1. Ch 3 review Quarter test 1 And Ch 3 TEST

2. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers Range: depends on the minimum and maximum The graph is a parabola.

3. if The graph of x 2 is shifted “h” units horizontally and “k” units vertically. opens: axis of symmetry: vertex: k is the range: V(h, k) / minimum x = h up x = h (h, k) minimum positive

4. if The graph of x 2 is shifted “h” units horizontally, “k” units vertically, and reflected over x-axis. opens: axis of symmetry: vertex: k is the range: V(h, k) / maximum x = h down x = h (h, k) maximum negative

5. Standard form: Vertex form: can “see” the transformations… The vertex form is easier to graph…to change from standard form to vertex form, either complete the square (YUCK!) or memorize this formula: and h Therefore, the vertex is at and the axis of symmetry is. k = f(h)

6. Example: Let. Find the vertex, axis of symmetry, the minimum or maximum value, and the intercepts. Use these to graph f(x). State the domain and range and give the intervals of increase and decrease. Then write the equation in vertex form and list the transformations that were made to the parent function, f(x) = x 2. 1 st identify a, b, and c: a = 3 b = 6 c = 1 Next find h and k: So, the vertex is (-1, -2) and the axis of symmetry is x = -1. Since a > 0, then the graph opens up and has a minimum value at -2. To find y-intercepts evaluate f(0): To find x-intercepts (roots/zeros) use the quadratic formula: -0.184 and -1.816 The intercepts are at (0, 1), (-0.184, 0), and (-1.816, 0).

7. To graph, plot the vertex, intercepts, utilize the axis of symmetry. V(-1, -2) y-int: (0,1) axis of sym: x = -1 So, to be symmetrical, another point will be at (-2, 1). Check using your graphing calculator! Domain: all real numbers Range: Decreasing: Increasing: Vertex form: f(x) = a(x - h) 2 + k a = 3 h = -1 k = -2 It is the graph of x 2 shifted left 1, vertically stretched by 3 and shifted down 2.

8. Example: Find the standard form equation of the quadratic function whose vertex is (1, -5) and whose y-intercept is -3. h k (0, -3) Vertex form: f(x) = a(x - h) 2 + k Fill in the information that was given: Solve for a… Write the equation in vertex form then simplify to standard form:

9. Power Functions The polynomial that the graph resembles (the end behavior model)… EX:The power function of the polynomial is…

10. Properties of Power Functions with Even Degrees 1. f is an even function a. The graph is symmetric with respect to the y-axis b. f(-x) = f(x) 2. Domain: all real numbers 3. The graph always contains the points (0, 0), (1, 1) and (-1, 1) 4. As the exponent increases in magnitude, the graph becomes more vertical when x 1; but for x near the origin, the graph tends to flatten out and be closer to the x-axis. The graph always contains the points (0, 0), (1, 1) and (-1, 1) * * Points used to make transformations

11. EX: Graph y = x 4, y = x 8 and y = x 12 all on the same screen. Let and be your viewing window. What do you notice?

12. Properties of Power Functions with Odd Degrees 1. f is an odd function a. The graph is symmetric with respect to the origin b. f(-x) = -f(x) 2. Domain: all real numbers Range: all real numbers 3. The graph always contains the points (0, 0), (1, 1) and (-1, -1) 4. As the exponent increases in magnitude, the graph becomes more vertical when x 1; but for x near the origin, the graph tends to flatten out and be closer to the x-axis. The graph always contains the points (0, 0), (1, 1) and (-1, -1) * * Points used to make transformations

13. EX: Graph y = x 3, y = x 7 and y = x 11 all on the same screen. Let and be your viewing window. What do you notice?

14. Graphs of polynomial functions are smooth (no sharp corners or cusps) and continuous (no gaps or holes…it can be drawn without lifting your pencil)… Is a polynomial graphIs not a polynomial graph

15. We can apply what we learned about transformations in Chapter 2 and what we just learned about power functions to graph polynomials…

16. EXAMPLE: Graph f(x) = 1 – (x – 2) 4 using transformations. Step 1: y = x 4 Step 2: y = (x – 2) 4 Step 3: y = - (x – 2) 4 Step 4: y = 1 – (x – 2) 4 Start with (0, 0), (1, 1) & (-1, 1) Shift right 2 units Reflect over x-axis Shift up 1 unit

17. EXAMPLE: Graph f(x) = 2(x + 1) 5 using transformations. Check your work with your graphing calculator. x 5 …(0, 0), (1, 1) & (-1, -1) (x + 1) 5 …shift left 1 unit 2(x + 1) 5 …vertical stretch by factor 2 multiply the y-values by 2

18. Zeros and the Equation of a Polynomial Function If f is a polynomial function and r is a real number for which f(r) = 0, then r is called a real zero of f, or a root of f. If r is a real zero/root of f then: a. r is an x-intercept of the graph of f, and b. (x – r) is a factor of f In other words…if you know a zero/root, then you know a factor…if you know a factor, then you know a zero/root EX:If (x – 4) is a factor, then 4 is a zero/root… If -3 is a zero/root, then (x + 3) is a factor…

19. EXAMPLE: Find a polynomial of degree 3 with zeros -4, 1, and 3. (Let a = 1) If x = -4, then the factor that solves to that is… If x = 1, then the factor that solves to that is… If x = 3, then the factor that solves to that is…

21. Rational Function A function of the form, where p and q are polynomial functions and q is not the zero polynomial. The domain is the set of all real numbers EXCEPT those for which the denominator q is zero. EX: * Enter in calculator as (x + 1)/(x – 2)...MUST put parentheses! *

22. Domain and Vertical Asymptotes To find the domain of a rational function, find the zeros of the denominator…this is where the denominator would be zero…this is where x cannot exist. The vertical asymptote(s) of a rational function are where x cannot exist…it is the virtual boundary line on the graph. Vertical asymptotes are defined by the equation ‘x =‘

23. How the graph reacts on either side of a vertical asymptote: Goes in opposite directions as it approaches the asymptote: Goes in the same direction as it approaches the asymptote: THE GRAPH WILL NEVER CROSS THE VERTICAL ASYMPTOTE!!!

24. EXAMPLE: Find the domain and vertical asymptotes of the rational functions. a. The graph will not exist where the denominator equals zero! x + 5 = 0 x = -5 When x = -5, the graph will not exist! The domain is and the VA is

25. EXAMPLE: Find the domain and vertical asymptotes of the rational functions. b. c. x 2 – 4 = 0 (x + 2)(x – 2) = 0 x = -2 x = 2 Domain: VA: x 2 + 1 = 0 x 2 = -1 x = not real Domain: all real #’s VA: none

26. Intercepts on the x and y axes To find the y-intercepts of a rational function, that is where x = 0, evaluate f(0). To find the x-intercepts of a rational function, first make sure the function is in lowest terms, that is the numerator and denominator have no common factors. Then, find the zeros of the numerator. factor top & bottom first!! The zeros of the numerator are the x-intercepts (zeros) of the rational function.

27. EXAMPLE: Find the x and y intercepts of the rational functions. a. No common factors…in lowest terms. y-intercept: x-intercept: The y-intercept is at and the x-intercepts are at and

28. EXAMPLE: Find the x and y intercepts of the rational functions. b. No common factors…in lowest terms. y-intercept: x-intercept: The y-intercept is at and there are no x-intercepts none…the numerator has no x in it to solve for!

29. EXAMPLE: Find the x and y intercepts of the rational functions. c. Cannot be factored…in lowest terms. y-intercept:x-intercept: The y-intercept is at 0 and the x-intercept is at 0.

30. Graphing Rational Functions Using Transformations a. Analyze the graph of 1 st find the domain and any vertical asymptotes: x 2 = 0 x = 0 Domain: VA: x = 0 Next, find the x & y-intercepts: x-intercepts: none y-intercepts: none

31. Graphing Rational Functions Using Transformations a. Analyze the graph of Is it even? To graph without using a calculator, identify a few points on the graph by plugging in x-values: It is an even function, so it is symmetric to the y-axis. f(1) = 1 f(-1) = 1 f(2) = ¼ f(-2) = ¼

32. Graphing Rational Functions Using Transformations b. Use transformations to graph Check the domain and any vertical asymptotes: (x – 2) 2 = 0 x – 2 =0 x = 2 Domain: VA: x = 2 Next, check the y-intercept: y-intercepts: 1.25 It is the graph of shifted right 2 and up 1. Shift the VA and points right 2 and up 1...

33. Graphing Rational Functions Using Transformations c. Analyze the graph of and use it to graph Find the domain and any vertical asymptotes: x - 2 = 0 x = 2 Domain: VA: x = 2 Next, find the x & y-intercepts: x-intercepts: 3 It is the graph of shifted right 2, reflected over the x-axis, and shifted up 1. x – 3 = 0 x = 3

34. Graphing Rational Functions Using Transformations c. Analyze the graph of and use it to graph Find the domain and any vertical asymptotes: x - 2 = 0 x = 2 Domain: VA: x = 2 Next, find the x & y-intercepts: x-intercepts: 3 y-intercepts: 1.5 It is the graph of shifted right 2, reflected over the x-axis, and shifted up 1.

35. Properties of Rational Functions

36. Holes (Points of Discontinuity) x-values for a rational function that cannot exist, BUT are not asymptotes. These occur whenever the numerator and denominator have a common factor. Must factor both top and bottom first!!

37. EXAMPLE: Find the domain and vertical asymptote(s). a hole occurs at x + 1 = 0 Domain: VA: x = -3 A hole occurs at x = -1

39. EXAMPLE: Find the domain and vertical asymptote(s). Domain: VA: x = -3 A hole occurs at x = -1 a hole occurs at x - 2 = 0 A hole occurs at x = 2 Domain: VA: x = -2

40. Horizontal Asymptotes describe a certain behavior of the graph as or as, that is its end behavior. How the graph behaves on the far ends of the x-axis. The graph of a function may intersect a horizontal asymptote. The graph of a function will never intersect a vertical asymptote. Always written as y =

42. Three Types of Rational Functions Balanced…the degree of the numerator and denominator are equal Horizontal Asymptote: H.A. The horizontal asymptote is where y = 2.

43. Three Types of Rational Functions Bottom Heavy…the degree of the denominator is larger than the degree of the numerator. Horizontal Asymptote: The horizontal asymptote is where y = 0.

44. Three Types of Rational Functions Top Heavy…the degree of the numerator is larger than the degree of the denominator Has NO HORIZONTAL ASYMPTOTE There is no horizontal asymptote. has an oblique asymptote instead…

45. Oblique (Slant) Asymptote an asymptote that is neither vertical nor horizontal, but also describes the end behavior of a graph. Has the equation “y =“ and has an x in it. It is found by dividing the polynomial: top bottom (quotient only) Top Heavy rational functions have oblique or slant asymptotes instead of a horizontal asymptote.

46. EXAMPLE: Find the oblique asymptote of Note: The textbook considers only linear equations oblique asymptotes… divide the polynomials using long division… ignore remainder The oblique asymptote is y = x 2 + 1.

47. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, and slant asymptotes. a. D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ Now find domain and vertical asymptotes x + 2 = 0x – 2 = 0 x = -2 x = 2 x = 2, x = -2 Balanced 1 st find the horizontal asymptote y = 2

48. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, and slant asymptotes. a. D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ x = 2, x = -2 y = 2 Find the x-intercepts: Find the y-intercepts: 2x + 1 = 0x + 1 = 0 x = -1/2 x = -1 -1/2, -1

49. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes. b. D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ Find domain and vertical asymptotes x + 1 = 0x – 1 = 0 x = -1 x = 1 x = 1, x = -1 Bottom-heavy y = 0 Find the x-intercepts: Find the y-intercepts: x - 3 = 0 x = 3 3 3

50. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes. c. D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ None Find the oblique asymptote: Top-heavy oblique asymptote: __________ y = 2x - 6

51. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes. c. D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ x = -3 None Find the x-intercepts: Find the y-intercepts: 2x 2 = 0 x = 0 0 oblique asymptote: __________ Find domain and vertical asymptotes x + 3 = 0 x = -3 0 y = 2x - 6

52. Real and Non-real Zeros of Polynomial Functions

53. The zeros of a polynomial function can be found by finding its factors. The real zeros (roots) are the x-values where the graph crosses the x-axis. In this section, you will be finding both real and non-real (imaginary) roots.

54. Can “SEE” Real roots x-intercepts Cannot “see” imaginary roots...must use algebraic method, such as the quadratic formula, to find them NON-REAL REAL

55. Remainder and Factor Theorems Recall: Division Algorithm for Polynomials dividend divisor quotient remainder If the remainder is zero (0) then, g(x) divides evenly into f(x) and EX: 12/4 = 3 with remainder 0, so 4 x 3 = 12 Remainder Theorem Let f be a polynomial function. If f(x) is divided by x – c, then the remainder is f(c). f(c) = the remainder!

56. EXAMPLE: Find the remainder if f(x) = x 3 – 4x 2 – 5 is divided by x – 3. THIS CAN BE DONE IN ONE OF 3 WAYS!! Using Synthetic Division: Remainder Using the Remainder Theorem: (3) 3 – 4(3) 2 – 5 = -14 f(3) = Using Graphing: Let y 1 = x 3 – 4x 2 - 5 When x = 3, y = -14 Look at table The remainder is -14.

57. Factor Theorem Let f be a polynomial function. Then x – c is a factor of f(x) iff f(c) = 0. 1. If f(c) = 0, then x – c is a factor of f(x). 2. If x – c is a factor of f(x), then f(c) = 0. Basically, if the remainder is zero, you have a factor and a zero/root…and if you have a factor or zero/root, the remainder will be zero…

58. EXAMPLE: Determine whether x – 1 is a factor of f(x) = 2x 3 – x 2 + 2x – 3. If so, then factor f(x). 1 st check for a remainder of 0… 2(1) 3 – (1) 2 + 2(1) – 3 = 0 f(1) = Since the remainder is 0, then (x – 1) is a factor of f(x). Now factor f(x) using synthetic division… 2x 2 + x + 3 cannot be factored any further...

59. Complex Zeros (Roots) of a Polynomial Function Fundamental Theorem of Algebra Every polynomial function f(x) of degree n has exactly n numbers of real + imaginary zeros...that is, there are exactly n complex zeros. Furthermore, a polynomial of odd degree has at least one real zero. WHY?? Goes in opposite directions, so it MUST go through the x-axis! Complex Roots (Conjugate Pairs) Theorem Let f be a polynomial function. If a + bi is a complex zero of f, then a – bi is also a zero of the function. Irrational AND Imaginary zeros must come in pairs!

60. 3 are listed, so there are 2 more...3 + 2 = degree 5 imaginary and irrational zeros must come in conjugate pairs...

61. Steps for finding zeros (roots) of polynomial functions: 1.Determine the number of real and non-real roots the function will have by graphing. 2.Find the real zeros (x-intercepts) on your graph. If no real zeroes, then polynomial WILL BE FACTORABLE. 3.Factor the function using synthetic division. Continue to factor until you get a quadratic factor. 4.Solve each of the factors for the roots. Answer in exact form (not decimals). where it crosses x-axis real TOTAL ZEROS = DEGREE f(c) = 0 or get a polynomial that is factorable... exact form First check to see if the polynomial can be factored by “normal” means!!

62. EXAMPLE: Factor and find the zeros of the polynomial function. a. Step 1: Graph and find # of real & non-real zeros crosses the x-axis 3 times, so there are 3 real and 0 non-real

63. EXAMPLE: Factor and find the zeros of the polynomial function. a. Step 2: Find the real zeros on your graph. Is 1 a zero? Does f(1) = 0? Is -6 a zero? Does f(-6) = 0? Yes All three are real and can be found using your calculator! Yes Is -1/2 a zero? Does f(-1/2) = 0? Yes So, f(x) factored is (x – 1)(2x + 1)(x + 6) and the zeros are x = -6, -1/2, 1

64. EXAMPLE: Factor and find the zeros of the polynomial function. b. crosses the x-axis once and touches once... What are the possibilities?

65. Odd multiplicity Even multiplicity 3 real: multiplicity of 1 and multiplicity of 2 + 2 non-real

66. EXAMPLE: Factor and find the zeros of the polynomial function. According to the graph, the real zeros are at x = -3 and x = 2 We must use synthetic division since we cannot factor by grouping. Choose one of the real zeros to use for the division. Let’s start with x = -3… b.

67. EXAMPLE: Factor and find the zeros of the polynomial function. b. If x = -3, then... Not a quadratic and not factorable by grouping, so divide/factor again using synthetic division and another real zero…

68. EXAMPLE: Factor and find the zeros of the polynomial function. b. Use x = 2 to factor x 4 – 4x 3 + 8x 2 – 16x + 16 further… So, f(x) factored is (x + 3)(x - 2)(x - 2)(x 2 + 4) = (x + 3)(x – 2) 2 (x 2 + 4)

69. EXAMPLE: Factor and find the zeros of the polynomial function. b. mult. 2 The zeros are -3, 2 (mult 2), 2i and -2i.

70. EXAMPLE: Factor and find the zeros of the polynomial function. c.