Download Chapter 2 Polynomial, Power, and Rational Functions

Chapter 2
Polynomial,
Power, and
Rational
Functions
Group Practice
1. Write an equation in slope-intercept form for a line with slope m  2 and
y -intercept 10.
2. Write an equation for the line containing the points (  2,3) and (3,4).
3. Expand ( x  6) .
4. Expand (2 x  3) .
5. Factor 2 x  8 x  8.
2
2
2
Slide 2- 3
Polynomial
Function
Let n be a nonnegative integer and let a , a , a ,..., a , a
0
1
n 1
2
n
be real numbers with
a  0. The function given by f ( x)  a x  a x  ...  a x  a x  a
n 1
n
n
n
is a polynomial function of degree n.
The leading coefficient is a .
n
n 1
2
2
1
0
Note:
 Remember
Polynomial
 Poly
= many
 Nomial= terms
 So
it literally means “many terms”
Name
Form
Zero Function
Undefined
Degree
f(x)=0
Constant Function f(x)=a (a≠0)
0
Linear Function
1
f(x)=ax+b (a≠0)
Quadratic Function f(x)=ax2+bx+c (a≠0)
2
Remember
 The
highest power (or highest degree) tells
you what kind of a function it is.
Example #1
 Which
of the follow is a function? If so,
what kind of a function is it?
 A)
 B)
𝑓 𝑥 = 3𝑥 2 − 5𝑥 +
6
7
𝑓 𝑥 = 𝑥 −4 + 3𝑥 + 4
 C) p x = 9𝑥 4 + 25𝑥 2
 D) 𝑦 = 15𝑥 − 2𝑥 6
 E) 𝑡 𝑥 = 8𝑥 6
Group talk: Tell me everything
about linear functions
Slide 2- 9
Average Rate of Change
(slope)
The average
rate of change of a function y  f ( x) between x  a and x  b,
a  b, is
f (b)  f (a)
.
ba
Rate of change is used in calculus. It can be expressing miles per
hour, dollars per year, or even rise over run.
Write an equation for the linear function f such that f (-1)  2 and f (2)  3.
Example
Answer
 Use
point-slope form
 (-1,2)
(2,3)
 Y-3=(1/3)(x-2)
Ultimate problem
 In
Mr. Liu’s dream, he purchased a 2014
Nissan GT-R Track Edition for $120,000. The
car depreciates on average of $8,000 a
year.
1)What is the rate of change?
2)Write an equation to represent this
situation
3) In how many years will the car be worth
nothing?
Answer
1) -8000
2) y=price of car, x=years
y= −8000𝑥 + 120000
3) When the car is worth nothing y=0
X=15, so in 15 years, the car will be worth
nothing.
Ultimate problem do it in your
group (based on 2011 study)

When you graduate from high school, the starting
median pay is $33,176. If you pursue a professional
degree (usually you have to be in school for 12
years after high school), your starting median pay
is $86,580.

1) Write an equation of a line relating median
income to years in school.

2) If you decide to pursue a bachelor’s degree (4
years after high school), what is your potential
starting median income?
Answer
 1)
y=median income, x=years in school
Equation: y= 4450.33x+33176
2) Since x=4, y=50,977.32
My potential median income is $50,977.32
after 4 years of school.
You are saying more school
means more money?!?!
Slide 2- 17
Characterizing the Nature of a
Linear Function
Point of View
Characterization
Verbal
degree 1
polynomial of
Algebraic
(m≠0)
f(x) = mx + b
Graphical
slope m
and y-intercept b
slant line with
Analytical
constant
function with
Linear Correlation
 When
you have a scatter plot, you can
see what kind of a relationship the dots
have.
 Linear
correlation is when points of a
scatter plot are clustered along a line.
Slide 2- 19
Linear Correlation
Slide 2- 20
Properties of the Correlation
Coefficient, r
1.
2.
3.
4.
5.
-1 ≤ r ≤ 1
When r > 0, there is a positive linear
correlation.
When r < 0, there is a negative linear
correlation.
When |r| ≈ 1, there is a strong linear
correlation.
When |r| ≈ 0, there is weak or no linear
correlation.
Slide 2- 21
1.
2.
3.
4.
Enter and plot the data (scatter plot).
Find the regression model that fits the problem
situation.
Superimpose the graph of the regression model
on the scatter plot, and observe the fit.
Use the regression model to make the predictions
called for in the problem.
Regression Analysis
Group Work: plot this with a
calculator. Example of
Regression
Price per Box
Boxes sold
2.40
38320
2.60
33710
2.80
28280
3.00
26550
3.20
25530
3.40
22170
3.60
18260
Slide 2- 23
Group Work
Describe how to transform the graph of f ( x)  x into the graph of
2
f ( x)  2  x  2   3.
2
Answer
 Horizontal
shift right 2
 Vertical shift up 3
 Vertical stretch by a factor of 2 or
horizontal shrink by a factor of 1/2
Group Work
 Describe
𝑓
𝑥 =−
the transformation
3
2
𝑥+4
2
+6
Answer
 Horizontal
shift left 4
 Vertical shift up 6
 Vertical stretch by a factor of 3/2 or
horizontal shrink by a factor of 2/3
 reflect over the x-axis
Slide 2- 27
Vertex Form of a Quadratic
Equation
Any quadratic function f(x) = ax2 + bx + c,
a≠0, can be written in the vertex form
f(x) = a(x – h)2 + k
The graph of f is a parabola with vertex
(h,k) and axis x = h, where h = -b/(2a) and
k = c – ah2. If a>0, the parabola opens
upward, and if a<0, it opens downward.
Group Work: where is
vertex?
3
2
𝑓
𝑥 =−
𝑥+4
𝑓
𝑥 =2 𝑥−1
2
2
+6
−3
Answer
 (-4,6)
 (1,-3)
Example: Use completing the
square to make it into vertex
Use the vertex form of a quadratic function to find the vertex and axis
form
of the graph of f ( x)  2 x  8 x  11. Rewrite the equation in vertex form.
2
Group Work
 Change
𝑓
this quadratic to vertex form
𝑥 = −3𝑥 2 + 6𝑥 − 5
Answer
𝑓
𝑥 = −3 𝑥 − 1
2
−2
Slide 2- 33
Characterizing the Nature of a
Point
of View
Characterization
Quadratic
Function
Verbal
polynomial of degree 2
Algebraic
f ( x)  ax  bx  c or
f ( x)  a ( x - h)  k (a  0)
2
2
Graphical
parabola with vertex (h, k ) and
axis x  h; opens upward if a > 0,
opens downward if a < 0;
initial value = y -intercept = f (0)  c
b  b  4ac
x-intercepts 
2a
2
Slide 2- 34
Vertical Free-Fall Motion
The height s and vertical velocity v of an object in free fall are given by
1
s (t )   gt  v t  s and v(t )   gt  v ,
2
where t is time (in seconds), g  32 ft/sec  9.8 m/sec is the acceleration
2
0
0
0
2
2
due to gravity , v is the initial vertical velocity of the object, and s is its
0
initial height.
0
Example
 You
are in MESA and we are doing bottle
rockets. You launched your rocket and
its’ total time is 8.95 seconds. Find out
how high your rocket went (in meters)
Flyin’ High
Answer

You first have to figure out how fast your
rocket is when launched. Remember the
velocity at the max is 0. Also the time to rise
to the peak is one-half the total time.
So 8.96/2 = 4.48s
0 = −9.8 4.48 + 𝑣0
𝑣0 = 43.904𝑚/𝑠

𝑠 4.48 = −

𝑠 4.48 = 98.34 𝑚



1
2
9.8 4.48
2
+ 43.904 4.48 + 0
Homework Practice
 Pg
182-184 #1-12, 45-50
 Pgs 182-184 #14-44e, 55, 58,61
Power Functions with
Modeling
Slide 2- 39
Power Function
Any function that can be written in the
form
f(x) = k·xa, where k and a are nonzero
constants,
is a power function. The constant a is the
power, and the k is the constant of
variation, or
constant of proportion. We say f(x) varies as
the ath power of x, or f(x) is proportional to
the
ath power of x.
Group Work
Write the following expressions using only positive integer powers.
1. x
2. r
3. m
5/3
-3
1.5
Write the following expressions in the form k  x using a single rational
number for the power of a.
a
4. 16 x
5.
3
x
27
3
Group Work: Answer the
following with these two
functions
3

𝑓 𝑥 =

Power:
Constant of variation:
Domain:
Range:
Continuous:
Increase/decrease:
Symmetric:
Boundedness:
Max/min:
Asymptotes:
End behavior:










𝑥
𝑓 𝑥 = 𝑥 −2
State the power and constant of variation for the function f ( x)  x ,
and graph it.
4
Example Analyzing Power
Functions
State the power and constant of variation for the function f ( x)  x ,
and graph it.
4
f ( x)  x  x  1 x so the power is 1/4 and
the constant of variation is 1.
4
1/ 4
1/ 4
Slide 2- 43
Slide 2- 44
Monomial Function
Any function that can be written as
f(x) = k or f(x) = k·xn, where k is a constant and n is a
positive integer, is a monomial function.
Slide 2- 45
Example Graphing Monomial Functions
Describe how to obtain the graph of the function f ( x)  3x from the graph
of g ( x)  x with the same power n.
3
n
Example Graphing Monomial
Functions
Describe how to obtain the graph of the function f ( x)  3x from the graph
of g ( x)  x with the same power n.
3
n
We obtain the graph of f ( x)  3x by vertically stretching the graph of
g ( x)  x by a factor of 3. Both are odd functions.
3
3
Slide 2- 46
Note:
 Remember,
it is important to know the
parent functions. Everything else is just a
transformation from it.
 Parent
functions can be found in chapter
1 notes.
Group Talk:

What are the characteristics of “even
functions”?

What are the characteristics of “odd
functions”?

What happen to the graphs when
denominator is undefined?

Clue: Look at all the parent functions.
Slide 2- 49
For any power function f(x) = k·xa, one of the
following three things happens when x < 0.
 f is undefined for x < 0.
 f is an even function.
 f is an odd function.
Graphs of Power Functions
Slide 2- 50
Graphs of Power Functions
Determine whether f is even,
odd, or undefined for x<0
 Usually,
it is easy to determine even or
odd by looking at the power. It is a little
different when the power is a fraction or
decimal.
 When
the power is a fraction or decimal,
you have to determine what happened
to the graph when x<0
Example:
 Determine
whether f is even, odd, or
undefined for x<0
1)𝑓 𝑥 = 𝑥
1
4
2)𝑓 𝑥 = 𝑥
2
3
3)𝑓 𝑥 = 𝑥
4
3
Answer
 1)
undefined for x<0
 2)
even
 3)
even
There is a trick!
 Odd
functions can only be an integer! It
can not be a fraction except when
denominator is 1
 Even
functions is when the numerator is
raised to an even power. Can be a
fraction
 If
the power is a fraction and the
numerator is an odd number, it is
undefined x<0
Homework Practice
 Pgs
196-198 #1-11odd, 17, 27, 30, 31, 39,
43, 48, 55, 57
Polynomial Functions of
Higher degree with
modeling
Group Talk
 How
do you determine how many
potential solutions you have on a graph?
What is a polynomial?
 Polynomial
means “many terms”
 Each monomial in the sum  a x , a x ,..., a  is a term of the polynomial.
n 1
n
n
n 1
0
 A polynomial function written in this way, with terms in descending degree,
is written in standard form.
 The constants a , a ,..., a  are the coefficients of the polynomial.
n
n 1
0
 The term a x is the leading term, and a is the constant term.
n
n
0
Group Review: Shifts
𝑓
𝑥 = −6 −𝑥 + 8
 What
2
−5
is the parent function?
 Give me all the shifts!
Answer

Parent function is 𝑓 𝑥 = 𝑥 2

Note: You have to factor out the negative in front of
2
the x 𝑓 𝑥 = −6 − 𝑥 − 8
−5

Horizontal shift right 8
Vertical shift down 5
Vertical Stretch by factor of 6
Horizontal shrink by a factor of 1/6
Flip over the x axis
Flip over the y axis





Group Work
Describe how to transform the graph of an appropriate monomial function
f ( x)  a x into the graph of h( x)  ( x  2)  5. Sketch h( x) and
n
n
compute the y -intercept.
4
Example Graphing
Transformations of Monomial
Functions
Describe how to transform the graph of an appropriate monomial function
f ( x)  a x into the graph of h( x)  ( x  2)  5. Sketch h( x) and
n
4
n
compute the y -intercept.
You can obtain the graph of h( x )  ( x  2)  5 by shifting the graph of
f ( x)   x two units to the left and five units up. The y -intercept of h( x)
4
4
is h(0)    2   5  11.
4
Slide 2- 62
Remember End Behavior?

What is End Behavior?

You have to find out the behavior when 𝑥 →
∞ 𝑎𝑛𝑑 𝑥 → −∞


lim 𝑓(𝑥) =
𝑥→∞
lim 𝑓 𝑥 =
𝑥→−∞
Find the end behavior for
all!
𝑓
𝑡 = 2𝑡 2
𝑓
𝑎 = −0.5𝑎7
𝑓
𝑔 = 5𝑔−5
𝑓
𝑠 = −3𝑥 15 Think about this one
Answer








lim 2𝑡 2 = ∞
𝑡→∞
lim 2𝑡 2 = ∞
𝑡→−∞
lim −0.5𝑎7 = −∞
𝑎→∞
lim −0.5𝑎7 = ∞
𝑎→−∞
lim 5𝑔−5 = 0
𝑔→∞
lim 5𝑔−5 = 0
𝑔→−∞
lim −3𝑥 15 = −3𝑥 15
𝑠→∞
lim −3𝑥 15 = −3𝑥 15
𝑠→−∞

Why do you think this happens?
Mr. Liu the trickster
 Find
the end behavior for:
𝑥 = 50𝑥 6 − 180𝑥 5 + 98𝑥 4 − 140𝑥 3 +
10𝑥 2 − 𝑥 + 35
𝑓
Answer
 You
only look at the term with the highest
power, which is the 6th power


lim 𝑓(𝑥) = ∞
𝑥→∞
lim 𝑓 𝑥 = ∞
𝑥→−∞
Determining if you have a
min/max
 Graph
this function
 𝑓 𝑡 = 𝑡3 + 𝑡
 Tell
me about this function
Answer
 It
is increasing for all domains
 Therefore there is no min/max
 There is one zero at t=0
Determine if you have a
min/max
 Graph
this function
 𝑓 𝑡 = 𝑡3 − 𝑡
 Tell
me about this function
Answer
 Graph
increases from (−∞, −0.38)
 Graph decreases from −0.38,0.58
 Graph increases from (0.58, ∞)
 Therefore there is a local max at x=-0.38
 There is a local min at x=0.58
 Three
zeros: x=-1, x=0 and x=1
Slide 2- 73
Potential Cubic Functions
(what it can look like)
Slide 2- 74
Quartic Function (what it can
look like)
Slide 2- 75
Local Extrema and Zeros of
A polynomial
function
of degree n has at most
Polynomial
Functions
n – 1 local extrema and at most n zeros.
For example
 If
you have a function that is to the 3rd
power


You may have potential of 3 zeros (3
solutions)
You may have 2 local extrema (either max
or min)
Now try this!
 Function


Zeros?
Extremas?
 Function


to the 5th power, how many…
to the 4th power, how many…
Zeros?
Extremas?
Remember I asked you guys
about the even powers vs odd
powers?
Here it is! More examples
Finding zeros
 Note:
very very very important to know
how to factor!!!!
Example
 Solve:
𝑓 𝑥 = 𝑥 3 − 𝑥 2 − 6𝑥 = 0
Group Work
Find the zeros of f ( x)  2 x  4 x  6 x.
3
2
Slide 2- 83
Multiplicity of a Zero of a
Function
If f Polynomial
is a polynomial function
and  x  c  is a factor of f
m
but  x  c 
m 1
is not, then c is a zero of multiplicity m of f .
Slide 2- 84
Example Sketching the Graph of a
Factored Polynomial
Sketch the graph of f ( x)  ( x  2) ( x  1) .
3
2
Slide 2- 85
Intermediate Value
If aTheorem
and b are real numbers with a < b and if f is
continuous on the interval [a,b], then f takes on
every value between f(a) and f(b). In other
words, if y0 is between f(a) and f(b), then y0=f(c)
for some number c in [a,b].
Note: That is important for
Calculus!
Homework Practice
 Pgs
209-210 #3, 6, 15-36, multiple of 3
Real zeros of polynomial
Functions
What’s division?
There are two ways to divide
polynomials
 Long
division
 Synthetic
division
Example:
 2𝑥 4
− 𝑥 3 − 2 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 2𝑥 2 + 𝑥 + 1
Work:
Group Work
Use long division to find the quotient and remainder when 2 x  x  3
is divided by x  x  1.
2
4
3
2x  x 1
x  x  1 2x  x  0x  0x  3
2x  2x  2x
 x  2x  0x  3
2
Answer
2
4
3
2
4
3
2
3
2
x  x
x
3
x
 2x
4
2
2
2
x
+ x3
 x 1
2x  2
 x  3   x  x  1  2 x  x  1 
3
2
2
2x  2
x  x 1
2
Remainder
theorem
If polynomial f ( x) is divided by x  k , then the remainder is r  f (k ).
What does the remainder
theorem say?
 Well,
it tells us what the remainder is
without us doing the long division!
 Basically,
you substitute what make the
denominator 0!
 EX:
if it was x-3, then you substitute x=3, so
it’s f(3)=r
I am so happy such that I
don’t have to do the long
division to find the remainder!
Example:
Find the remainder when f ( x)  2 x  x  12 is divided by x  3.
2
Answer
r  f (3)  2  3   3  12 =33
2
Group Work
 Find the
 6𝑥 3 − 5𝑥
remainder
+ 5 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 𝑥 − 4
Synthetic Division
 Divide
2𝑥 3 − 3𝑥 2 − 5𝑥 − 12 𝑏𝑦 𝑥 − 3
Divide 3x  2 x Work
 x  5 by x  1 using synthetic division.
Group
3
2
Slide 2- 103
Example Using Synthetic
Divide 3x  2 x  x  5 by x  1 using synthetic division.
Division
3
1 3
2
2
1
5
1 3
3
3
3x  2 x  x  5
3
 3x  x  2 
 x  1
 x  1
3
2
2
2
3
1
1
1
2
5
2
3
Again
 Divide
𝑥 8 − 1 𝑏𝑦 𝑥 + 2
Rational Zeros Theorem
 This
is P/Q
Slide 2- 106
Rational Zeros Theorem
Suppose f is a polynomial function of degree n  1 of the form
f ( x)  a x  a x  ...  a , with every coefficient an integer
n 1
n
n
n 1
0
and a  0. If x  p / q is a rational zero of f , where p and q have
0
no common integer factors other than 1, then
p is an integer factor of the constant coefficient a , and
0
q is an integer factor of the leading coefficient a .
n
In Another word
P
are the factors of the last term of the
polynomial
Q
are the factors of the first term of the
polynomial
 Use
Synthetic division to determine if that
is a zero
Example:
𝑓
𝑥 = 𝑥 3 − 3𝑥 2 + 1
Group Work: Find Rational
Zeros
𝑓
𝑥 = 3𝑥 3 + 4𝑥 2 − 5𝑥 − 2
Slide 2- 110
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f ( x)  2 x  7 x  8 x  14 x  8.
4
3
2
Finding the polynomial
 Degree
3, with -2,1 and 3 as zeros with
coefficient 2
Answer
 2(x+2)(x-1)(x-3)
Group Work
 Find
polynomial with degree 4, coefficient
of 4 with 0, ½, 3 and -2 as zeros
Answer
 4x(x-1/2)(x-3)(x+2)
Homework Practice
 Pgs
223-224 # 1, 4, 5, 7, 15, 18, 28, 35, 36,
49, 50, 57
Complex Zeros and the
Fundamental Theorem of
Algebra
Bell Work
Perform the indicated operation, and write the result in the form a  bi.
1.  2  3i    1  5i 
2.  3  2i  3  4i 
Factor the quadratic equation.
3. 2 x  9 x  5
Solve the quadratic equation.
4. x  6 x  10  0
List all potential rational zeros.
5. 4 x  3 x  x  2
2
2
4
2
Slide 2- 118
Fundamental Theorem of
A Algebra
polynomial function of degree n has n complex
zeros (real and nonreal). Some of these zeros may
be repeated.
Slide 2- 119
Linear Factorization
If f (Theorem
x) is a polynomial function of degree n  0, then f ( x) has precisely
n linear factors and f ( x)  a( x  z )( x  z )...( x  z ) where a is the
1
2
n
leading coefficient of f ( x) and z , z ,..., z are the complex zeros of f ( x).
1
2
n
The z are not necessarily distinct numbers; some may be repeated.
i
In another word
 The
highest degree tells you how many
zeros you should have (real and nonreal)
and how many times it may cross the xaxis (solutions)
 Very

Important!!!
If you have a nonreal solution, it comes in
pairs. One is the positive and one is
negative (next slide is an example)
Example: Find the
polynomial
𝑓

𝑥 = (𝑥 − 4𝑖)(𝑥 + 4𝑖)
Note: This is linear factorization
 How
many real zeros?
 How
many nonreal zeros?
 What’s
the degree of polynomial?
Group Work: Find the
polynomial
𝑓

𝑡 = (𝑡 − 1)(𝑡 + 2)(𝑡 − 𝑖)(𝑡 + 𝑖)
Note: This is called linear factorization
 How
many real zeros?
 How
many nonreal zeros?
 What’s
the degree of polynomial?
Group work
 Find
the polynomial with -1, 1+i, 2-i as zeros
Answer
 (x+1)(x-(1+i))(x+(1+i))(x-(2-i))(x+(2-i))
(x+1)(x-1-i)(x+1+i)(x-2+i)(x+2-i)
or
Slide 2- 125
Group Work
Write a polynomial of minimum degree in standard form with real
coefficients whose zeros include 2,  3, and 1  i.
Group work: Finding Complex
Zeros
 Z=1-2i
is a zero of 𝑓 𝑥 4𝑥 4 + 17𝑥 2 + 14𝑥 + 65
Find the remaining zeros and write it in its
linear factorization
Write the function as a
product of linear factorization
and as real coefficient
𝑓
𝑥 = 𝑥 4 + 3𝑥 3 − 3𝑥 2 + 3𝑥 − 4
Answer
 (x-1)(x+4)(x-i)(x+i)
 As
Real coefficient
 (𝑥
− 1)(𝑥 + 4)(𝑥 2 + 1)
Slide 2- 129
Example Factoring a Polynomial
Write f ( x)  3x  x  24 x  8 x  27 x  9 as a product of linear and
irreducible quadratic factors, each with real coefficients.
5
4
3
2
Slide 2- 130
Example Factoring a
Write f ( x)  3x  x  24 x  8 x  27 x  9 as a product of linear and
Polynomial
irreducible quadratic factors, each with real coefficients.
5
4
3
2
The Rational Zeros Theorem provides the candidates for the rational
zeros of f . The graph of f suggests which candidates to try first.
Using synthetic division, find that x  1/ 3 is a zero. Thus,
f ( x)  3 x  x  24 x  8 x  27 x  9
5
4
3
2
1

  x    3 x  8 x  9 
3

1

 3  x    x  9  x  1
3

4
2
2
2
1

 3  x    x  3 x  3 x  1
3

2
Homework Practice
 Pg
234 #1, 3, 5, 14, 17-20 ,37, 38, 6, 11, 15,
21, 23, 27-29, 33,43, 51
Graphs of Rational
Functions
Slide 2- 133
Rational Functions
Let f and g be polynomial functions with g ( x)  0. Then the function
f ( x)
given by r ( x) 
is a rational function.
g ( x)
Note: Vertical Asymptote
 You
look at the restrictions at the
denominator to determine the vertical
asymptote
Slide 2- 135
Group Work
Find the domain of f and use limits to describe the behavior at
value(s) of x not in its domain.
2
f ( x) 
x2
Answer
 Remember
you always see what can’t X
be (look at the denominator)
 D:
−∞, −2 𝑢(−2, ∞)
Note: Horizontal Asymptote

If the power of the numerator is < power of
denominator then horizontal asymptote is y=0

If the power of the numerator is = power of
denominator then horizontal asymptote is the
coefficient

If the power of numerator is > power of
denominator, then there is no horizontal
asymptote
Note 2
 If
numerator degree > denominator
degree. You may have a slant
asymptote.
 You
have to use long division to
determine the function
Example: Find the horizontal
asymptote
𝑓
𝑓
𝑓
𝑥 =
3𝑥 2 −5
5𝑥 3
𝑥 =
𝑥7
𝑥5
𝑥 =
6𝑥 3 +2𝑥+5
𝑥 3 −5𝑥+6
Answer
 Y=0
 None
 Y=6
Slant asymptote example
𝑓
𝑥 =
𝑥3
𝑥 2 −9
Slide 2- 142
Example Finding Asymptotes of Rational
Functions
Find the asymoptotes of the function f ( x) 
2( x  3)( x  3)
.
( x  1)( x  5)
Slide 2- 143
Example Finding Asymptotes
2( x  3)( x  3)
Find the
of the function
f ( x) 
.
ofasymoptotes
Rational
Functions
( x  1)( x  5)
There are vertical asymptotes at the zeros of the denominator:
x  1 and x  5.
The end behavior asymptote is at y  2.
Example Graphing a Rational
Function
x 1
Find the asymptotes and intercepts of f ( x) 
 x  2  x  3
and graph f ( x).
Slide 2- 144
Example Graphing a Rational
Function
x 1
Find the asymptotes and intercepts of f ( x) 
 x  2  x  3
and graph f ( x).
The numerator is zero when x  1 so the x-intercept is 1. Because f (0)  1/ 6,
the y -intercept is 1/6. The denominator is zero when x  2 and x  3, so
there are vertical asymptotes at x  2 and x  3. The degree of the numerator
is less than the degree of the denominator so there is a horizontal asymptote
at y  0.
Slide 2- 145
Ultimate Problem
3𝑥 2 −2𝑥+4
𝑥 2 −4𝑥+5

𝑓 𝑥 =

Domain:
Range:
Continuous:
Increase/decrease:
Symmetric:
Y-intercept:
X-intercept:
Boundedness:
Max/min:
Asymptotes:
End behavior:










Homework Practice
 Pg
245 #3, 7, 11-19, 21, 23, 25
Solving Equations and
inequalities
Slide 2- 149
Example
Solving by Clearing
2
Solve x   3.
x
Fractions
Slide 2- 150
Example
Eliminating
1
2x
2
Solve the equation


.
x  3 x  1Solutions
x  4x  3
Extraneous
2
Group Work
𝑥
4
𝑥
+ = 10
Group Work

2𝑥
1
+
𝑥−1
𝑥−3
=
2
𝑥 2 −4𝑥+3
Slide 2- 153
Example Finding a Minimum
Find the dimensions of the rectangle with minimum perimeter if its area is 300
squarePerimeter
meters. Find this least perimeter.
Solving inequalities
 Solving
inequalities, it would be good to
use the number line and plot all the zeros,
then check the signs.
Slide 2- 155
Example Finding where a
Polynomial is Zero, Positive, or
Let f ( x)  ( x  3)( x  4) . Determine the real number values of x that
Negative
cause f ( x) to be (a) zero, (b) positive, (c) negative.
2
Slide 2- 156
Example Solving a Polynomial
Solve x  6 x  2  8 x graphically.
Inequality Graphically
3
2
Slide 2- 157
Example Solving a Polynomial
Solve x  6 x  2  8 x graphically.
Inequality Graphically
3
2
Rewrite the inequality x  6 x  8 x  2  0. Let f ( x)  x  6 x  8 x  2
and find the real zeros of f graphically.
3
2
3
2
The three real zeros are approximately 0.32, 1.46, and 4.21. The solution
consists of the x values for which the graph is on or below the x-axis.
The solution is (, 0.32]  [1.46, 4.21].
Slide 2- 158
Example
Creating a Sign Chart
x 1
Let r ( x) 
. Determine the values of x that cause r ( x) to be
 3Rational
x  1
for x a
Function
(a) zero, (b) undefined, (c) positive, and (d) negative.
Slide 2- 159
Example Solving an Inequality
Solve ( x  2) x  1  0.
Involving a Radical
Group Work
𝑓
𝑥 =
2𝑥+1
(𝑥+3)(𝑥−1)
determine when it’s a)
zero b) undefined c) positive d) negative
Group Work
 Solve
5
𝑥+3
−
2
𝑥−2
>0
Group Work
 𝑠𝑜𝑙𝑣𝑒
𝑥−8
𝑥−2
≤0
Group Work
 (𝑥
+ 2) 𝑥 ≥ 0
Homework Practice
 253-254
35, 39
 264
#3, 9, 11,15, 17, 27, 28, 31, 32, 34,
#1, 6, 8, 13, 21, 28, 33, 36, 47