Example: A golf ball is hit with an initial velocity of 130 feet per second at an inclination of 45°
to the horizontal. In physics, it is established that the height h of the golf ball is given by the
function 2
2
32
130
xh x x
where x is the horizontal distance that the golf ball has traveled is.
a) Determine the height of the golf ball after it has traveled 100 feet, 300 feet, and 500 feet.
b) How far was the golf ball hit?
c) Using a graphing calculator, graph the function .h h x
d) How far has the ball traveled when it reaches its maximum height? What is its maximum
height?
SM3H 3.3 Notes Properties of a Function
Even and Odd Functions: The words even and odd, when applied to a function f , describe the
symmetry that exists for the graph of the function.
Even Function: A function f is even if, for every number x in its domain, the number x is
also in the domain and f x f x . Even functions are symmetric with respect to the y-axis.
Odd Function: A function f is odd if, for every number x in its domain, the number x is also
in the domain and f x f x . Odd functions are symmetric with respect to the origin.
Examples: Determine whether each graph is an even function, odd function, or neither.
a) b) c)
Examples: Determine whether each of the following functions is even, odd, or neither. Then
determine whether the graph is symmetric with respect to the y-axis, the origin, or neither.
a) 2( ) 2f x x b) 3 24 1f x x x
c) 3f x x x d) 5f x x
Increasing, Decreasing, and Constant Graphs: If you look from left to right along the graph of
the function, you will notice parts are rising, parts are falling and parts are horizontal. In such
cases, the function is described as increasing, decreasing, or constant, respectively.
Definitions:
A function f is increasing if for any choice of 1x and 2x , where 1 2x x , then 1 2f x f x .
A function f is decreasing if for any choice of 1x and 2x , where 1 2x x , then 1 2f x f x .
A function f is constant if for any choice of 1x and 2x , where 1 2x x , then 1 2f x f x .
Example: Determine where the graph is increasing, decreasing, or constant.
Local Maxima and Minima: When a graph is increasing to the left of a point on the graph, and decreasing to the right of that
point on the graph, then the value is a local maximum. When a graph is decreasing to the left of
a point on the graph, and increasing to the right of that point on the graph, then the value is a
local minimum.
Note: If a question asks “Where…”, “On what interval(s)…”, or “At what number(s)…”,
it is asking for x-coordinates. If it asks “What is...” or “Find the value of…”, it is asking
for a y-coordinate.
Example:
a) At what number(s), if any, does f have a local maximum?
b) What are the local maxima?
c) At what number(s), if any, does f have a local minimum?
d) What are the local minima?
e) List the intervals where f is increasing and the intervals where f is decreasing.
Using a Graphing Calculator to Find Local Minima and Maxima
To find the exact value at which a function f has a local maximum or local minimum usually
requires calculus. However, a graphing calculator may be used to approximate these values.
1. Press Y = and enter function.
2. Press GRAPH.
3. Enter the domain by pressing WINDOW. Xmin = smallest #, Xmax = largest #.
4. Press GRAPH.
5. Press ZOOM, and choose 0: ZoomFit.
6. Press 2ND TRACE (CALC) and choose 3: minimum or 4: maximum, enter.
7. Move the arrows until you are left of the minimum or maximum and press enter.
8. Move the arrows until you are right of the minimum or maximum and press enter.
9. Move the arrows until you are near the minimum or maximum and press enter.
10. The Y= ___ on the bottom right gives the minimum or maximum.
Example: Use a graphing calculator to graph 3 3 2f x x x for 2 2x . Approximate
where f has a local maximum and where f has a local minimum. Also find the values of the
minimum and maximum.
Find the Average Rate of Change of a Function
To find the average rate of change of a function between any two points on its graph, calculate
the slope of the line containing the two points.
If a and b, a ≠ b, are in the domain of a function y = f (x), the average rate of change of f from a
to b is defined as:
Average rate of change = ( ) ( )y f b f a
a bx b a
The symbol y above is “the change in y,” and x is the “change in x.” The average rate of
change of f is the change in y divided by the change in x.
Example:
Find the average rate of change of 2( ) 3f x x for the following intervals:
a) From 1 to 3 or [1, 3] b) From 1 to 5 or [1, 5]
c) From 1 to 7 or [1, 7]
SM3H 3.4 Notes – Piecewise Functions Graphing Piecewise-Defined Functions Sometimes a function is defined differently on different parts of its domain. When functions are
defined by more than one equation, they are called piecewise-defined functions.
Remember how to graph: Line Quadratic Square root
Here is how to write a piecewise function.
0( )
3 0
x if xf x
if x
This means:
When you graph a piecewise function, think of the points where the graph changes to the second
function as a fence. If you have _____or____ the function does NOT live on the fence. If you have
_____or_____ then the function lives on the fence.
EX. A) find domain (try to do it without graphing), B) Find the intercept(s), C) Graph the function,
D) Find the range.
2
1 1( )
x +1 1
x if xf x
if x
Domain:
Intercept(s):
Range:
Sometimes we want to find the value at a given coordinate. You can do this using a graph.
EX. A)
B)
C)
D)
Examples: For the following functions: a) Graph the function. b) Find the domain and range of the function. c) Locate any intercepts.
1) 3 if 1
2 if 1
x xf x
x x
2)
2
2 if 4 0
2 if 0
xf x
x x
Find: f(-4) f(0) f(4) Find: f(-2) f(0) f(6)
3)
3 if 5 2
if 0 4
2 6 if 4
x x
f x x x
x x
4) 12
if 2
5 if 2
if 2
x x
f x x
x x
Find: f(-3) f(1) f(8) Find: f(-5) f(2) f(-6)
Write a definition(equation) for each piecewise function.
Here is a real -life example for using a piecewise function.
A doctor's fee is based on the length of time.
A) up to 6 minutes it costs $50
B) over 6 to 15 minutes costs $80
C) over 15 minutes $80 plus $5 per minute above 15 minutes
Write the equation.
How much it would cost at 12 minutes? How much would it cost for 20 minutes?
SM3H 3.5 Notes – Transformations
Graphical Transformations
2y x
2( 3)y x
21y x
2 4 5y x x
Transformations – functions that map real numbers to real numbers
Rigid Transformations – leave the size and shape of a graph unchanged (horizontal and vertical
translations, reflections)
Non-rigid transformations – distort shape of graph (horizontal and vertical stretches and
shrinks)
Vertical Translations:
Vertical translation shifts graph up or down.
y = f(x) + k up by k units. y= f(x) - k down by k units
Ex. y x
4y x
Horizontal Translations:
Horizontal translation shifts graph left or right.
y = f(x - h) to right by h units. y = f(x + h) to left by h units.
y x
2y x
Reflections:
Points (x, y) and (x, -y) are reflections of each other across the x-axis.
Across the x-axis: y = f(x) y = - f(x)
Points (x, y) and (-x, y) are reflections of each other across the y-axis.
Across the y-axis: y = f(x) y = f(-x)
Finding Equations of Reflections:
Find an equation for the reflection of 2
3 7( )
2
xf x
x
across each axis.
Reflection across x-axis: y = -f(x)
2 2
3 7 3 7( )
2 2
x xf x
x x
Reflections across y – axis: y = f(-x)
2 2
3 7 3 7( )
22
x
x
xf x
x
Stretches and Shrinks
Let a & b be positive real numbers. Then the following transformations result in stretches or
shrinks of the graph of y = f(x).
Horizontal Stretches or Shrinks
( )y f bx , b > 0 a shrink by a factor of 1
b if b >1
a stretches by a factor of 1
b if 0 < b < 1
Vertical Stretches or Shrinks
y = a f(x) a stretch by a factor of a if a > 1
a shrink by a factor of a if 0 < a < 1
Examples:
Transform the given function by a) a vertical stretch by a factor of 3, and b) a horizontal
shrink by a factor of 1/2.
( ) 4f x x
a) 3 ( ) 3 4f x x
b) (2 ) 2 4f x x
Sequence of transformations:
1) Reflections, 2) Stretches and shrinks and 3) shifts
Describe a basic graph and a sequence of transformations that can be used to produce a graph of
the given function.
2 3y x
Starting with y x , reflect across the x-axis, vertical stretch by factor of 2, and shift left 3.
A new graph is obtained from the series of transformations on the given graph; write the
equation for the new graph.
2;y x a vertical stretch by a factor of 4, then a shift right 6 units.
24( 6)y x
Examples: Graph the following:
a) 3
1 2f x x b) 2 1 3g x x
c) 3 2f x x d) 3 2g x x
e)
3
2h x
x
Summary of Graphing Transformations:
To Graph: Draw the Graph of ( )y f x and: Functional Change to
( ) :y f x
Reflection About the x-axis
( )y f x Reflect the graph of f about the x-axis. Multiply ( )f x by –1.
Reflection About the y-axis
( )y f x Reflect the graph of f about the y-axis. Replace x by .x
Vertical Stretches &
Compressions
( ), 0y a f x a
Multiply each y-coordinate of ( )y f x by
a.
This stretches the graph of f vertically if
1.a
This compresses the graph of f vertically if
0 1.a
Multiply ( )f x by a.
Horizontal Stretches &
Compressions
( ) 0,y f bx b
Divide each x-coordinate of ( )y f x by b.
This stretches the graph of f horizontally if
0 1.b
This compresses the graph of f horizontally
if 1.b
Replace x by .bx
Vertical Shifts
( ) , 0y f x k k Raise the graph of f by k units. Add k to ( )f x
( ) , 0y f x k k Lower the graph of f by k units. Subtract k from ( )f x
Horizontal Shifts
( ), 0y f x h h Shift the graph of f to the right by h units. Replace x by .x h
( ), 0y f x h h Shift the graph of f to the left by h units. Replace x by .x h
SM3H 3.6 Notes – Properties of Rational Functions
Asymptotes - Can’t Touch This
Horizontal Vertical Oblique
Assignment 3.6A
Domain (restrictions)
Find the Domain
1. 2
( )3
xf x
x
2.
2
2
6 8( )
6
x xf x
x x
List the vertical asymptotes from example 1 and 2
6.1N – Graphing Rational Equations #1
A. Find the vertical asymptotes. Remember: It is the same as finding the restrictions.
B. Find the x-intercepts (this is the same as finding the zeros). Write the intercepts as ordered pairs.
C. Find the y-intercepts (let x=0). Write intercepts as ordered pairs.
1. 2
( )3
xf x
x
2.
2
4( )
4 12
xf x
x x
3. 2
2
6 8( )
6
x xf x
x x
4.
2
4( )
16f x
x
D. Find the horizontal asymptotes
2
2( )
1
xf x
x
The horizontal asymptote roughly tells you what happens to the graph as x get
really, really large. So, let’s look at a table that shows the function 2
2( )
1
xf x
x
when x is equal to some very big numbers (numbers far from zero).
x 2
2( )
1
xf x
x
-100,000 -0.0000099…
-10,000 -0.0000999…
-1000 -0.0009979…
-100 -0.0097990…
-10 -0.0792079…
-1 0.5
0 2
1 1.5
10 0.1188118…
100 0.0101989…
1,000 0.0010019…
10,000 0.0001000…
100,000 0.0000100…
1. As x gets large what value is our graph(y-value) approaching?
2. So the horizontal asymptote is?
In the function2
2( )
1
xf x
x
, the degree of the denominator is bigger than the degree of the numerator, and the
horizontal asymptote was y = 0 (the x-axis). This property is always true: If the degree on x in the denominator is larger
than the degree on x in the numerator, then the denominator, being "bigger", pulls the fraction down to the x-axis (y=0)
when x gets big. (Dividing by a larger and larger number makes the fraction get closer and closer to zero.)
What happens if the degrees are the same in the numerator and denominator?
Let’s look at the function:2
2
2 11( )
9
xf x
x
When x = 1,000, y = ___________________
When x= 10,000, y = __________________
When x = 100,000, y = _________________
3. As x gets large what value is our graph(y-value) approaching?
4. So the horizontal asymptote is?
When the degrees of the numerator and the denominator are the same like in the function2
2
2 11( )
9
xf x
x
, then the
horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:
y = (numerator's leading coefficient) / (denominator's leading coefficient)
What happens if the degree is greater in the numerator than in the denominator?
Let’s look at the function:23 2
( )1
xf x
x
Use long division:
2
2
3 3
1 3 0 2
3 3
3 2
3 3
1
x
x x x
x x
x
x
5. So the asymptote is?
6. Draw in the asymptote and label it.
7. This type of asymptote is called a ___________________________________.
If the degree of the numerator is exactly one more than the degree of the denominator, like the
function23 2
( )1
xf x
x
then the asymptote is called a "slant" (or "oblique") asymptote. The equation for the slant
asymptote is the polynomial part of the rational that you get after doing the long division.
*You will only be required to identify if the function has an oblique asymptote, not find the equation
of the line.
Horizontal asymptote summary:
If degree of numerator is __________________ the degree of the denominator, then the asymptote is y=_______.
If degree of numerator is _____________________ the degree of the denominator, then the asymptote is y=__________________________________________________.
If degree of numerator is ___________________ the degree of the denominator, then the asymptote is ___________________________________________________.
Practice finding the horizontal asymptotes on the following functions:
1. 1
( )2 6
xf x
x
2. 2
2
3 4( )
4 3
x xf x
x x
3.
2
4 12( )
12
xf x
x x
4. 3 2
2
2( )
4 12
x x xf x
x x
5. 2
3( )
4 4 8f x
x x
6.
2
2
4( )
6 9
x xf x
x x
E. Make a sign array using the following information. (Hint: * goes on the sign array)
2( )
3
xf x
x
*vertical asymptote: 3x
horizontal asymptote: 1y
*x-intercept(s) (ordered pair):
( 2,0)
y-intercept(s) (ordered pair):
2(0, )
3
F. Find the vertical asymptotes, horizontal asymptotes, x-intercepts, y-intercepts and make a sign array
for the following equation.
2
2
6( )
4 4 24
x xf x
x x
*vertical asymptote:
_____________
horizontal asymptote:
___________
*x-intercept(s) (ordered pair):
__________
y-intercept(s) (ordered pair):
__________
STEPS
1.
2.
Rules:
3.
4.
5.
6.
7.
1.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
2.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
3.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
4.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
5.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
6.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
7.
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
A rational function is a function of the form
,p x
R xq x
where p and q are polynomial
functions and q is not the zero polynomial.
Finding the Domain of a Rational Function
The domain of a rational function is the set of all real numbers except those for which the
denominator q is 0.
Examples: Find the domain of each rational function.
a) 22 3
3
xR x
x
b) 2
3
9
xR x
x
c)
2
2 7 12
xR x
x x
Note: It is important to understand that 2
3
9
xR x
x
and
1
3f x
x
are not the same.
They have different domains. Their graphs are nearly identical, but the graph of the first
function has a hole in it at 3,x while the graph of the second function does not.
A rational function
p xR x
q x is in lowest terms if p x and q x have no common factors.
Finding the Intercepts of a Rational Function
Warning: Always write the rational function in lowest terms before finding the x-intercepts
of the graph. Otherwise, you may end up listing values that are actually holes in the graph.
When finding the y-intercepts, remember that if zero is not in the domain, there is no y-
intercept
To find the x-intercepts (real zeros) of the graph of a rational function, we set 0R x and
solve for .x
Notice that if
0,p x
R xq x
p x must equal zero. It is the numerator that tells us about the
x-intercepts.
To find the x-intercepts (real zeros), simplify the rational function and set the numerator
equal to zero.
To find the y-intercept, plug 0x into either the simplified or the unsimplified function. If
zero is not in the domain of the function, there is no y-intercept.
Examples: Find the x- and y- intercepts of each rational function.
a) 2
2
16
xR x
x
b)
2
2
2 35
11 30
x xR x
x x
c)
2 8
xR x
x x
Example: Analyze the graph of 2
1.H x
x (Find the domain, range, intercepts, and any other
important features).
Example: Using transformations, graph
2
11.
2R x
x
Asymptotes In the previous example, notice that as the values of x become more negative, that is, as x
becomes unbounded in the negative direction
, read "as approaches negative infinity"x x the values of R x approach 1. We
conclude the following:
1. As ,x the values of 1.R x
2. As 2,x the values of .R x
3. As ,x the values of 1.R x
The graph goes near, but does not reach the lines 2x and 1.y These lines are called
asymptotes.
Let R denote a function.
If, as x or as ,x the values of R x approach some fixed number ,L then
the line y L is a horizontal asymptote of the graph of R.
If, as x approaches some number ,c the values of R x approach or , then the
line x c is a vertical asymptote of the graph of .R
limx
R x L
limx
R x L
limx c
R x
limx c
R x
limx c
R x
limx c
R x
There is a third type of asymptote called an oblique asymptote. An oblique asymptote occurs
when the graph’s end behavior follows an oblique line (a linear function of the form
, 0y ax b a ). A graph may also approach some function such as a parabola.
Note: The graph of a function may intersect a horizontal or oblique asymptote at some other
point, but the graph will never intersect a vertical asymptote.
Find the Vertical Asymptotes of a Rational Function
A rational function
,p x
R xq x
in lowest terms will have a vertical asymptote at x r if r is
a real zero of the denominator .q That is, if x r is a factor of the denominator q of the rational
function in lowest terms, it will have the vertical asymptote .x r
Warning: If the rational function is not in lowest terms, this theorem will result in an
incorrect listing of vertical asymptotes.
Example: Find the vertical asymptotes, if any, of the graph of each rational function.
a) 2
1
9
xR x
x
b) 2
2
3 10
xR x
x x
c) 3 210 9
xR x
x x x
d)
2
2 4
xR x
x
Find the Horizontal or Oblique Asymptotes of a Rational Function
If a rational function R x is proper, that is if the degree of the numerator is less than the
degree of the denominator, then the line 0y is a horizontal asymptote.
If a rational function
p xR x
q x is improper, that is, if the degree of the numerator is
greater than or equal to the degree of the denominator, we can use long division to write the
rational function as the sum of a polynomial f x plus the proper rational function
.r x
q x
As x or as ,x
0,r x
q x and the graph of R x approaches the graph of f x .
o The possibilities include:
1. If ,f x b a constant, the line y b is a horizontal asymptote of the graph of
R.
2. If , 0,f x ax b a the line y ax b is an oblique (slanted) asymptote of
the graph of .R
3. In all other cases, the graph of R approaches the graph of ,f and there are no
horizontal or oblique asymptotes.
Examples: Find the horizontal or oblique asymptotes, if any, of the graph of the function.
a) 2
5
3 10
xR x
x x
b)
2
2
6 2 5
3 2
x xH x
x
c) 3 2
2
8 2 6
2 3
x xH x
x
d)
4 2
2
3 4 7
2 5
x xH x
x x
Summary:
Given a rational function
,p x
R xq x
1. The domain is found by setting the denominator not equal to zero and solving. (Do not
simplify first).
2. The x-intercepts are found by setting the numerator to zero and solving. (Simplify first).
3. The y-intercept is found by setting 0x and simplifying. (You can do this before or
after simplifying the function). Remember, if zero is not in the domain, the function has
no y-intercept.
4. Vertical asymptotes are found by setting the denominator to zero and solving (Simplify
first).
5. Horizontal or Oblique Asymptotes:
a) If proper, where ,degree of numerator degree of denominator the graph has a
horizontal asymptote at 0.y
b) If improper, where ,degree of numerator degree of denominator the graph has
a horizontal asymptote at the line leading coefficient of the numerator
.leading coefficient of the denominator
y
c) If improper, where degree of numerator degree of denominator (by one), the
graph has an oblique asymptote of y f x found by performing long division.
d) If improper, where degree of numerator degree of denominator (by more than
one), then R has neither a horizontal nor an oblique asymptote, but the end
behavior can be determined using long division.
SM3H 3.7 Notes – Graphing Rational Functions
. Review the following:
vertical asymptotes
The denominator cannot equal zero (undefined) so set the denominator equal to zero and solve
for x, you will then know what x cannot equal . Therefore that is an asymptote at
horizontal asymptotes
degree of the polynomials
degree of n degree of d then
degree of n degree of d then
degree of n degree of d then asymptotes is oblique (slant)
answer will look like
x-intercepts
To find the x-intercepts (when y=0), set the numerator equal to zero and solve for x. Write
answer as an ordered pair.
y-intercept
To find the y-intercept (when x=0), substitute 0 in for all the x’s and simplify. Write answer as
an ordered pair.
domain
Include everything except the vertical asymptotes and write in interval notation.
sign array
Make a sign array (plus minus number line). This determines where the graph is positive and
negative.
B. DRAW GRAPH
Graph your intercepts and asymptotes. Use sign array to draw graph. Remember positive is
above the x-axis and negative is below the x-axis.
Analyzing the Graph of a Rational Function 1. Factor the numerator and denominator of the rational function and find the
domain by setting the denominator not equal to zero. (Do this before
simplifying).
2. Simplify the rational function.
3. Find any holes in the graph. Set any factors that appeared in both the
numerator and denominator of the unsimplified function equal to zero to find
the x-coordinate of the hole. Plug the x-coordinate into the simplified
function to find the y-coordinate of the hole.
4. Find the x-intercepts by setting the numerator of the simplified function
equal to zero. Find the y-intercept by plugging in zero for .x
5. Find the vertical asymptotes by setting the denominator of the simplified
rational function equal to zero.
6. Find the horizontal asymptotes or oblique asymptotes:
a) If the degree of the numerator < the degree of the denominator, then
the horizontal asymptote is 0.y
b) If the degree of the numerator = the degree of the denominator, then
the horizontal asymptote is leading coefficient of numerator
.leading coefficient of denominator
y
c) If the degree of the numerator > the degree of the denominator, then
the graph has an oblique asymptote (if the degree of the numerator is
one more than the degree of the denominator), or the graph
approaches a quadratic or higher degree function (if the degree of the
numerator is more than one higher than the degree of the
denominator). To determine the end behavior, use long division.
7. Use the x-intercepts and vertical asymptotes as critical points to divide the
graph into intervals and determine where it is above the x-axis (y is positive)
and where it is below the x-axis (y is negative). Do this by choosing a
number x in each interval and evaluating R there.
8. Analyze the behavior of the graph near each asymptote.
9. Put together all of the information to graph the function.
1. 2
( )1
xf x
x
vertical asymptote:
horizontal or oblique asymptote:
x-intercept:
y-intercept:
Domain:
2. 2
1( )
6
xf x
x x
vertical asymptote:
horizontal or oblique asymptote:
x-intercept:
y-intercept:
Domain:
3. 2
2
2 2( )
4
xf x
x
vertical asymptote:
horizontal or oblique asymptote:
x-intercept:
y-intercept:
Domain:
4. 2 2 3
( )2
x xf x
x
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
5. 2
3( )
9
xf x
x
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
6. 2
4( )
2 24
xf x
x x
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
7. 2 42
( )4
x xf x
x
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
3.1 Notes Functions
8. 3
2
1( )
2
xf x
x x
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
7. 2
2
12( )
4
x xf x
x
vertical asymptote
horizontal or oblique asymptote
x-intercept
y-intercept
Domain
Example: Find a rational function that might have the graph shown below.
Note: If the graph goes the same direction on both sides of an asymptote (approaches on both sides or
approaches on both sides), the related factor in the denominator has an even multiplicity. If the graph
goes in opposite directions on the two sides of an asymptote (approaches on one side and on the
other), the related factor in the denominator has an odd multiplicity.
Example: Finding the Least Cost of a Can
Reynolds Metal Company manufactures aluminum cans in the shape of a right circular cylinder with a capacity
of 500 cubic centimeters. The top and bottom of the can are made of special aluminum alloy that costs 0.05¢ per
square centimeter. The sides of the cans are made of material that costs 0.02¢ per square centimeter.
a) Express the cost C of material for the can as a function of the radius r of the can.
b) Find any vertical asymptotes. Discuss the cost of the can near any vertical asymptotes.
c) Use a graphing calculator to graph the function .C C r
d) What value of r will result in the least cost? What is the least cost?
