Chapter 2 Polynomial, Power, and Rational Functions
Group Practice
2
2
1. Write an equation in slope-intercept form for a line with slope 2 and
-intercept 10.
2. Write an equation for the line containing the points ( 2,3) and (3,4).
3. Expand ( 6) .
4. Expand (2 3) .
5.
m
y
x
x
2Factor 2 8 8.x x
Polynomial Function
Slide 2- 3
0 1 2 1
1 2
1 2 1 0
Let be a nonnegative integer and let , , ,..., , be real numbers with
0. The function given by ( ) ...
is a .
The
n n
n n
n n n
n a a a a a
a f x a x a x a x a x a
polynomial function of degree
leading coeffi
n
is .n
acient
Note: Remember Polynomial
Poly = many Nomial= terms
So it literally means “many terms”
Name Form DegreeZero Function f(x)=0
Undefined
Constant Function f(x)=a (a≠0) 0
Linear Function f(x)=ax+b (a≠0) 1
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) C) D) E)
Group talk: Tell me everything about linear functions
Average Rate of Change (slope)
Slide 2- 9
The average rate of change of a function ( ) between and ,
( ) ( ), is .
y f x x a x b
f b f aa b
b a
Rate of change is used in calculus. It can be expressing miles per hour, dollars per year, or even rise over run.
ExampleWrite an equation for the linear function such that (-1) 2 and (2) 3.f f f
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 situation3) In how many years will the car be worth nothing?
Answer1) -80002) y=price of car, x=yearsy
3) When the car is worth nothing y=0X=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 schoolEquation: y= 4450.33x+33176
2) Since x=4, y=50,977.32My potential median income is $50,977.32 after 4 years of school.
You are saying more school means more money?!?!
Characterizing the Nature of a Linear FunctionPoint of View
Characterization
Verbal polynomial of degree 1
Algebraic f(x) = mx + b (m≠0)
Graphical slant line with slope m
and y-intercept b
Analytical function with constant
nonzero rate of change m: f is
increasing if m>0, decreasing if m<0; initial
value of the function = f(0) = b
Slide 2- 17
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.
Linear Correlation
Slide 2- 19
Properties of the Correlation Coefficient, r1. -1 ≤ r ≤ 12. When r > 0, there is a positive linear
correlation.3. When r < 0, there is a negative linear
correlation.4. When |r| ≈ 1, there is a strong linear
correlation.5. When |r| ≈ 0, there is weak or no
linear correlation.
Slide 2- 20
Regression Analysis1. Enter and plot the data (scatter plot).2. Find the regression model that fits the problem
situation.3. Superimpose the graph of the regression model
on the scatter plot, and observe the fit.4. Use the regression model to make the
predictions called for in the problem.
Slide 2- 21
Group Work: plot this with a calculator. Example of RegressionPrice 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
2
2
Describe how to transform the graph of ( ) into the graph of
( ) 2 2 3.
f x x
f x x
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 the transformation
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
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.
Slide 2- 27
Group Work: where is vertex?
Answer (-4,6) (1,-3)
Example: Use completing the square to make it into vertex form
2
Use the vertex form of a quadratic function to find the vertex and axis
of the graph of ( ) 2 8 11. Rewrite the equation in vertex form. f x x x
Group Work Change this quadratic to vertex form
Answer
𝑓 (𝑥 )=−3 (𝑥−1 )2−2
Characterizing the Nature of a Quadratic FunctionPoint of View Characterization
Slide 2- 33
2
2
Verbal polynomial of degree 2
Algebraic ( ) or
( ) ( - ) ( 0)
Graphical parabola with vertex ( , ) and
axis ; opens upward if >
f x ax bx c
f x a x h k a
h k
x h a
0,
opens downward if < 0;
initial value = -intercept = (0)
-intercept
a
y f c
x
2 4
s2
b b ac
a
Vertical Free-Fall Motion
Slide 2- 34
2
0 0 0
2 2
The and vertical of an object in free fall are given by
1( ) and ( ) ,
2where is time (in seconds), 32 ft/sec 9.8 m/sec is the
,
s v
s t gt v t s v t gt v
t g
height velocity
acceleration
due to gravity0 0
is the of the object, and is its
.
v initial vertical velocity s
initial height
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
Homework Practice Pg 182-184 #1-12, 45-50 Pgs 182-184 #14-44e, 55, 58,61
Power Functions with Modeling
Power FunctionAny function that can be written in the
formf(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.
Slide 2- 39
Group Work
5 / 3
-3
1.5
3
3
Write the following expressions using only positive integer powers.
1.
2.
3.
Write the following expressions in the form using a single rational
number for the power of .
4. 16
5. 27
a
x
r
m
k x
a
x
x
Group Work: Answer the following with these two functions
Power: Constant of variation: Domain: Range: Continuous: Increase/decrease: Symmetric: Boundedness: Max/min: Asymptotes: End behavior:
4State the power and constant of variation for the function ( ) ,
and graph it.
f x x
Example Analyzing Power Functions
Slide 2- 43
4State the power and constant of variation for the function ( ) ,
and graph it.
f x x
1/ 4 1/ 44( ) 1 so the power is 1/4 and
the constant of variation is 1.
f x x x x
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- 44
Slide 2- 45
Example Graphing Monomial Functions
3Describe how to obtain the graph of the function ( ) 3 from the graph
of ( ) with the same power .n
f x x
g x x n
Example Graphing Monomial Functions
Slide 2- 46
3Describe how to obtain the graph of the function ( ) 3 from the graph
of ( ) with the same power .n
f x x
g x x n
3
3
We obtain the graph of ( ) 3 by vertically stretching the graph of
( ) by a factor of 3. Both are odd functions.
f x x
g x x
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.
Graphs of Power FunctionsFor 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.
Slide 2- 49
Graphs of Power Functions
Slide 2- 50
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)2)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”1
1 0 Each monomial in the sum , ,..., is a of the polynomial.
A polynomial function written in this way, with terms in descending degree,
is written in .
The constan
n n
n na x a x a
term
standard form
1 0
0
ts , ,..., are the of the polynomial.
The term is the , and is the constant term.n n
n
n
a a a
a x a
coefficients
leading term
Group Review: Shifts
What is the parent function? Give me all the shifts!
Answer Parent function is
Note: You have to factor out the negative in front of the x
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 Work4
Describe how to transform the graph of an appropriate monomial function
( ) into the graph of ( ) ( 2) 5. Sketch ( ) and
compute the -intercept.
n
nf x a x h x x h x
y
Example Graphing Transformations of Monomial Functions
Slide 2- 62
4
Describe how to transform the graph of an appropriate monomial function
( ) into the graph of ( ) ( 2) 5. Sketch ( ) and
compute the -intercept.
n
nf x a x h x x h x
y
4
4
4
You can obtain the graph of ( ) ( 2) 5 by shifting the graph of
( ) two units to the left and five units up. The -intercept of ( )
is (0) 2 5 11.
h x x
f x x y h x
h
Remember End Behavior? What is End Behavior?
You have to find out the behavior when
Find the end behavior for all!
Think about this one
Answer
Why do you think this happens?
Mr. Liu the trickster Find the end behavior for:
Answer You only look at the term with the
highest power, which is the 6th power
Determining if you have a min/max Graph this function
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
Tell me about this function
Answer Graph increases from ( Graph decreases from Graph increases from ( 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
Potential Cubic Functions (what it can look like)
Slide 2- 73
Quartic Function (what it can look like)
Slide 2- 74
Local Extrema and Zeros of Polynomial FunctionsA polynomial function of degree n has at most
n – 1 local extrema and at most n zeros.
Slide 2- 75
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 to the 5th power, how many…
Zeros? Extremas?
Function 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:
Group Work3 2Find the zeros of ( ) 2 4 6 .f x x x x
Multiplicity of a Zero of a Polynomial Function
Slide 2- 83
1
If is a polynomial function and is a factor of
but is not, then is a zero of of .
m
m
f x c f
x c c f
multiplicity m
Slide 2- 84
Example Sketching the Graph of a Factored Polynomial
3 2Sketch the graph of ( ) ( 2) ( 1) .f x x x
Intermediate Value TheoremIf a 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].
Slide 2- 85
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:
2 𝑥4− 𝑥3 −2𝑑𝑖𝑣𝑖𝑑𝑒𝑏𝑦 2𝑥2+𝑥+1
Work:
Group Work 4 3
2
Use long division to find the quotient and remainder when 2 3
is divided by 1.
x x
x x
Answer
2
2 4 3 2
4 3 2
3 2
3 2
2
2
4 3 2 2
2
2 11 2 0 0 3
2 2 2
2 0 3
+ 3
1
2 2
2 22 3 1 2 1
1
x xx x x x x x
x x x
x x x
x x x
x x
x x
x
xx x x x x x
x x
Remainder theoremIf polynomial ( ) is divided by , then the remainder is ( ).f x x k 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: 2Find the remainder when ( ) 2 12 is divided by 3.f x x x x
Answer 2
( 3) 2 3 3 12 =33r f
Group Work Find the remainder
Synthetic Division Divide
Group Work3 2Divide 3 2 5 by 1 using synthetic division.x x x x
Example Using Synthetic Division
Slide 2- 103
3 2Divide 3 2 5 by 1 using synthetic division.x x x x
1 3 2 1 5
3
1 3 2 1 5
3 1 2
3 1 2 3
3 2
23 2 5 33 2
1 1
x x xx x
x x
Again Divide
Rational Zeros Theorem This is P/Q
Rational Zeros Theorem
Slide 2- 106
1
1 0
0
Suppose is a polynomial function of degree 1 of the form
( ) ... , with every coefficient an integer
and 0. If / is a rational zero of , where and have
no common integ
n n
n n
f n
f x a x a x a
a x p q f p q
0
er factors other than 1, then
is an integer factor of the constant coefficient , and
is an integer factor of the leading coefficient .n
p a
q a
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 )=𝑥3 −3 𝑥2+1
Group Work: Find Rational Zeros
𝑓 (𝑥 )=3 𝑥3+4 𝑥2− 5𝑥−2
Slide 2- 110
Example Finding the Real Zeros of a Polynomial Function
4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x
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
2
2
Perform the indicated operation, and write the result in the form .
1. 2 3 1 5
2. 3 2 3 4
Factor the quadratic equation.
3. 2 9 5
Solve the quadratic equation.
4. 6 10 0
List all potential ra
a bi
i i
i i
x x
x x
4 2
tional zeros.
5. 4 3 2x x x
Fundamental Theorem of Algebra A polynomial function of degree n has n complex
zeros (real and nonreal). Some of these zeros may be repeated.
Slide 2- 118
Linear Factorization Theorem
Slide 2- 119
1 2
1 2
If ( ) is a polynomial function of degree 0, then ( ) has precisely
linear factors and ( ) ( )( )...( ) where is the
leading coefficient of ( ) and , ,..., are the complex zen
n
f x n f x
n f x a x z x z x z a
f x z z z
ros of ( ).
The are not necessarily distinct numbers; some may be repeated.i
f x
z
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 x-axis (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
Note: This is linear factorization
How many real zeros?
How many nonreal zeros?
What’s the degree of polynomial?
Group Work: Find the polynomial
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))
or (x+1)(x-1-i)(x+1+i)(x-2+i)(x+2-i)
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 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 )=𝑥4 +3 𝑥3 − 3𝑥2+3 𝑥− 4
Answer (x-1)(x+4)(x-i)(x+i)
As Real coefficient
Slide 2- 129
Example Factoring a Polynomial
5 4 3 2Write ( ) 3 24 8 27 9as a product of linear and
irreducible quadratic factors, each with real coefficients.
f x x x x x x
Example Factoring a Polynomial
Slide 2- 130
5 4 3 2Write ( ) 3 24 8 27 9as a product of linear and
irreducible quadratic factors, each with real coefficients.
f x x x x x x
The Rational Zeros Theorem provides the candidates for the rational
zeros of . The graph of suggests which candidates to try first.
Using synthetic division, find that 1/ 3 is a zero. Thus,
( ) 3
f f
x
f x x
5 4 3 2
4 2
2 2
2
24 8 27 9
1 3 8 9
3
1 3 9 1
3
1 3 3 3 1
3
x x x x
x x x
x x x
x x x x
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
Rational Functions
Slide 2- 133
Let and be polynomial functions with ( ) 0. Then the function
( )given by ( ) is a .
( )
f g g x
f xr x
g x
rational function
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 and use limits to describe the behavior at
value(s) of not in its domain.
2( )
2
f
x
f xx
Answer Remember you always see what can’t X
be (look at the denominator)
D:
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
Answer Y=0
None
Y=6
Slant asymptote example
𝑓 (𝑥 )= 𝑥3
𝑥2− 9
Slide 2- 142
Example Finding Asymptotes of Rational Functions
2( 3)( 3)Find the asymoptotes of the function ( ) .
( 1)( 5)
x xf x
x x
Example Finding Asymptotes of Rational Functions
Slide 2- 143
2( 3)( 3)Find the asymoptotes of the function ( ) .
( 1)( 5)
x xf x
x x
There are vertical asymptotes at the zeros of the denominator:
1 and 5.
The end behavior asymptote is at 2.
x x
y
Example Graphing a Rational Function
Slide 2- 144
1
Find the asymptotes and intercepts of ( ) and graph ( ).2 3
xf x f x
x x
Example Graphing a Rational Function
Slide 2- 145
1
Find the asymptotes and intercepts of ( ) and graph ( ).2 3
xf x f x
x x
The numerator is zero when 1 so the -intercept is 1. Because (0) 1/ 6,
the -intercept is 1/6. The denominator is zero when 2 and 3, so
there are vertical asymptotes at 2 and 3. The degree
x x f
y x x
x x
of the numerator
is less than the degree of the denominator so there is a horizontal asymptote
at 0.y
Ultimate Problem
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
Example Solving by Clearing Fractions
Slide 2- 149
2Solve 3.x
x
Example Eliminating Extraneous Solutions
Slide 2- 150
2
1 2 2Solve the equation .
3 1 4 3
x
x x x x
Group Work
𝑥+4𝑥
=10
Group Work
2𝑥𝑥−1
+1
𝑥− 3=
2
𝑥2 − 4 𝑥+3
Example Finding a Minimum Perimeter
Slide 2- 153
Find the dimensions of the rectangle with minimum perimeter if its area is 300
square 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.
Example Finding where a Polynomial is Zero, Positive, or Negative
Slide 2- 155
2Let ( ) ( 3)( 4) . Determine the real number values of that
cause ( ) to be (a) zero, (b) positive, (c) negative.
f x x x x
f x
Example Solving a Polynomial Inequality Graphically
Slide 2- 156
3 2Solve 6 2 8 graphically.x x x
Example Solving a Polynomial Inequality Graphically
Slide 2- 157
3 2Solve 6 2 8 graphically.x x x
3 2 3 2Rewrite the inequality 6 8 2 0. Let ( ) 6 8 2
and find the real zeros of graphically.
x x x f x x x x
f
The three real zeros are approximately 0.32, 1.46, and 4.21. The solution
consists of the values for which the graph is on or below the -axis.
The solution is ( ,0.32] [1.46,4.21].
x x
Example Creating a Sign Chart for a Rational Function
Slide 2- 158
1
Let ( ) . Determine the values of that cause ( ) to be3 1
(a) zero, (b) undefined, (c) positive, and (d) negative.
xr x x r x
x x
Example Solving an Inequality Involving a Radical
Slide 2- 159
Solve ( 2) 1 0.x x
Group Work determine when it’s a) zero b)
undefined c) positive d) negative
Group Work Solve
Group Work
𝑠𝑜𝑙𝑣𝑒𝑥− 8
|𝑥−2|≤ 0
Group Work
(𝑥+2)√𝑥≥ 0
Homework Practice 253-254 #3, 9, 11,15, 17, 27, 28, 31,
32, 34, 35, 39
264 #1, 6, 8, 13, 21, 28, 33, 36, 47