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