1. Chapter 2 Polynomial, Power, and Rational Functions

2. Group Practice

3. Polynomial Function

4. Note: Remember Polynomial Poly = many Nomial= terms
So it literally means “many terms”

5. Name Form Degree Zero 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

6. Remember
The highest power (or highest degree) tells you what kind of a function it is.

7. Example #1
Which of the follow is a function? If so, what kind of a function is it?
A) 𝑓 𝑥 =3 𝑥 2 −5𝑥+ 6 7
B) 𝑓 𝑥 = 𝑥 −4 +3𝑥+4
C) p x = 9 𝑥 𝑥 2
D) 𝑦=15𝑥−2 𝑥 6
E) 𝑡 𝑥 = 8 𝑥 6

8. Group talk: Tell me everything about linear functions

9. Average Rate of Change (slope)
Rate of change is used in calculus. It can be expressing miles per hour, dollars per year, or even rise over run.

10. Example

11. Answer
Use point-slope form
(-1,2) (2,3)
Y-3=(1/3)(x-2)

12. 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?

13. Answer
1) ) y=price of car, x=years y=−8000𝑥 ) When the car is worth nothing y=0 X=15, so in 15 years, the car will be worth nothing.

14. 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?

15. Answer 1) y=median income, x=years in school
Equation: y= x+33176
2) Since x=4, y=50,977.32
My potential median income is $50, after 4 years of school.

16. You are saying more school means more money?!?!

17. Characterizing the Nature of a Linear Function
Point 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

18. 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.

19. Linear Correlation

20. Properties of the Correlation Coefficient, r
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.

21. Regression Analysis 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.

22. 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

23. Group Work

24. 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

25. Group Work
Describe the transformation
𝑓 𝑥 =− 𝑥

26. 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

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.

28. Group Work: where is vertex?
𝑓 𝑥 =− 𝑥
𝑓 𝑥 =2 𝑥−1 2 −3

29. Answer
(-4,6)
(1,-3)

30. Example: Use completing the square to make it into vertex form

31. Group Work
Change this quadratic to vertex form
𝑓 𝑥 =−3 𝑥 2 +6𝑥−5

32. Answer
𝑓 𝑥 =−3 𝑥−1 2 −2

33. Characterizing the Nature of a Quadratic Function
Point of View Characterization

34. Vertical Free-Fall Motion

35. 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

36. 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=− 𝑣 0
𝑣 0 =43.904𝑚/𝑠
𝑠 4.48 =−
𝑠 4.48 =98.34 𝑚

37. Homework Practice Pg 182-184 #1-12, 45-50
Pgs #14-44e, 55, 58,61

38. Power Functions with Modeling

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.

40. Group Work

41. Group Work: Answer the following with these two functions
𝑓 𝑥 = 3 𝑥 𝑓 𝑥 = 𝑥 −2
Power:
Constant of variation:
Domain:
Range:
Continuous:
Increase/decrease:
Symmetric:
Boundedness:
Max/min:
Asymptotes:
End behavior:

43. Example Analyzing Power Functions

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.

45. Example Graphing Monomial Functions

46. Example Graphing Monomial Functions

47. 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.

48. 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.

49. Graphs of Power Functions
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.

50. Graphs of Power Functions

51. 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

52. 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

53. Answer
1) undefined for x<0
2) even
3) even

54. 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

55. Homework Practice
Pgs #1-11odd, 17, 27, 30, 31, 39, 43, 48, 55, 57

56. Polynomial Functions of Higher degree with modeling

57. Group Talk
How do you determine how many potential solutions you have on a graph?

58. What is a polynomial?
Polynomial means “many terms”

59. Group Review: Shifts 𝑓 𝑥 =−6 −𝑥+8 2 −5 What is the parent function?
𝑓 𝑥 =−6 −𝑥+8 2 −5
What is the parent function?
Give me all the shifts!

60. Answer Parent function is 𝑓 𝑥 = 𝑥 2
Note: You have to factor out the negative in front of the x 𝑓 𝑥 =−6 − 𝑥−8 2 −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

61. Group Work

62. Example Graphing Transformations of Monomial Functions

63. Remember End Behavior? What is End Behavior?
You have to find out the behavior when 𝑥→∞ 𝑎𝑛𝑑 𝑥→−∞
lim 𝑥→∞ 𝑓(𝑥) =
lim 𝑥→−∞ 𝑓 𝑥 =

64. Find the end behavior for all!
𝑓 𝑡 =2 𝑡 2
𝑓 𝑎 =−0.5 𝑎 7
𝑓 𝑔 =5 𝑔 −5
𝑓 𝑠 =−3 𝑥 15 Think about this one

65. Answer lim 𝑡→∞ 2 𝑡 2 =∞ lim 𝑡→−∞ 2 𝑡 2 =∞ lim 𝑎→∞ −0.5 𝑎 7 =−∞
Why do you think this happens?

66. Mr. Liu the trickster Find the end behavior for:
𝑓 𝑥 =50 𝑥 6 −180 𝑥 𝑥 4 −140 𝑥 𝑥 2 −𝑥+35

67. Answer
You only look at the term with the highest power, which is the 6th power
lim 𝑥→∞ 𝑓(𝑥) =∞
lim 𝑥→−∞ 𝑓 𝑥 = ∞

69. Determining if you have a min/max
Graph this function
𝑓 𝑡 = 𝑡 3 +𝑡
Tell me about this function

70. Answer It is increasing for all domains Therefore there is no min/max
There is one zero at t=0

71. Determine if you have a min/max
Graph this function
𝑓 𝑡 = 𝑡 3 −𝑡
Tell me about this function

72. Answer Graph increases from (−∞,−0.38) Graph decreases from −0.38,0.58
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

73. Potential Cubic Functions (what it can look like)

74. Quartic Function (what it can look like)

75. Local Extrema and Zeros of Polynomial Functions
A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.

76. 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)

77. Now try this! Function to the 5th power, how many…
Zeros?
Extremas?
Function to the 4th power, how many…

78. Remember I asked you guys about the even powers vs odd powers?

79. Here it is! More examples

80. Finding zeros
Note: very very very important to know how to factor!!!!

81. Example
Solve: 𝑓 𝑥 = 𝑥 3 − 𝑥 2 −6𝑥=0

82. Group Work

83. Multiplicity of a Zero of a Polynomial Function

84. Example Sketching the Graph of a Factored Polynomial

85. Intermediate Value Theorem
If 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].

86. Note: That is important for Calculus!

87. Homework Practice
Pgs #3, 6, 15-36, multiple of 3

88. Real zeros of polynomial Functions

89. What’s division?

90. There are two ways to divide polynomials
Long division
Synthetic division

91. Example:
2 𝑥 4 − 𝑥 3 −2 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 2 𝑥 2 +𝑥+1

92. Work:

93. Group Work

94. Answer

95. Remainder theorem

96. 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

97. I am so happy such that I don’t have to do the long division to find the remainder!

98. Example:

99. Answer

100. Group Work
Find the remainder
6 𝑥 3 −5𝑥+5 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 𝑥−4

101. Synthetic Division
Divide 2 𝑥 3 −3 𝑥 2 −5𝑥−12 𝑏𝑦 𝑥−3

102. Group Work

103. Example Using Synthetic Division

104. Again
Divide 𝑥 8 −1 𝑏𝑦 𝑥+2

105. Rational Zeros Theorem
This is P/Q

106. Rational Zeros Theorem

107. 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

108. Example:
𝑓 𝑥 = 𝑥 3 −3 𝑥 2 +1

109. Group Work: Find Rational Zeros
𝑓 𝑥 =3 𝑥 3 +4 𝑥 2 −5𝑥−2

110. Example Finding the Real Zeros of a Polynomial Function

111. Finding the polynomial
Degree 3, with -2,1 and 3 as zeros with coefficient 2

112. Answer
2(x+2)(x-1)(x-3)

113. Group Work
Find polynomial with degree 4, coefficient of 4 with 0, ½, 3 and -2 as zeros

114. Answer
4x(x-1/2)(x-3)(x+2)

115. Homework Practice
Pgs # 1, 4, 5, 7, 15, 18, 28, 35, 36, 49, 50, 57

116. Complex Zeros and the Fundamental Theorem of Algebra

117. Bell Work

118. Fundamental Theorem of Algebra
A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

119. Linear Factorization Theorem

120. 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)

121. 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?

122. 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?

123. Group work
Find the polynomial with -1, 1+i, 2-i as zeros

124. 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)

125. Group Work

126. Group work: Finding Complex Zeros
Z=1-2i is a zero of 𝑓 𝑥 4 𝑥 𝑥 2 +14𝑥+65 Find the remaining zeros and write it in its linear factorization

127. Write the function as a product of linear factorization and as real coefficient
𝑓 𝑥 = 𝑥 4 +3 𝑥 3 −3 𝑥 2 +3𝑥−4

128. Answer
(x-1)(x+4)(x-i)(x+i)
As Real coefficient
(𝑥−1)(𝑥+4)( 𝑥 2 +1)

129. Example Factoring a Polynomial

130. Example Factoring a Polynomial

131. Homework Practice
Pg 234 #1, 3, 5, 14, ,37, 38, 6, 11, 15, 21, 23, 27-29, 33,43, 51

132. Graphs of Rational Functions

133. Rational Functions

134. Note: Vertical Asymptote
You look at the restrictions at the denominator to determine the vertical asymptote

135. Group Work

136. Answer
Remember you always see what can’t X be (look at the denominator)
D: −∞,−2 𝑢(−2,∞)

137. 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

138. Note 2
If numerator degree > denominator degree. You may have a slant asymptote.
You have to use long division to determine the function

139. Example: Find the horizontal asymptote
𝑓 𝑥 = 3 𝑥 2 −5 5 𝑥 3
𝑓 𝑥 = 𝑥 7 𝑥 5
𝑓 𝑥 = 6 𝑥 3 +2𝑥+5 𝑥 3 −5𝑥+6

140. Answer
Y=0
None
Y=6

141. Slant asymptote example
𝑓 𝑥 = 𝑥 3 𝑥 2 −9

142. Example Finding Asymptotes of Rational Functions

143. Example Finding Asymptotes of Rational Functions

144. Example Graphing a Rational Function

145. Example Graphing a Rational Function

146. Ultimate Problem 𝑓 𝑥 = 3 𝑥 2 −2𝑥+4 𝑥 2 −4𝑥+5 Domain: Range:
𝑓 𝑥 = 3 𝑥 2 −2𝑥+4 𝑥 2 −4𝑥+5
Domain:
Range:
Continuous:
Increase/decrease:
Symmetric:
Y-intercept:
X-intercept:
Boundedness:
Max/min:
Asymptotes:
End behavior:

147. Homework Practice
Pg 245 #3, 7, 11-19, 21, 23, 25

148. Solving Equations and inequalities

149. Example Solving by Clearing Fractions

150. Example Eliminating Extraneous Solutions

151. Group Work
𝑥+ 4 𝑥 =10

152. Group Work
2𝑥 𝑥−1 + 1 𝑥−3 = 2 𝑥 2 −4𝑥+3

153. Example Finding a Minimum Perimeter

154. Solving inequalities
Solving inequalities, it would be good to use the number line and plot all the zeros, then check the signs.

155. Example Finding where a Polynomial is Zero, Positive, or Negative

156. Example Solving a Polynomial Inequality Graphically

157. Example Solving a Polynomial Inequality Graphically

158. Example Creating a Sign Chart for a Rational Function

159. Example Solving an Inequality Involving a Radical

160. Group Work
𝑓 𝑥 = 2𝑥+1 (𝑥+3)(𝑥−1) determine when it’s a) zero b) undefined c) positive d) negative

161. Group Work
Solve 5 𝑥+3 − 2 𝑥−2 >0

162. Group Work
𝑠𝑜𝑙𝑣𝑒 𝑥−8 𝑥−2 ≤0

163. Group Work
(𝑥+2) 𝑥 ≥0

164. Homework Practice
#3, 9, 11,15, 17, 27, 28, 31, 32, 34, 35, 39
264 #1, 6, 8, 13, 21, 28, 33, 36, 47