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Unit 2
Polynomial and Rational Functions
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Given the following function,
f (x)  2x  5x  2
2
find

f (2) 
f (0) 
f (2) 
Unit 2 Lesson 1
2.1 Quadratic Functions
2.2 Polynomial Functions
Lesson Objectives
In this lesson presentation, you will learn how to:
define, sketch & analyze graphs of quadratic & polynomial
functions.
define and identify the vertex, minimum and maximum values of
a quadratic& polynomial functions.
use the LEADING COEFFICENT TEST to determine how a
polynomial function rises or falls.
identify the zeros of both of quadratic & polynomial functions.
use the Intermediate Value Theorem to help identify the real
zeros of a polynomial function.
2.1 Quadratic Functions
a, b, and c are real numbers where a0
f (x)  ax  bx  c
2
When graphed called a Parabola
Axis of Symmetry
– all parabolas are
symmetric (line where
it’s cut in half)
Vertex – axis intersects
the parabola
If a is positive…
opens up
If a is negative…
opens down
f ( x)  ax , a 0
f ( x)  ax , a0
2
2
Minimum: (0,0)
Maximum: (0,0)
Axis of Symmetry for both: x=0
f (x)  ax  bx  c
2
f ( x)  ax  bx  c, a 0
2
If a is positive…opens up
f ( x)  2 x  2 x  1

2
Vertex is lowest point.
Axis of symmetry
f ( x)  ax 2  bx  c, a0
If a is negative…opens down
f ( x)   x
2
Vertex is highest point.

Base function or squaring function.
f (x)  ax
2
How does each function change?
1) f ( x)  x 2  5
2) f ( x)   x  5 
3) f ( x)  5 x
2
Up 5 units
2
Left 5 units
Get’s skinny
Summarize: How do they change?
If c>0, then move up.
If c<0, then move down.
If (x+#), then move left.
If (x-#), then move right.
If “a” is integer, then get skinny.
If “a” is fraction, then get wide.
Practice
Given the following functions, graph each on your
graph paper and identify the following:
a) Domain:
b) Range:
c) Decreasing on:
d) Increasing on:
e) Turn up or down:
Example 1.
f ( x)   x  6
Example 2.
f ( x)  ( x  2)2  4
2
Practice
Given the following function,
identify the following:
Domain:
Range:
Decreasing on:
Increasing on:
Turn up or down:
Standard Form of a Quadratic
f ( x )  a ( x  h)  k
2
Vertex is (h, k)
and
a0
Axis is the vertical line x = h
If a < 0, then the parabola opens down
If a > 0, then the parabola opens up
Identify the vertex of a Quadratic
Given a function
f ( x)  2 x  8 x  5
2
Write the function in Standard Form by completing the
square to find the vertex.
f ( x)  2 x 2  8 x  5
Example 1.
 2( x 2  4 x)  5
*Factor out the 2
 2( x  4 x  4  4)  5
*b=4, add &
 2( x 2  4 x  4)  2(4)  5
*Regroup terms
2
 2( x  2)  3
2
The vertex is (-2,-3)
subtract (4/2)2 =4
*Write in
Standard Form.
Identify the Minimum & Maximum Values
Given a function
f (x)  ax 2  bx  c
b
You can find the minimum if a>0, x  
2a
b
You can
find the maximum if a<0 x  
2a
Example: Given the following quadratic equation
find coordinates of the relative minimum or
maximum point. f ( x)  4 x 2  24 x  2
2.2 Graphs of Polynomial Functions
Features of Polynomial Functions:
Continuous
Smooth Curves
NOT Features of Polynomial Functions:
NOT Continuous
NOT Smooth
Functions with 1 degree are Linear.
Functions with 2 degrees are Quadratic.
Functions with 3rd or greater are Polynomial.
f ( x)  x 2
f ( x)  x 3
Functions with an even degree will look like our
quadratics. (e.g.)
4
8
f ( x)  x
f ( x)  x
Functions with an odd degree will look like our
11
5
cubic. (e.g.)
f ( x)  x
f ( x)  x
A polynomial of the nth degree has the form:
f ( x)  an x  an 1 x
n
n 1
 ...  a2 x  a1 x  a0
2
n is a positive integer
The greater the value of n the flatter the
graph near (0,0)
The Leading Coefficient Test
A functions ability to rise or fall can be determined by the
functions degree (even or odd) and by it’s leading
coefficient.
f ( x)  an x  an 1 x
n
n 1
 ...  a2 x  a1 x  a0
2
When n is odd: When
an  0
f ( x)  
as
f ( x)  
as
x
x  
When n is odd: When an 0
f ( x)  
as
x  
f ( x)  
as
x
When n is even: When
an  0
f ( x)  
as
f ( x)  
as
x  
x
When n is even: When
an 0
f ( x)  
as
f ( x)  
as
x  
x
Independent Practice:
Take a minute to see if you can develop four
different polynomial functions that would behave
as we just discussed. Write down your functions
and check your work in you calculator.
Practice
Use the Leading Coefficient Test to determine the
right and left side behavior of the function.
Example 1.
Example 2.
f ( x)  x  5 x  4
4
2
f ( x)  x  x
5
Finding Zeros of Polynomial Functions
Set the function equal to zero and factor.
3
2
(e.g)
f ( x)  2 x  4 x  16 x
0  2 x  4 x  16 x
2
0  2 x( x  2 x  8)
0  2 x( x  2)( x  4)
3
2x  0
x0
2
x2 0
x2
x40
x  4
Check on the graphing calculator.
2.3 Real Zeros of Polynomial Functions
Disclaimer:
There are many ways for us to determine the zeros
of a polynomial function. Please be patient and
as the lesson progresses through the various
topics the math should become easier for you!
Topics will include:
*Long Division of Polynomials
*Synthetic Division
*Remainder & Factor Theorems
*Rational Zero Test
*Descartes's Rule of Signs
Long Division of Polynomials
f ( x)  6 x  19 x  16 x  4
3
2
Graph the function to identify a zero that is an integer.
You can see that x=2 is a zero, as a result (x-2) is a
factor of the polynomial.
We will divide the entire function by (x-2).
6x 7x 2
2
2
Divide!
1) Multiply: 6 x 2 ( x  2)
2
2) Subtract.
x  2 (6 x 19 x  16 x  4)
3
6 x  12 x
3
7 x  16 x
3) Multiply: 7 x( x  2)
7 x  14 x
4) Subtract.
2
2
2x  4
2x  4
0
5) Multiply: 2( x  2)
6) Subtract.
Now we know that:
6 x  19 x  16 x  4  ( x  2)(6 x  7 x  2)
3
2
2
So we factor and solve for zero!
0  ( x  2)(6 x  7 x  2)
0  ( x  2)(2 x  1)(3x  2)
2
0  x2
x2
0  2x 1
1
x
2
We have three x-intercepts at:
0  3x  2
2
x
3
1
2
x  2, x  , x 
2
3
Synthetic Division (the shortcut to long division)
We will use Synthetic Division to divide the following:
x  10 x  2 x  4by( x  3)
4
Solve to get divisor:
2
Leading coefficients (NOTE: zero because there is no x cubed.
x3  0
-3
1
1
0
-10
-3
9
-3
-1
-2
4
3 -3
1
1
Remainder: 1
1) Add terms in first column, and multiply the result by -3.
2) Repeat this process until the last column.
-3
1
1
Quotient:
0
-3
-3
-10
9
-1
-2 4
3 -3
1 1
x  3x  x  1
3
2
Remainder:
1
( x  3)
original problem = quotient + remainder
x  10 x  2 x  4
1
3
2

 x  3x  x  1 
( x  3)
( x  3)
4
2
Remainder & Factor Theorems
The Remainder Theorem (the remainder from synthetic division):
If a polynomial
f ( x ) is divided by x-k,
the remainder is r  f ( k ) .
Practice 1. Use the Remainder Theorem to
evaluate the given function at x = -2.
f ( x)  3 x  8 x  5 x  7
3
2
Start by doing synthetic division!
-2
3
3
8 5
-6 -4
2 1
-7
-2
-9
Remainder is -9,  f (2)  9 .
(-2,-9) is a point on the graph of our polynomial.
Check by substituting in -2 into our original function.
Rational Zero Test
If the polynomial
f ( x)  an x  an 1 x
n
n 1
 ...  a2 x  a1 x  a0
2
has integer coefficients, every rational zero of f has
the form:
p
Rational Zero 
q
p: factor of constant term
q: factor of leading coefficient
Example 1. Find the rational zero(s) of the given function.
f ( x)  x  x  1
3
p 1
q 1
The factors of 1 are +1 and -1
Therefore: Rational Zeros: +1 or -1
Test each possible zero algebraically.
f (1)  (1)  (1)  1  1
3
f (1)  (1)  (1)  1  3
3
Since -1, and 3 are not zero, this polynomial does
not have a rational zero.
If the leading coefficient is not 1 our possible zeros
will increase significantly.
*We will use our graphing calculators to plot the
zeros for these situations.
If you would like to see a problem working out
please reference pg.119 in your textbook.
Practice. Using your graphing calculator find the
rational zeros of the following function.
f ( x)  2 x  3 x  8 x  3
3
2
Variation in sign: Two consecutive (nonzero)
coefficients have opposite signs.
Example: Do the given functions have a difference of signs?
+ to +
+ to +
f ( x)  3 x  2 x  6 x  9
3
2
NO!!!
+ to +
+ to -
+ to -
f ( x)  7 x  8 x  6 x  5
3
2
- to +
YES!!!
Descartes's Rule of Signs
Let
f ( x)  an x  an 1 x
n
n 1
 ...  a2 x  a1 x  a0
2
Be a polynomial with real coefficients and
a0  0 .
1. The number of positive real zeros of f is either
equal to the number of variations in sign of f(x) or
less than the number by an even integer.
2. The number of negative real zeros of f is either
equal to the number of variations in sign of f(-x) or
less than the number by an even integer.
Example.
Describe the possible real zeros of the given function.
+ to 3
+ to +
f ( x)  7 x  8 x  6 x  5
2
- to +
- to 3
This polynomial has
either two or no
positive real zeros.
- to +
f ( x)  7 x  8x  6 x  5
2
- to -
This one negative
real zeros.
2.4 The Imaginary Unit i
i  1
If a and b are real numbers:
a + bi is a complex number in standard form
If b = 0, a +bi = a
If b ≠ 0, a +bi is an imaginary number
If b ≠ 0, bi is a pure imaginary number.
Addition & Subtraction with Complex Numbers
Just like combining like terms in Algebra!
Example1: Solve / Simplify.
(5 - i) + (9 + 5i)
Example2: Solve / Simplify.
10 + (3 + 6i) - (-2 + 2i)
Multiplication with Complex Numbers
Remember: i
Example1: Multiply.
Example 2: Multiply.
2
25
 1
81
(3 + 2i)(5 + 4i)
Complex Conjugates
When two complex numbers are multiplied and
their solution is a real number.
(remember the difference of squares?)
Example of Complex Conjugates:
(5 + 2i)(5 – 2i)
Find the complex conjugate for the following
complex number and multiply.
(7 - 4i)
Division with Complex Numbers
Example1: Write the following quotient in
standard form.
(2  3i)
(4  2i)
2.5 The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n,
where n > 0, the f has at least one zero in the
complex number system.
Linear Factorization Theorem
If f(x) is a polynomial of degree n where n >
0, f has precisely n linear factors.
f ( x)  an ( x  c1 )( x  c2 )  ( x  cn )
Where c1 , c2 , , cn are complex numbers.
Example 1. Using the Ft of A and LFT, prove that the
given 3rd degree polynomial function does indeed
have three exact zeros in the complex number system.
f ( x)  x  9 x
3
Factor completely.
Find your zeros.
x  9 x  x( x  9)
3
2
x( x  3i )( x  3i )  0
x0
x  3i
x  3i
Conjugate Pairs (think: difference of squares)
(a  bi )and (a  bi )
FACT:
Complex zeros occur in conjugate pairs!
(think about the last example!)
If a  bi is a zero of a function, then the
conjugate a  bi is also a zero.
Example 2. Using you knowledge of the FTofA, LFT, and
conjugate pairs find all of the zeros of the given
function if 1 + 3i is a zero.
f ( x)  x  3x  6 x  2 x  60
4
3
2
If 1 +3i is a zero we know that 1 – 3i is a zero.
∴ If x = (1 + 3i), then x – (1 + 3i) is a factor.
Multiply.
Distribute.
 x  (1  3i) x  (1  3i)
 x 1  3i x 1  3i
Multiply.
( x 1)  3i( x 1)  3i   ( x 1)
2
 9i
2
( x  1)  9i  ( x  1)( x  1)  9i
2
2
2
( x  2 x  1)  9(1)  x  2 x  10
2
2
Now, use long division to divide.
x  2 x  10 x  3x  6 x  2 x  60
2
4
3
2
( x  2 x  10)( x  x  6)
2
2
Factor!
Remember this…
x  1  3i
x  1  3i
( x  3)( x  2)
x  3, x  2
Please practice this following problems with you partner
so that I can walk throughout the room and provide
help and assistance as needed.
Pg.141 #13, & 47
(use the answers in the back of the book for help!)
2.6 Rational Functions & Asymptotes
Rational Function
N ( x)
f ( x) 
D( x)
Remember: D(x) ≠ 0
As f(x) approaches D(x) = 0 the function will
increase or decrease rapidly!
Reciprocal Function - most basic rational function.
1
f ( x) 
x
We know that x ≠ 0, so…
Use your calculator to complete the following chart
as x gets closer to 0
f(x)
f(x)
x
x
-1
-.5
-.1
-.01
-.001
∴ →0
1
.5
.1
.01
.001
∴ →0
Complete the following for the rational function.
Domain: (-∞,0) and (0, ∞)
Range: (-∞,0) and (0, ∞)
Increasing: NOT Increasing!
Decreasing: on(-∞,0) and
(0, ∞)
Zeros/Intercepts: NONE!
Even/Odd/ Neither:
Odd
Vertical asymptote: y-axis
Horizontal asymptote: x-axis
Asymptotes
Vertical Asymptotes
If f(x) → a
OR f(x) → -∞
x = a (line)
as
x→a
(from the right or left)
Horizontal Asymptotes
If f(x) → b
OR f(x) → -∞
y = b (line)
as
x→a
Finding Asymptotes
for a rational function
n 1
N ( x) an x  an1 x  ...  a2 x  a1 x  a0
f ( x) 

m
m 1
2
D( x) bm x  bm1 x  ...  b2 x  b1 x  b0
n
2
Vertical asymptotes - at the zeros of D(x)
Finding Asymptotes
for a rational function
n
n 1
2
a
x

a
x

...

a
x
 a1 x  a0
N ( x)
n
n 1
2
f ( x) 

m
m 1
2
D( x) bm x  bm1 x  ...  b2 x  b1 x  b0
Horizontal asymptotes – compare degrees of both N(x) and D(x)
*If n < m, the x-axis (y=0) is an asymptote
*If n > m, there is NO horizontal asymptote
an
*If n = m, then the line y 
is an asymptote
bm
(an and bm are leading coefficinets.)
Practice 1. Find all asymptotes of the function.
2 x2
f ( x)  2
x 1
Find vertical asymptotes by setting denominator
equal to zero.
2
x 1  0
( x  1)( x  1)  0
x  1
x 1
We have two vertical asymptotes at x = -1 and x = 1
Practice 1. Find all asymptotes of the function.
2 x2
f ( x)  2
x 1
Find horizontal asymptotes by checking exponents.
Since n = m, we look at the leading coefficients.
2
2
1
y 2
We have one horizontal asymptote at y = 2
Practice 2. The following function will have two
horizontal asymptotes.
 x  10

f ( x)  
, x  0
 x2

1
 y 1
1
1
x  10
f ( x) 
x 2
 x  10

f ( x)  
, x  0
 x  2

1
 y  1
 1
1
We have two horizontal asymptotes at
y = 1 for x values less than zero.
y = -1 for x values greater than or equal to zero
2.7 Graphs of Rational Functions
WHAT DO I DO????
1. Simplify f, if possible.
2. Find & plot y-intercept *IF ANY?*
evaluate f(0)
3. Find zeros *IF ANY?* of numerator
solve N(x) = 0 and plot the y-intercepts
WHAT DO I DO? Continued…
4. Find zeros *IF ANY?* of denominator
solve D(x) = 0
sketch the vertical asymptote(s)
5. Find & sketch the horizontal asymptote(s)
*IF ANY?*
6. Plot at least one point between each xintercept and vertical asymptote.
7. Draw smooth curves between the points
and asymptotes.
Practice 1. Using your knowledge from Chapter 2,
follow the previously mentioned steps to sketch a
graph of the following rational function by hand.
(NO CALCULATORS!)
x
f ( x) 
Simplify for later…
x x2
2
x
f ( x) 
( x  1)( x  2)
0
0
Find the y-intercept: f (0)  2
0 02
Find zeros for numerator N(x) = 0
x
f ( x)  2
 N ( x)  x
x x2
y-intercept: (0,0)
x  0
x-intercept: (0,0)
Find vertical asymptotes D(x) = 0
x
f ( x) 
 D( x)  ( x  1)( x  2)
( x  1)( x  2)
0  ( x  1)( x  2)
x  1
x2
Vertical asymptotes are at x = -1 and x = 2
Check the degree of the numerator and
denominator to determine if there are horizontal
x
asymptotes.
f ( x) 
x x2
2
The degree the numerator is less than the degree
of the denominator so we have a Horizontal
asymptote at y = 0
f(x)
x
Complete the
following to
determine other
points.
-3
-1
-.5
1
2
3
Since we have found the following, sketch a graph.
y-intercept: (0,0)
x-intercept: (0,0)
Vertical Asymptotes: x = -1 and x = 2
Horizontal Asymptotes: y = 0
Extra Points: x
f(x)
-3
-1
-.0
UND
-.5
1
2
.4
.5
UND
3
.75
Slant Asymptotes - if the degree of
the numerator is one degree larger than
the denominator then you have a slant
asymptote.
WE USE LONG DIVISON TO HELP US!
Example 1. Find the slant (oblique) asymptote.
x  x2
f ( x) 
x 1
2
x
x 1 x  x  2
2
x x
02
2
2
x
x 1
When we divide, we end up with x,
the line y = x is a oblique asymptote.
2.8 Exploring Data: Quadratic Models
Sometimes a linear model is not always the
best for a grouping of data...
(see pg.161 for more than one example)
Given the following set of data, we
will plot a scatter plot and use the
QuadReg function in our calculators
to determine a quadratic equation
of the curve.
The following table shows monthly average high
temperature for Yorktown, VA
Month
Avg. Monthly Temp.
January
47°
February
51°
March
59°
April
69°
May
76°
June
84°
July
87°
August
85°
September
80°
October
70°
November
61°
December
51°
Input the data into L1 & L2.
Use 1,2,3,4… to represent
the months of the year.
Buttons:
*Stat
*Select EDIT
*L1 = Months
*L2 = Temperatures
Month
Avg. Monthly
Temp.
January
47°
February
51°
March
59°
April
69°
May
76°
June
84°
July
87°
August
85°
September
80°
October
70°
November
61°
December
51°
Buttons:
*2nd Function “y=” (Stat Plot)
Buttons:
*Enter, turn ON, & select mark
Buttons:
*Window change your widow
so that it will fit our data
Buttons:
*Graph
Buttons:
*#5 QuadReg
Buttons:
*Stat, right one to Calc