1. Unit 2 Polynomial and Rational Functions

2. Do Now Given the following function, find

3. Unit 2 Lesson 1 2.1 Quadratic Functions 2.2 Polynomial Functions

4. 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.

5. 2.1 Quadratic Functions a, b, and c are real numbers where a0
When graphed called a Parabola

6. Axis of Symmetry – all parabolas are symmetric (line where it’s cut in half)
Vertex – axis intersects the parabola

7. If a is positive… If a is negative… opens up opens down
Minimum: (0,0) Maximum: (0,0) Axis of Symmetry for both: x=0

8. If a is positive…opens up
If a is positive…opens up If a is negative…opens down Vertex is lowest point. Vertex is highest point Axis of symmetry

9. Base function or squaring function. How does each function change. 1)
Base function or squaring function. How does each function change? 1) Up 5 units 2) Left 5 units 3) Get’s skinny

10. Summarize: How do they change. If c>0, then move up
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.

11. 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 Example 2.

12. Practice Given the following function, identify the following: Domain: Range: Decreasing on: Increasing on: Turn up or down:

13. Standard Form of a Quadratic
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

14. Identify the vertex of a Quadratic Given a function Write the function in Standard Form by completing the square to find the vertex. Example *Factor out the *b=4, add & subtract (4/2)2 = *Regroup terms *Write in Standard Form.
The vertex is (-2,-3)

15. Identify the Minimum & Maximum Values Given a function You can find the minimum if a>0, You can find the maximum if a<0
Example: Given the following quadratic equation find coordinates of the relative minimum or maximum point.

16. 2.2 Graphs of Polynomial Functions
Features of Polynomial Functions:
Continuous Smooth Curves
NOT Features of Polynomial Functions:
NOT Continuous NOT Smooth

17. Functions with 1 degree are Linear
Functions with 1 degree are Linear. Functions with 2 degrees are Quadratic. Functions with 3rd or greater are Polynomial.
Functions with an even degree will look like our quadratics. (e.g.)
Functions with an odd degree will look like our cubic. (e.g.)

18. The greater the value of n the flatter the graph near (0,0)
A polynomial of the nth degree has the form:
n is a positive integer
The greater the value of n the flatter the graph near (0,0)

19. 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.

20. When n is odd: When

21. When n is even: When

22. 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 Example 2.

23. Finding Zeros of Polynomial Functions
Set the function equal to zero and factor.
(e.g)
Check on the graphing calculator.

24. 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

25. Long Division of Polynomials
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).

26. Divide!
1) Multiply:
2) Subtract.
3) Multiply:
4) Subtract.
5) Multiply:
6) Subtract.

27. Now we know that:
So we factor and solve for zero!
We have three x-intercepts at:

28. Synthetic Division (the shortcut to long division)
We will use Synthetic Division to divide the following:
Solve to get divisor: Leading coefficients (NOTE: zero because there is no x cubed. -3 9 3 -3 1 -3 -1 1 1
Remainder: 1
1) Add terms in first column, and multiply the result by -3.
2) Repeat this process until the last column.

29. -3 1 0 -10 -2 4 -3 9 3 -3 1 -3 -1 1 1 Quotient: Remainder: -3 9 3 -3 1 -3 -1 1 1
Quotient:
Remainder:
original problem = quotient + remainder

30. Remainder & Factor Theorems
The Remainder Theorem (the remainder from synthetic division):
If a polynomial is divided by x-k,
the remainder is
Practice 1. Use the Remainder Theorem to evaluate the given function at x = -2.
Start by doing synthetic division!

31. -6 -4 -2 3 2 1 -9
Remainder is -9,
(-2,-9) is a point on the graph of our polynomial.
Check by substituting in -2 into our original function.

32. Rational Zero Test If the polynomial
has integer coefficients, every rational zero of f has the form:
Rational Zero
p: factor of constant term
q: factor of leading coefficient

33. Example 1. Find the rational zero(s) of the given function.
The factors of 1 are +1 and -1
Therefore: Rational Zeros: +1 or -1
Test each possible zero algebraically.
Since -1, and 3 are not zero, this polynomial does not have a rational zero.

34. 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.

35. Variation in sign: Two consecutive (nonzero) coefficients have opposite signs.
Example: Do the given functions have a difference of signs?
+ to to +
NO!!!
+ to +
+ to to -
YES!!!
- to +

36. Descartes's Rule of Signs
Let
Be a polynomial with real coefficients and .
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.

37. This polynomial has either two or no positive real zeros.
Example.
Describe the possible real zeros of the given function.
+ to to +
This polynomial has either two or no positive real zeros.
- to +
- to to +
This one negative real zeros.
- to -

38. 2.4 The Imaginary Unit i 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.

39. 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)

40. Multiplication with Complex Numbers
Remember:
Example1: Multiply.
Example 2: Multiply. (3 + 2i)(5 + 4i)

41. Example of Complex Conjugates: (5 + 2i)(5 – 2i)
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)

42. Find the complex conjugate for the following complex number and multiply.

43. Division with Complex Numbers
Example1: Write the following quotient in standard form.

44. 2.5 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.

45. Linear Factorization Theorem
If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors.
Where are complex numbers.

46. 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.
Factor completely.
Find your zeros.

47. Conjugate Pairs (think: difference of squares)
FACT:
Complex zeros occur in conjugate pairs!
(think about the last example!)
If is a zero of a function, then the conjugate is also a zero.

48. If 1 +3i is a zero we know that 1 – 3i is 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.
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.
Multiply.

49. Now, use long division to divide.

50. Factor!
Remember this…
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!)

51. 2.6 Rational Functions & Asymptotes
Remember: D(x) ≠ 0
As f(x) approaches D(x) = 0 the function will increase or decrease rapidly!

52. Reciprocal Function - most basic rational function.
We know that x ≠ 0, so…
Use your calculator to complete the following chart as x gets closer to 0 x
f(x) -1
-.5
-.1
-.01
-.001
∴ →0 x
f(x) 1 .5 .1
.01
.001
∴ →0

53. Complete the following for the rational function.
Domain:
Range:
Increasing:
Decreasing:
Zeros/Intercepts:
Even/Odd/ Neither:
Vertical asymptote:
Horizontal asymptote:
(-∞,0) and (0, ∞)
NOT Increasing!
on(-∞,0) and (0, ∞)
NONE!
Odd
y-axis
x-axis

54. Asymptotes Vertical Asymptotes x = a (line)
If f(x) → a OR f(x) → -∞ as x → a
(from the right or left)
Horizontal Asymptotes y = b (line)
If f(x) → b OR f(x) → -∞ as x → a

55. Finding Asymptotes for a rational function
Vertical asymptotes - at the zeros of D(x)

56. Finding Asymptotes for a rational function
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
*If n = m, then the line is an asymptote
(an and bm are leading coefficinets.)

57. Practice 1. Find all asymptotes of the function.
Find vertical asymptotes by setting denominator equal to zero.
We have two vertical asymptotes at x = -1 and x = 1

58. Practice 1. Find all asymptotes of the function.
Find horizontal asymptotes by checking exponents.
Since n = m, we look at the leading coefficients.
We have one horizontal asymptote at y = 2

59. Practice 2. The following function will have two horizontal asymptotes.
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

60. 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

61. 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 x-intercept and vertical asymptote.
7. Draw smooth curves between the points and asymptotes.

62. 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!)
Simplify for later…
Find the y-intercept:

63. y-intercept: (0,0) x-intercept: (0,0)
Find zeros for numerator N(x) = 0
y-intercept: (0,0) x-intercept: (0,0)
Find vertical asymptotes D(x) = 0
Vertical asymptotes are at x = -1 and x = 2

64. Complete the following to determine other points.
Check the degree of the numerator and denominator to determine if there are horizontal asymptotes.
The degree the numerator is less than the degree of the denominator so we have a Horizontal asymptote at y = 0 x
f(x) -3 -1
-.5 1 2 3
Complete the following to determine other points.

65. 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
-.0 -1
UND
-.5 .4 1 .5 2 3
.75

67. 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.

68. When we divide, we end up with x,
Remainder
When we divide, we end up with x,
the line y = x is a oblique asymptote.

69. 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.

70. The following table shows monthly average high temperature for Yorktown, VA
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

71. Use 1,2,3,4… to represent the months of the year.
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

72. Buttons:
*2nd Function “y=” (Stat Plot)
Buttons:
*Enter, turn ON, & select mark
Buttons:
*Window change your widow so that it will fit our data

73. Buttons:
*Graph
Buttons:
*Stat, right one to Calc
Buttons:
*#5 QuadReg